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Distinguishing reionization models using the largest cluster statistics of the
21-cm maps
Aadarsh Pathak1★, Satadru Bag2, Suman Majumdar1,3, Rajesh Mondal4, Mohd Kamran1, Prakash Sarkar5
1Department of Astronomy, Astrophysics & Space Engineering, Indian Institute of Technology Indore, Indore 453552, India
2Korea Astronomy and Space Science Institute, Daejeon, Republic of Korea
3Department of Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, U. K.
4The Oskar Klein Centre and The Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden
5Ramakrishna Mission English School, Sidhgora, Jamshedpur
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The evolution of topology and morphology of ionized or neutral hydrogen during different stages of the Epoch of Reionization
(EoR) have the potential to provide us a great amount of information about the properties of the ionizing sources during this era.
We compare a variety of reionization source models in terms of the geometrical properties of the ionized regions. We show that
the percolation transition in the ionized hydrogen, as studied by tracing the evolution of the Largest Cluster Statistics (LCS), is
a robust statistic that can distinguish the fundamentally different scenarios – inside-out and outside-in reionization. Particularly,
the global neutral fraction at the onset of percolation is significantly higher for the inside-out scenario as compared to that for the
outside-in reionization. In complementary to percolation analysis, we explore the shape and morphology of the ionized regions
as they evolve in different reionization models in terms of the Shapefinders (SFs) that are ratios of the Minkowski functionals
(MFs). The shape distribution can readily discern the reionization scenario with extreme non-uniform recombination in the IGM,
such as the clumping model. In the rest of the reionization models, the largest ionized region abruptly grows only in terms of
its third SF – ‘length’ – during percolation while the first two SFs – ‘thickness’ and ‘breadth’ – remain stable. Thus the ionized
hydrogen in these scenarios becomes highly filamentary near percolation and exhibit a ‘characteristic cross-section’ that varies
among the source models. Therefore, the geometrical studies based on SFs, together with the percolation analysis can shed light
on the reionization sources.
Key words: intergalactic medium – dark ages, reionization, first stars – large-scale structure of Universe – cosmology: theory.
1 INTRODUCTION
In the history of our Universe, the Epoch of Reionization (EoR) is the
period when the neutral hydrogen (H i) in the Inter-Galactic Medium
(IGM) was gradually ionized by the radiations from the first sources
(see e.g. Furlanetto et al. 2006;Choudhury et al. 2009;Pritchard &
Loeb 2012). In recent times, although the EoR has obtained a boost of
attention, still our present understanding of this epoch is very limited
due to the scarcity of observations of this epoch. So far, it has been
possible to obtain a limited amount of insight into this epoch via a
number of indirect observations such as the CMBR (Komatsu et al.
2011;Planck Collaboration et al. 2016), quasar absorption spectra
of the Ly𝛼photons at high redshifts (Fan et al. 2003;Goto et al.
2012;Becker et al. 2015;Barnett et al. 2017) and the luminosity
function and clustering properties of the Ly𝛼emitters (Ouchi et al.
2010;Ota et al. 2017;Zheng et al. 2017). These observations suggest
that the H ireionization was an extended process and had most likely
ended by redshift 𝑧∼6. Since these observations can not trace the
Hidistribution at different IGM ionization stages, thus are unable
to resolve a number of fundamental issues such as the topological
evolution of the Hiin the IGM due to reionization, the morphology
★E-mail: aadarshbritia@gmail.com
of the ionized regions at a particular IGM ionization stage and how
the different reionizing source characteristics are affecting the time
evolution of the Hidistribution.
The H i21-cm signal, originating when the electron and proton
in the ground state of H ichange their spin states from parallel to
anti-parallel, promises to act as a direct probe of the H idistribution
in the IGM and thus can potentially answer many of these funda-
mental issues related to the EoR (Furlanetto et al. 2006;Pritchard
& Loeb 2012). This signal can directly probe the H idistribution in
the IGM at different cosmic times and hence in principle can trace
the reionization history. In order to detect this signal, a number of
first generation radio telescopes such as GMRT1(Paciga et al. 2013),
LOFAR2(Mertens et al. 2020), MWA3(Barry et al. 2019), PAPER4
(Kolopanis et al. 2019) and HERA5(DeBoer et al. 2017;The HERA
Collaboration et al. 2021) are operational. The sensitivity of the first
generation of radio telescopes however are very low, thereby the inter-
1http://www.gmrt.ncra.tifr.res.in
2http://www.lofar.org/
3http://www.mwatelescope.org/
4http://eor.berkeley.edu/
5https://reionization.org/
©2022 The Authors
arXiv:2202.03701v1 [astro-ph.CO] 8 Feb 2022
2Pathak et al.
ferometric detection of the H i21-cm signal has not been possible yet.
Due to their low sensitivity, these telescopes are targeting to detect
the signal via Fourier statistics instead of trying to make tomographic
images of the EoR 21-cm signal. The future Square Kilometre Array
(SKA)6(Koopmans et al. 2015;Mellema et al. 2015) is expected to
have enough sensitivity to make high resolution tomographic images
of the EoR 21-cm signal. Once the tomographic images are produced
through these future observations, it will open up new avenue for bet-
ter insights into the EoR, the images will contain both the amplitude
and phase information of the 21-cm field.
So far the analysis of EoR 21-cm signal is mostly done via vari-
ous Fourier statistics such as the power spectrum (Bharadwaj & Ali
2004;Barkana & Loeb 2005;Lidz et al. 2008;Mao et al. 2012;
Majumdar et al. 2013,2014,2016;Pober et al. 2014;Mondal et al.
2015,2016;Patil et al. 2017;Giri et al. 2019;Mertens et al. 2020),
multi-frequency angular power spectrum (Datta et al. 2007;Mon-
dal et al. 2017b;Mondal et al. 2019,2020), bispectrum (Bharadwaj
& Pandey 2005;Yoshiura et al. 2015;Shimabukuro et al. 2016;
Majumdar et al. 2018,2020;Watkinson et al. 2019,2021a,b;Trott
et al. 2019;Hutter et al. 2020;Saxena et al. 2020;Kamran et al.
2021a,b;Mondal et al. 2021;Tiwari et al. 2021), trispectrum (Lewis
2011;Mondal et al. 2015;Shaw et al. 2019) etc. The EoR 21-cm
signal fluctuations, which are mainly determined by the sizes, distri-
butions and connectivity between the ionized regions, is expected to
be highly non-Gaussian. Thus one would expect the signal statistics,
which are of higher order than the power spectrum, to contain more
details about this non-Gaussian signal. However, the estimation of
higher order statistics is computationally more involved and their
interpretation is also difficult. Further, these higher order Fourier
statistics e.g. bispectrum, trispectrum etc. contain information about
the fluctuations of the signal at different length scales and correlation
between them. However, by definition, they do not contain the phase
information of the signal.
There are a number of complementary methods which deal with
the signal in real space and have been employed directly on the simu-
lated tomographic images to probe the morphology of the 21-cm field
and its evolution during the EoR through the analysis of topology
and geometry of this field. Among them, the widely used methods
are the Minkowski Functionals (MFs) (Friedrich et al. 2011;Hong
et al. 2014;Yoshiura et al. 2017;Bag et al. 2018,2019) which have
been used to track the reionization history, the Minkowski tensors
(Kapahtia et al. 2019) which are the generalized tensorial form of
MFs and can encapsulate the direction information. These are also
methods based on percolation theory (Iliev et al. 2006;Iliev et al.
2014;Furlanetto & Oh 2016;Bag et al. 2018,2019), granulome-
try (Kakiichi et al. 2017) and persistence theory (Elbers & van de
Weygaert 2019) which have been used for the theoretical study of
the topological phases of H ii regions during EoR. In addition to
these, a method based on the Betti numbers (Giri & Mellema 2020;
Kapahtia et al. 2021) provide the number of connected regions, tun-
nels and cavities to describe the state of the IGM. A recent study of
Gorce et al. (2021) used a method of local variance which probes
the reionization history of the observed patches of the sky as well as
trace the ionization morphology. However, in most of these earlier
works it is widely been considered that modelled reionizing sources
are standard fiducial in nature and the inference drawn through these
methods depend on the detection of a large number of H ii regions
at any stage consisting of wide variety of sizes. A recent study of
Bag et al. (2018) on the contrary focuses on the detection of only the
6http://www.skatelescope.org/
largest ionized region in order to get better insights of the percolation
process. For this they have used the “largest cluster statistic” along
with a shape finding algorithm.
The insights about the percolation of H ii regions, from its onset
to the stage when all of the ionized regions are interconnected to
form a single large cluster, will depend on the ionizing source and
IGM properties. Motivated by this, in this work, we aim to study the
morphology and topology of the largest ionized regions (at differ-
ent stages of the EoR) while considering a number of reionization
scenarios with different source and IGM properties. In order to ex-
plore the morphology of the largest H ii regions and its evolution
with cosmic time, we use the percolation technique in addition with
the Shapefinders which are defined as the ratio of the Minkowski
functional (Sahni et al. 1998). These Shapefinders can be used to
analyze the shape of the ionized regions. Thus, this method allows
us to gather information related to individual ionized regions along
with the information about how these regions form very large inter-
connected network at the cosmological scales. For our analysis, we
have employed the advanced shape diagnostic tool SURFGEN2
(Bag et al. 2019;Bag & Liivamägi 2021).
The size and shape of the H ii regions around the ionizing sources,
the onset of connecting tunnels between these H ii regions to form
a comparatively larger H ii region and their further growth are all
closely dependent on the ionizing source and IGM properties. Our
aim here is to study the impact of the different source properties on
the morphology of reionization. For this purpose, we consider dif-
ferent simulated reionization scenarios. The source models in these
simulated reionization scenarios are different from each other in two
fundamental ways: a) how the number of ionizing photons emitted
by the sources are related to their host halo mass and b) how the
rest frame energy of the ionizing photons are distributed. For all of
these scenarios we follow the evolution of the topology of the largest
ionized region with the cosmic time.
This paper is organized as follows: In section 2, we discuss the
simulated EoR 21-cm maps for various reionization scenarios which
we have used for this work. Section 3briefly describes our methods
for analysing these simulated 21-cm maps, including the percolation,
Minkowski functionals and Shapefinders along with the SURFGEN2
code. In section 4, we discuss our findings regarding the evolution of
largest ionized region. Finally, in section 5, we summarize our result.
Throughout the paper, we have used the cosmological parameters
satisfying the WMAP five year data release h=0.7 , Ω𝑚=0.27,
ΩΛ=0.73,Ω𝑏ℎ2=0.0226 as mentioned in literature (Komatsu
et al. 2009).
2 SIMULATION
Simulating the reionization is essentially a challenging task due to
the requirements of high dynamic range. The fundamental problem
in simulating reionization from the first principles is essentially to
capture the large scale cosmology and small scale astrophysics. This
requires one to simulate the reionization in large enough cosmologi-
cal volume (∼1 Gpc3) so that the impact of large scale matter density
fluctuations are properly taken into account. At the same time, it re-
quires one to resolve the sources of reionization (typical galaxies;
∼10 kpc in size) so that their properties are mimicked correctly.
One can use a 3D radiative transfer simulation which works on
the principle of ray tracing by following the ionization fronts in
the IGM and based on which one can check for different physical
processes taking place during the EoR (Ricotti 2002;Thomas et al.
2009;Iliev et al. 2014;Gnedin 2014;Ghara et al. 2015). But it is
MNRAS 000,1–16 (2022)
Probing EoR with 21-cm LCS 3
almost computationally impossible to explore the multi-dimensional
reionization parameter space using these simulations. So, in order
to reduce the computational complexities, one can fairly choose the
semi-numerical technique (Zahn et al. 2007;Choudhury et al. 2009;
Mesinger et al. 2011;Majumdar et al. 2013,2014;Mondal et al. 2015,
2017a) which generally compares the average number of photons
with the average number of neutral hydrogen present in a smoothing
volume rather than performing a full radiative transfer calculations.
These simulations are based on the excursion set formalism proposed
by Furlanetto et al. (2004) where the ionization map generated at any
redshift is mostly dependent on the underlying matter distribution
and collapsed structures at that redshift. In this paper, we have used
the semi-numerically simulated 21-cm maps from Majumdar et al.
(2016) for our analysis.
The semi-numerical method used in Majumdar et al. (2016) mainly
involves three steps. First, it generates the dark matter distribution at
any desired redshift by using 𝑁-body dark matter gravity only sim-
ulations. Second, it identifies the collapsed dark matter halos within
the simulated matter distribution which can be accomplished with the
help of algorithms like FoF or spherical smoothing. Finally, it con-
siders the halos as the most probable hosts of the sources of ionizing
photons and uses excursion set formalism to generate the ionization
field which is later converted into the 21-cm field. Majumdar et al.
(2016) have simulated the signal in a cube with 500 ℎ−1Mpc =714
Mpc (in comoving scale) in length along each side. The underlying
𝑁-body output for their simulation used CUBEP3Mcode (Harnois-
Déraps et al. 2013) ran as a part of the PRACE4LOFAR project
(PRACE projects 2012061089 and 2014102339). For reionization
modeling, they have used 69123particles of mass 4.0×107𝑀on
a138243mesh and the fields are then interpolated on a 6003grid
points. The minimum halo mass used in their reionization simulation
is 2.09 ×109𝑀.
2.1 Source Models and Reionization scenarios
The evolution of ionized regions in the IGM depends on the character-
istics of the sources producing ionizing photons and IGM properties.
Thus depending upon the type of possible source characteristics and
IGM properties one can construct several reionization scenarios. In
this paper, we are using six such simulated reionization scenarios
from Majumdar et al. (2016). In the following section, we describe
the four sources models which are taken in different combinations to
simulate these six scenarios. For a detailed discussion on the same
we refer the interested reader to Majumdar et al. (2016).
2.1.1 Source models
The four source models shaping the six reionization scenarios con-
sidered here are:
(i) Ultraviolet photons (UV photons): The galaxies residing in the
collapsed dark matter halos are considered to be the most probable
sources to produce ionizing photons in the form of UV radiation.
Here, it is assumed that the total number of emitted ionizing photons
follow the relation :
𝑁𝛾(𝑀ℎ)=𝑁ion
𝑀ℎΩ𝑏
𝑚𝑝Ω𝑚
(1)
where, 𝑁ion represents the number of photons entering in the IGM
per baryon in collapsed objects, 𝑀ℎis the halo mass and 𝑚𝑝is the
mass of proton.
Reionzation
scenarios
UV UIB SXR PL,n
Fiducial 100 % - - 1
Clumping 100 %
& Non
Uniform
Re-
comb.
- - 1
UIB Domi-
nated
20 % 80 % - 1
SXR Domi-
nated
20 % - 80 % 1
UV + SXR +
UIB
50 % 10 % 40 % 1
PL (n=3) 100 % - - 3
Table 1. Contribution of different source models in our reionization scenarios.
Table taken from Majumdar et al. (2016).
(ii) Uniform Ionizing Background (UIB photons) : This source
model assumes the sources like AGNs, X-ray binaries etc as the most
probable ones to produce the hard X-ray photons, following a similar
relationship as Equation (1), which due to its long mean free path
ultimately form an uniform ionizing photon background.
(iii) Soft X-ray photons (SXR) : This source model produces soft
X-ray photons, following a formalism similar to Equation (1), which
creates uniform ionizing background within a region limited by the
mean free path of those photons.
(iv) Power Law mass dependent efficiency (PL) : Here, the number
of ionizing UV photons produced by the sources residing in the
collapsed dark matter halos is proportional to the 𝑛𝑡 ℎ power of the
halo mass following the relation :
𝑁𝛾(𝑀ℎ) ∝ 𝑀𝑛
ℎ.(2)
In the scenario considered here, the chosen value of power law index
is 3. Here, 𝑁𝛾(𝑀ℎ)is the total number of ionizing photons emitted
and 𝑀ℎis the corresponding halo with mass 𝑀.
2.1.2 Reionization Scenarios
We use the 21-cm maps of six different reioniziation scenarios from
Majumdar et al. (2016) for our analysis here. These scenarios are built
by different combinations of the above mentioned source models and
their main characteristics are summarized in the Table 1.
The fiducial scenario considers 100% ionizing UV photon con-
tribution from the galaxies residing in the halos of mass ≥2.09 ×
109𝑀. In a similar fashion, the clumping and PL (𝑛=3) scenar-
ios also consider 100% UV photon contribution from halos but are
slightly different from the fiducial. Clumping is the only scenario
where non-uniform density dependent recombination has been taken
into account. On the other hand, in PL (𝑛=3) scenario, high mass
halos have a higher weightage in UV photon production than low
mass halos as they follow Equation (2) instead of (1) with 𝑛 > 1.
Scenarios like UIB dominated, SXR dominated and UV+SXR+UIB
have a mixed contribution of different types of ionizing photons as
tabulated in Table 1. The UIB dominated and SXR dominated sce-
MNRAS 000,1–16 (2022)
4Pathak et al.
Figure 1. 21-cm map of six reionization scenarios taken from Majumdar et al. (2016) at neutral fraction ¯𝑥Hi=0.5.
narios have 80% of ionizing photons in the form of hard and soft
X-rays respectively.
In this work, while analyzing the topology of the 21-cm maps, we
consider the completely ionized regions only7, defined as 𝜌Hi(x)=0
in the simulation8. The neutral fraction ( ¯𝑥Hi) and ionized filling factor
(FF) at a specific redshift are defined as
¯𝑥Hi(𝑧)=¯𝜌Hi(𝑧)/ ¯𝜌H(𝑧),(3)
and
FF =total volume of all the ionized regions
simulation volume (4)
respectively. In other words, FF and (1−¯𝑥Hi)essentially measure
the fraction of hydrogen mass and volume that is ionized at a given
redshift/time.
In the reionization scenarios considered here, the proportionality
constant 𝑁ion was tuned such that all of them have the same mass
averaged neutral fraction ( ¯𝑥Hi(𝑧)) at a given time/redshift, i.e. they
follow the same reionization history. This is illustrated in Figure 2
which shows the evolution of the neutral fraction with redshift for all
of the scenarios. Since different scenarios follow the same ¯𝑥Hi(𝑧),
we discuss our results a functions of ¯𝑥Hiinstead of redshift. This
7We consider the regions with 𝜌Hi(x)>0as ‘neutral’ which includes
partially ionized regions too.
8In reality, the low frequency radio interferometers will measure the bright-
ness temperature which is proportional to the neutral hydrogen density after
mean subtraction, i.e. (𝜌Hi(x) − ¯𝜌Hi). The ionized region, defined here as
𝜌Hi(x)=0, would be mapped to the minima of the observed brightness
temperature field that can be easily traced.
will help us in understanding how the 21-cm topology/morphology
is related to the neutral fraction in different reionization scenarios.
Note that despite ¯𝑥Hibeing same for all scenarios at a given red-
shift, the total volume of the ionized regions may not be the same
in the different scenarios, as demonstrated in the Figure 3. For a
given ¯𝑥Hi(or redshift), UIB dominated scenario results in least FF
whereas PL (𝑛=3) produces highest FF. Since in the UIB domi-
nated scenario, the hard x-ray photons can escape to longer distances
and can effectively produce an uniform ionizing background. This
leads to an so called ‘outside-in’ reionization, where the low density
regions ionize first and then the high density regions follow the suit.
Therefore, in this scenario a large volume will be partially ionized in
contrast to PL or Fiducial scenarios. In the Fiducial, Clumping or PL
scenarios, the reionization will be ‘inside-out’ in nature, where UV
radiations from the collapsed halos escapes to longer distances only
after they have substantially ionized their local IGM. The SXR Dom
and the UV+SXR+UIB lie somewhere in between the ‘inside-out’
and ‘outside-in’ scenarios.
3 METHODS
3.1 Percolation
When the first luminous objects form at the cosmic dawn (near 𝑧∼15
Choudhury & Ferrara 2006) they start to ionize their surrounding
neutral hydrogen field. As reionization progresses, these small ion-
ized hydrogen bubbles grow both in size and number and slowly
they start to overlap too. But at some point in time, depending on
the reionization scenario, these bubbles suddenly coalesce together to
MNRAS 000,1–16 (2022)
Probing EoR with 21-cm LCS 5
Figure 2. Reionization history. Global neutral fraction as a function of red-
shift. All six reionization scenarios considered here follow this same reion-
ization history.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.2
0.4
0.6
0.8
Filling Factor
Fiducial
Clumping
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
7.2
7.3
7.4
7.5
7.7
7.8
7.9
8.2
8.38.5
8.89.2
9.6
10.3
Figure 3. Filling Factor (FF) of ionized hydrogen is plotted against the ¯𝑥Hifor
all reionization scenarios. The contribution of FF is significant in the entire
neutral fraction range for scenarios like fiducial, clumping, PL (n=3) while it
become significant at the later stage of reionization in the scenarios like UIB
dom. and SXR dom.
form a large connected single ionized region. This abrupt topological
change in the ionization field can be viewed as a ‘phase’ transition
and we call it percolation transition Klypin & Shandarin (1993);
Yess & Shandarin (1996). In this work, we track the largest ionized
region with redshift for different reionization scenarios. We identify
the onset of percolation in a reionization scenario, when the largest
ionized region stretches from one face of the simulation box to the
opposite face. The largest ionized region is then infinitely extended
due to periodic boundary condition of our simulation volume. This
is illustrated in Figure 4for the fiducial model. The largest ionized
region is shown at three stages, well before, just before and just after
percolation transition in the panels from left to right respectively.
One can see that the largest region grows rapidly from the middle
panel to the right one with minute change in the neutral fraction
at percolation. Just after percolation, as shown in the right panel, it
extends through out the space in all directions.
How the percolation transition happens in different reionization
scenarios can be studied more comprehensively using the Largest
Cluster Statistics (LCS) Yess & Shandarin (1996); Klypin & Shan-
darin (1993); Sahni et al. (1997) defined as ,
LCS =volume of the largest ionized region
total volume of all the ionized regions .(5)
Therefore, LCS of the ionized region can be regarded as the frac-
tion ionized volume residing inside the largest ionized region. At
percolation transition, we expect that LCS increases sharply with
filling factor or ¯𝑥Hi. But when and how percolation takes place in
an ionized hydrogen field depend on the reionization process itself.
Therefore, we study the evolution of LCS (with FF and ¯𝑥Hi) for
different reionization models to understand the effects of different
ionization processes on the percolation transition and eventually to
distinguish the models.
3.2 Minkowski functionals and Shapefinders
We complement the percolation analysis with the study of mor-
phology of individual ionized regions using Minkowski function-
als (MFs). A closed two dimensional surface has the following four
Minkowski functionals Mecke et al. (1994) –
(i) Volume enclosed: 𝑉,
(ii) Surface area: 𝑆,
(iii) Integrated mean curvature (IMC):
𝐶=1
2∮1
𝑅1
+1
𝑅2𝑑𝑆 , (6)
(iv) Euler characteristic (or Gaussian mean curvature):
𝜒=1
2𝜋∮1
𝑅1𝑅2
𝑑𝑆 . (7)
Here 𝑅1and 𝑅2are the two principal radii of curvature at a point on
the surface. Euler characteristic (the fourth Minkowski functional)
measures the topology of a surface and it can be further expressed in
terms of the genus (G) of the surface as follows,
𝐺=1−𝜒/2≡ (no.of tunnels)−(no.of isolated surfaces) + 1.(8)
For example, an isolated closed surface with 𝑁𝑐cavities and 𝑁𝑡
tunnels passing through it would have genus=𝑁𝑡−𝑁𝑐.
The ratios of these Minkowski functionals are introduced as
‘Shapefinders’ in Sahni et al. (1998) to assess the shape of an ob-
ject (such as a cluster or a void). In three dimensions, we have three
Shapefinders, namely,
Thickness : 𝑇=3𝑉/𝑆 , (9)
Breadth : 𝐵=𝑆/𝐶 , (10)
Length : 𝐿=𝐶/(4𝜋).(11)
All of the Shapefinders – 𝑇, 𝐵, 𝐿 – have dimension of length, and they
are good measures of the extensions of an object in 3-dimensions,
MNRAS 000,1–16 (2022)
6Pathak et al.
(a) ¯𝑥Hi=0.80, before percolation (b) ¯𝑥Hi=0.75, onset of percolation (c) ¯𝑥Hi=0.72, just after percolation
Figure 4. The largest ionized region in the fiducial model has been shown within the simulation volume in the three panels at three neutral fractions,
¯𝑥HI =0.80,0.75,0.72 from left to right, characterising well before, just before and just after percolation. At percolation, the largest regions grows abruptly for
minuscule changes in the ¯𝑥Hi, as seen in the middle to right panel. These plots are shown in grid units of the simulation which has (600)3grid points and a
physical size of (714.28Mpc)3.
hence the name – thickness, breadth and length9. The Shapefinders
are defined to be spherically normalized, i.e. 𝑉=(4𝜋/3)𝑇 𝐵𝐿 .
Using the Shapefinders one can further determine the morphology
of an object (such as ionized regions), by means of the following
dimensionless quantities
Planarity : 𝑃=𝐵−𝑇
𝐵+𝑇,(12)
Filamentarity : 𝐹=𝐿−𝐵
𝐿+𝐵.(13)
As the names suggest, 𝑃and 𝐹characterize the ‘planarity’ and
‘filamentarity’ of an object and 0≤𝑃, 𝐹 ≤1by construction. A
sphere will have 𝑃'𝐹'0, while 𝑃∼𝐹∼1for a ribbon. A
planar object (such as a sheet) 𝑃𝐹whereas the reverse is true for
a filament which has 𝐹𝑃. In the context of this work, studying
Shapefinders of ionized regions at different stages of reionization,
together with percolation analysis, would shed light on the evolution
of geometry, morphology and topology of the ionization field.
3.3 The SURFGEN2 code
To assess the shape of the ionized regions in terms of Shapefinders,
we employed SURFGEN2, which is a more advanced version of
SURFGEN algorithm, originally developed by Sheth et al. (2003);
Sheth & Sahni (2005); Sheth (2006) for studying the largest scale
structure of the universe. The improvements in SURFGEN2 have
been explained in Bag et al. (2018,2019) in more details.
•The code first identifies all the isolated ionized regions, follow-
ing the definition 𝜌Hi(x)=0, in the simulated 𝜌Hifield using the
‘Friends-of-Friends’ (FoF) algorithm. Note that, SURFGEN2 finds
the regions consistent with periodic boundary conditions.
•Next, the code models the surface of each ionized region using
the advanced ‘Marching Cube 33’ triangulation scheme (Lorensen
& Cline 1995), which circumvents the issues associated with the
original Marching Cube algorithm (Chernyaev 1987).
•Finally, SURFGEN2 determines the Minkowski functionals and
9One finds 𝐿≥𝐵≥𝑇in general. However, if the natural order 𝑇6𝐵6𝐿
is violated for a region, we choose the the largest one as 𝐿and the smallest
as 𝑇to restore the order. In the some rare cases a region may have 𝐶 < 0.
We redefine 𝐶→ |𝐶|to ensure that all the Shapefinders are positive in those
cases.
Shapefinders of each ionized region separately from the triangle
vertices that we have from the previous step.
•We follow the above steps separately at every stage of all the
reionization scenarios considered here to determine the shape statis-
tics of ionized regions in those different reionization models.
•Note that, by tracking the largest ionized region, we can compute
the LCS using (5) and study its evolution in different reionization
scenarios.
4 RESULTS
4.1 Distinguishing different reionization scenarios using cluster
statistics and percolation
The cluster statistics can be very useful in order to distinguish differ-
ent reionization models, particularly the two fundamental scenarios –
inside-out and outside-in reionization. First we focus on the number
of ionized regions (𝑁𝐶) which has been plotted against the neu-
tral fraction ( ¯𝑥Hi) and the filling factor (FF) respectively in the left
and right panels of Figure 5for different reionization scenarios. As
reionization progresses, ¯𝑥Hidecreases from unity in the left panel
whereas the ionized filling factor increases in the right panel. At the
beginning of reionization, 𝑁𝐶increases with time as new ionized
pockets start to appear as evident from both panels. On the other
hand, at the advanced stages of reionization, ionized regions overlap
extensively and the number of ionized regions decrease. Therefore,
somewhere in between, 𝑁𝐶exhibits a maximum in all the reioniza-
tion models as illustrated in both the panels. However, the way 𝑁𝐶
reaches to its maximum is different for different reionization scenar-
ios (the percolation transition takes place near the maxima of the
respective models, as we would find out below). For example, UIB
dominated model (shown in black color) has the highest number of
ionized regions whereas 𝑁𝐶in the PL (𝑛=3) model (shown in or-
ange color) is the smallest throughout. Interestingly, as demonstrated
in the left panel, the maxima in 𝑁𝐶occur at different ¯𝑥Hivalues for
different models e.g. at early phases for PL (n=3), Fiducial, Clumping
models while at late phases for the UIB dominated model (others in
between). In contrast, the maxima in 𝑁𝐶for different models are lo-
cated at similar values of ionized filling factor, 𝐹 𝐹 ∼10%, as shown
in the right panel. Note that different models have different filling
factor by the end of the redshift range we consider in this work.
Next, we analyse the percolation (explained in section 3.1) by
following the Largest Cluster Statistics (LCS) at multiple ¯𝑥Hi. Let us
MNRAS 000,1–16 (2022)
Probing EoR with 21-cm LCS 7
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8
Filling Factor
Fiducial
Clumping
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
Figure 5. The number of ionized regions (𝑁𝐶) for different reionization models has been shown against the neutral fraction ( ¯𝑥Hi) and the ionized filling factor
(FF) in the left and right panel respectively. Note in the right panel that different models go up to different filling factor by the end of the redshift range we
consider in this work.
first focus on the results for the fiducial reionization scenario. Figure
6shows the evolution of LCS with ¯𝑥Hiand filling factor (FF) in the
left and right panel respectively. The vertical dashed lines in both
panels show the onset of percolation when the largest ionized region
abruptly grows in all directions and extend throughout the simulation
volume. During the percolation transition, a very little change in ¯𝑥Hi
or FF results in a sharp rise in LCS as illustrated in the respective
panels. Note that percolation transition itself can be defined through
this formally discontinuous growth in LCS (Klypin & Shandarin
1993). We find that percolation transition takes place for the fiducial
model at ¯𝑥Hi≈0.75 and FF ≈9.6% (corresponds to 𝑧≈8.1) which
are consistent with the earlier findings in the literature (Iliev et al.
2006;Chardin et al. 2012;Furlanetto & Oh 2016;Bag et al. 2018)
where slightly different models were considered. Post percolation,
most of the individual ionized regions are assimilated into one large
ionized region and the LCS quickly saturates near unity.
Next, we compare the percolation transitions in different reioniza-
tion scenarios in figure 7. The left and right panels show the evolution
of LCS with ¯𝑥Hiand ionized filling factor (FF) respectively for dif-
ferent source models. The onset of percolation in these models is
marked by the vertical lines with respective colours in both panels.
We find that the LCS exhibits the sharp increase at the percolation
transition for all the models, similar to what we observe for fiducial
model above. However, the ionized hydrogen in different scenarios
percolates at different values of ¯𝑥Hibut at only slightly different FF
values. The first two columns of the table 2present the critical neutral
fractions and the ionized FF where percolation take place for each
reionization source model.
Focusing on the LCS vs ¯𝑥Hicurves in the left panel of figure 7, one
can get a wealth of information regarding the nature and properties of
the source models or reionization scenarios. (1) Firstly, in all reion-
ization scenarios the ionized hydrogen percolates at very different
global neutral fraction values. This clearly demonstrates that perco-
lation is strongly connected to the 21-cm topology. (2) Although we
can see a sharp ascent in the LCS for all reionization scenarios at
the percolation transition but the shape of the LCS curves is different
for different scenarios. For example, in PL (n=3) model, the large
ionized regions could easily connect and eventually percolate at a
relatively earlier stage of reionization. In contrast, the percolation
transition in UIB dominated model takes place at an advance stage
of reionization because of the ionized regions being relatively finer
and the filling factor being low. In summary, ionized hydrogen in
the scenarios like fiducial, clumping, PL (n=3), percolates at higher
neutral fractions because of the inside-out reionization in these cases.
On the other hand, in the outside-in reionization scenarios, like UIB
dominated, SXR dominated etc., percolation takes place at much
lower neutral fractions once the filling factor becomes significant. In
view of figure 3one finds that the filling factor in these scenarios
could grow substantially only after the respective percolation transi-
tions. (3) For PL (n=3) (shown by the orange curve), LCS starts with
a larger value at the beginning of the EoR (i.e. before percolation)
compared to the other models. This is the manifestation of the fact
that this reionization scenario creates larger (in size) but fewer ion-
ized regions as compare to other scenarios, as shown in figure 5.(4)
The clumping model shows percolation at very early stage of the EoR
(nearly 𝑧≈8.515) since this is the only scenario where non-uniform
recombination has been taken into the account. Although this non-
physical scenario could not be distinguished from the others using
the percolation analysis, the shape diagnostic tools – Shapefinders –
can isolate it from the rests, as we demonstrate below.
From the LCS vs FF curves, presented in the right panel of figure 7,
we observe similar characteristics. However, percolation transitions
in all the models take place in a narrow range of ionized filling
factor, FF ∈ (10 −20%). Therefore, we find that LCS vs FF curves
are not as sensitive to the reionization scenarios as the LCS vs ¯𝑥Hi
curves, making the former curves not quite suitable for distinguishing
reionization models.
We note that the cluster statistics ideally should be accompanied
by the uncertainties in order to distinguish various source models. A
proper error estimation requires analysing a large number of realiza-
tions of the simulations for each scenario at each redshift. Simulating
many high resolution realizations for every case is beyond the scope
MNRAS 000,1–16 (2022)
8Pathak et al.
Reionzation sce-
narios
Critical
¯𝑥HI
𝐹 𝐹𝐶Planarity Filamentarity
Fiducial 0.75 0.096 0.162 0.998
Clumping 0.81 0.075 0.858 0.914
PL (n=3) 0.75 0.181 0.041 0.988
UIB Dominated 0.33 0.116 0.269 0.979
SXR Dominated 0.55 0.072 0.177 0.993
UV + SXR + UIB 0.59 0.099 0.104 0.999
Table 2. All reionization scenarios are compared at percolation in terms of
the critical neutral fraction, planarity and filamentarity of the largest ionized
region.
of the present paper. Nevertheless, we crudely estimate the errors
in appendix Aon filling factor, number of ionized regions per unit
volume and LCS by dividing the large primary simulation volume
in 8 equal smaller volumes those are themselves large enough to
obey the cosmological principles. The uncertainties we find in LCS
are reasonably tight, larger for inside-out models as compared to the
outside-in models. Remarkably, we find that LCS with the tight er-
rorbars is very much suitable for distinguishing the source models,
especially between the inside-out and outside-in scenarios with high
certainty.
We have further checked the consistency of our findings from the
LCS analysis by comparing the results with the Bubble Size Distri-
bution (BSD) in different scenarios. As demonstrated in Appendix
B, the BSD shows a bimodal distribution where the largest ionized
bubble is largely separated from the bulk after percolation for all
reionization scenarios which is quite consistent with the LCS evolu-
tion.
4.2 Shape of the largest ionized region
In this section, we discuss the evolution of the topology, morphology
and shape of the largest ionized region based on the Minkowski
functionals and the Shapefinders (see the section 3.2). The left panel
of figure 8illustrates how the topology of the largest ionized region in
different reionization scenarios evolves with time. The genus value of
the largest ionized region has been plotted against the neutral fraction
in different scenarios till the percolation transition10 in the respective
scenarios. Note that we shift the x-axis by the critical neutral fraction
(at the onset of percolation) for each model. We observe that as the
reionization progresses the largest ionized regions in all the scenarios
become more multiply connected and their genus increases. As we
find out later, the ‘length’ (the 3rd Shapefinder) of the largest region
also increases with reionization. The right panel of figure 8shows
the evolution of the genus per unit ‘length’ for the largest regions
in these scenarios. In all the scenarios, except the clumping model,
genus per unit ‘length’ remains stable.
It is evident from the figure 8that the genus of the largest region
in the clumping model is much higher (by several orders) than that in
the other scenarios despite the fact that ionized hydrogen percolates
10 Because of the periodic boundary condition, the physical shape of a per-
colating region cannot be defined. Therefore, the genus value of a region
just after percolation is also not well defined. However, well beyond percola-
tion when the ionized region covers most of the simulation volume, one can
reliably estimate the genus value in terms of per unit volume.
earlier in time in the clumping model. Because of the aggressive
non-uniform recombination in the clumping scenario, there remain
many pockets of neutral hydrogen which tunnels through the ionized
regions giving rise to the high genus values. Most of these tunnels are
found to be negatively curved and hence they lead to decrease in the
overall integrated mean curvature (the 3rd MF). This in turn breaks
the natural order of the Shapefinders, i.e. 𝑇 < 𝐵 < 𝐿 and results in
higher planarity value that, one can argue, is not physical. Therefore,
we do not include the Shapefinders results for the clumping model in
the rest of the main paper, if not mentioned otherwise. We analyse the
results for the clumping model separately in the appendix Cwhere
we show how this ‘non-physical’ high planarity can distinguish the
clumping model from the rests.
The planarity (𝑃) and filamentarity (𝐹) of the largest ionized re-
gions in different reionization scenarios (except the clumping model)
have been shown in figure 9near the percolation transitions in the
ionized segment for the respective models. The filamentarity of the
largest ionized regions in all the models increases as reionization
progresses and it reaches to almost unity on approaching the per-
colation transition. In contrast, the planarity of the largest regions
always remains low, 𝑃.0.2considering all the models. Thus one
can conclude that the largest ionized regions in all the reionization
scenarios become highly filamentary near the percolation transition,
consistent with what Bag et al. (2018) found out for a single reioniza-
tion scenario. The planarity and filamentarity of the largest ionized
region at the onset of percolation have been given in the third and
fourth columns of the table 2respectively for all the ionization source
models.
For a filamentary object, 𝑇×𝐵can be regarded as the effective
‘cross-section’ (of the filament). Bag et al. (2018) pointed out that
the largest ionized region (in their simulation) abruptly grows only
in terms of its third Shapefinder , ‘length (𝐿)’, during the percolation
transition while its ‘cross-section’ remains stable. Here we put this
claim to test for different scenarios. The top panel of figure 10 shows
the cross-section (𝑇×𝐵) of the largest ionized regions in different
scenarios against the neutral fraction near the percolation in the
respective models. Their lengths have been plotted in log-scale in the
bottom panel. The vertical dashed lines in both panels correspond to
the onset of percolation in each model.
Comparing both panels one can find out that the lengths are in
general several orders of magnitude higher than the cross sections.
Indeed, we find that the cross-section of largest ionized regions in
different models does not increase much during percolation whereas
their length rises sharply. Therefore, the claim of Bag et al. (2018) that
the largest region grows in-terms of its length only during percolation
has been somewhat followed in all the models. One can further
observe in the top panel of the figure 10 that this stable ‘characteristic’
cross-section has the highest value for the PL (n=3) model while
the lowest value for the UIB dominated model among the ones we
consider in this work. For UIB dominated reionization source model,
the ionized bubbles are much smaller in size (and more uniformly
distributed in the space) as compared to, say, the fiducial or the
PL (n=3) model. Therefore, the inter-connected filamentary largest
region arising because of merger of these tiny bubbles in the UIB
dominated model also has smaller cross-section as compared to other
models. On the other hand, since PL (n=3) produces large ionized
bubbles to start with, the largest region also possesses wider effective
cross-section.
MNRAS 000,1–16 (2022)
Probing EoR with 21-cm LCS 9
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Largest Cluster Statistics (LCS)
Fiducial
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Filling Factor
7.2
7.3
7.4
7.5
7.7
7.8
7.9
8.2
8.38.5
8.89.2
9.6
10.3
10.3
9.6
9.2
8.8
8.5
8.3
8.2
7.9
7.87.7
7.57.4
7.3
7.2
Figure 6. The left and right panels show LCS of the ionized hydrogen in the fiducial model as a function of the neutral fraction ( ¯𝑥Hi) and the filling factor (FF)
respectively. In both panels, the vertical dashed line represents the onset of percolation where LCS rises abruptly in this reionization scenario.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Largest Cluster Statistics
0.0 0.2 0.4 0.6 0.8
Filling Factor
Fiducial
Clumping
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
Figure 7. The percolation transition has been compared for all six reionization models in terms of LCS. The left panel shows the LCS as a function of ¯𝑥Hi
whereas the right panel illustrates how LCS evolves with the filling factor (FF) for the different reionization scenarios. In both panels, the vertical lines represent
the percolation transitions in the reionization models with corresponding colours.
4.3 Shape distribution of ionized regions in different
reionization scenarios
In this section we study the shape (together with topology and mor-
phology) distribution of the individual ionized regions in the different
source models at the onset of the percolation transitions. We divide
the regions into 8 volume bins. The errorbars represent the standard
deviation which illustrate the scatter of respective quantities in each
bin.
The top panel of figure 11 shows the volume averaged genus
values of ionized regions falling in different volume bins for all the
source models (excluding the clumping model). For all the models,
larger regions are more multiply connected with higher average genus
values. The PL (n=3) model produces large ionized regions and the
MNRAS 000,1–16 (2022)
10 Pathak et al.
0.00 0.05 0.10 0.15 0.20
0
1000
2000
3000
4000
5000
6000
7000
Genus
Fiducial
Clumping
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
0.00 0.05 0.10 0.15 0.20
Fiducial
Clumping
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
Figure 8. Genus of largest ionized region is compared in the left panel for all reionization scenarios up to the percolation transition in linear scale ( ¯𝑥Hi=¯𝑥Hicrit ).
The right panel which shows the genus per unit ‘length’ in log-scale for the largest regions illustrates that for all scenarios the genus is almost proportional to
the length except for the clumping model.
Figure 9. Planarity and filamentarity of the largest ionized region in different
models have been compared near the respective percolation transitions. The
solid and dashed curves represent the reionization scenarios following inside-
out and outside-in reionization respectively. The curves shown with thick lines
(solid and dashed) represent the filamentarity and the curves with thin lines
represent the planarity of the largest ionized region in different models. Note
that percolation transition takes place at different ¯𝑥Hicrit for different models,
thus at ¯𝑥Hi−¯𝑥Hicr it =0(shown by the dashed ver tical line) for all the models.
growth of genus with volume is shallower than that in the other
models. On the other hand, for the SXR dominated model average
genus value increases more rapidly for the higher volume bins. In
general we observe the trend that for outside-in models the larger
ionized regions tend to be more multiply connected. Note that the
UIB dominated model produces small ionized regions, hence the
curve is truncated at much lower volume.
The bottom two panels show the volume averaged planarity and
filamentarity, respectively, of the ionized regions belonging to dif-
ferent volume bins. For all the models, the ionized regions have very
low planarity. But the filamentarity tends to increase with the vol-
ume. Interestingly, for the PL (n=3) model the growth of filamentarity
-0.2 -0.1 0.0 0.1 0.2
Fiducial
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
Figure 10. These panels show the result for all reionization scenarios. In top
panel, cross section (TXB) is plotted against the neutral fraction ( ¯𝑥HI −¯𝑥HIcrit ).
In bottom panel, length is plotted against the neutral fraction ¯𝑥HI −¯𝑥HIcrit .
One can observe the similar kind of behaviour in length and cross-section
as observed for fiducial for our largest ionized region for all reionization
scenarios in terms of their evolution with decreasing neutral fraction.
with volume is the slowest among all the scenarios we consider here.
On the other hand, the ionized regions in the UIB dominated model
have overall higher filamentarity due to their narrower cross-sections.
MNRAS 000,1–16 (2022)
Probing EoR with 21-cm LCS 11
0.2
0.4
0.6
0.8
1.0
0.0
0.1
0.2
0.3
0.4
0.5
Figure 11. The three panels from top to bottom are plotted to study the shape
of the largest ionized regions where planarity (volume averaged h𝑃i𝑉) , fila-
mentarity (volume averaged h𝑃i𝑉) and genus (volume averaged h𝑃i𝑉) has
been plotted as a function of volume bins (averaged) in Mpc3respectively.
Each plot shows the variation of above mentioned quantities for all reioniza-
tion scenarios and error bars in each plot depicts the standard deviation which
could be observed for ionized region in each bin.
However, we conclude that the large ionized regions in all model tend
to be filamentary with high filamentarity and low planarity.
5 SUMMARY AND DISCUSSION
We have studied the percolation transition in the ionized hydrogen
by tracing the largest cluster statistics (LCS) in the 3D redshifted
21-cm maps of EoR using a suite of semi-numerical simulations.
We compared different reionization scenarios in terms of the geo-
metrical properties of the ionized regions and specifically the way
they evolve in the vicinity of percolation transition. We used SUR-
FGEN2 algorithm which implements the shape finding techniques
like Minkowski functionals and Shapefinders in conjunction with the
percolation analysis to explore the morphology of the 21-cm field.
The key points of our work are summarized below:
•The number of isolated ionized regions formed by the different
reionization scenarios essentially informs about the nature of the
undergoing reionization process. Scenario like UIB dominated has
the largest value of 𝑁𝐶which illustrates the fact that the low dense
regions (voids in between the filaments) which are mainly larger in
numbers are ionized first and hence obeys outside-in reionization.
While on the other hand, small value of 𝑁𝐶in scenario like PL
(𝑛=3) indicates that the highly dense regions are being ionized first
and hence it will extensively follow inside-out reionization.
•The largest ionized region before percolation transition is almost
indistinguishable from other ionized regions for all reionization sce-
narios. LCS starts to grow at the onset of percolation transition in
the ionized hydrogen. Since the evolution of LCS is sensitive to the
reionization scenarios, it can be used to distinguish different source
models. We can further classify the scenarios in two broad categories,
inside-out and outside-in reionization. For the inside-out reionization
scenarios (that include the fiducial, PL (𝑛=3) and clumping models)
percolation transition in the ionized hydrogen takes place at an earlier
phase (larger ¯𝑥Hi) in contrast to the outside-in scenarios (UIB, SXR
dominated and UV+SXR+UIB scenarios) where ionized hydrogen
percolates only at low ¯𝑥Hivalues, i.e. at advanced stages of reion-
ization. These findings based on LCS are found to be consistent with
that from the Bubble Size Distributions.
•We also find that the cross section of the largest ionized region
in all the scenarios (except for the clumping model) remains stable
at the percolation transition when the largest ionized region abruptly
grows, mostly in terms of its third Shapefinder – ‘length’. Therefore,
the filamentary largest ionized region at percolation exhibits a char-
acteristic cross-section, its value depends on the reionization source
model. The only exception is the clumping model where the extreme
inhomogeneity in the IGM results in a boost in the planarity of the
ionized regions (including the largest one) that might not be entire
physical. However, one can discern such non-uniform recombination
models by assessing the Shapefinders in the ionized hydrogen.
•The genus of largest ionized region for all scenarios increases as
the reionization progresses which confirms the multi-connectedness
we observe at the later stage of reionization. Except of the unusual
behaviour for Clumping scenario, rest of the reionization scenarios
show a near linear relationship of genus with length.
We estimated the uncertainties (shown via error bars in different
plots) in our LCS analysis by dividing the simulation volume into
eight smaller sub-volumes and estimating the variance of various
quantities from these sub-volumes. However, due to the computa-
tional cost of running a large N-body simulation with fine mass
resolution, we are limited to a small number of realizations which
lead to a crude approximation of the estimated uncertainties. We have
discussed about this in details in Appendix A.
In follow up, we plan to study the impact of cosmic variance,
system noise and residual foregrounds in the 21-cm maps on our
inferences drawn from LCS and Shapefinders of ionized regions and
we would also like to explore whether these geometrical tools can
be used to put constraints on reionization history via the future SKA
observations of the EoR.
6 ACKNOWLEDGEMENTS
SB thanks Varun Sahni, Santanu Das for their contributions in de-
veloping SURFGEN2 in its initial phase. RM is supported by the
MNRAS 000,1–16 (2022)
12 Pathak et al.
Wenner-Gren Postdoctoral Fellowship. The entire analysis of the
simulated 21-cm maps presented here were done using the comput-
ing facilities available with the Cosmology with Statistical Inference
(CSI) research group at IIT Indore.
7 DATA AVAILABILITY
The data and codes used for the analysis in this work can be made
available on reasonable request to the authors.
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APPENDIX A: ESTIMATING UNCERTAINTIES IN THE
LCS USING SUB-VOLUMES OF THE SIMULATED SIGNAL
In section 4.1 we distinguish the inside-out reionization scenarios
from the outside-in models using cluster statistics and by compar-
ing the percolation transitions in the different reionization scenarios.
However, one needs to account for the uncertainties in the respec-
tive quantities, especially on the largest cluster statistics (LCS) vs
¯𝑥Hicurves. A proper error estimation requires analysing many re-
alizations in all the scenarios, however simulating large number of
high resolution realizations would be extremely computationally ex-
pensive and beyond the scope of the present paper. In this section,
we crudely estimate the errors on the important quantities by break-
ing our large (714 Mpc)3simulation volume into 8 equal smaller
(357 Mpc)3boxes preserving the resolution. Since the dimensions
of the smaller boxes are sufficiently large to satisfy the cosmological
principles of isotropy and homogeneity, one can effectively consider
the smaller volumes as independent realizations. Therefore, all the
cosmological phenomena including the percolation process can be
compared between the larger and the smaller simulation volumes.
Strictly speaking, here we do not follow the periodic boundary con-
dition (PBC) properly. However, since the percolation transition hap-
pens abruptly in the whole volume, the effect of PBC can be ignored
for this exercise.
We study the smaller 8 boxes separately and, as expected, we find
that the quantities like filling factor, LCS etc are statistically consis-
tent between the larger (714 Mpc)3box and the smaller boxes. This
allows us to estimate the uncertainties in various quantities from
their standard deviations. In figure A1, we show the ionized filling
factor against the neutral fraction (¯𝑥Hi) for the different scenarios but
this time with the uncertainties calculated using the smaller boxes.
The solid and dashed curves represent the results coming from our
primary large simulation volume that is in extremely good agree-
ment with the results of the 8 smaller boxes analysed separately. The
shaded regions with respective colours represent the uncertainty es-
timated from the standard deviation of the individual results for these
smaller boxes. Note that the uncertainty is larger for the inside-out
reionization scenarios (maximum for the PL (𝑛=3) model) where
typically larger ionized bubbles are produced. In contrast, for the
outside-in models we get negligible error on the filling factor (min-
imum for the UIB dominated model) owing to the fact that smaller
ionized bubbles formed in these scenarios leading to less variation
among different realizations.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.2
0.4
0.6
0.8
Filling Factor
Fiducial
Clumping
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
7.2
7.3
7.4
7.5
7.7
7.8
7.9
8.2
8.38.5
8.89.2
9.6
10.3
Figure A1. Same as figure 3but with the uncertainties (shown by the shaded
regions) estimated using the standard deviation of the filling factor obtained
from the smaller boxes analysed separately.
For a fair comparison, we show the evolution of the number of
ionized regions per unit volume (𝑁C/𝑉) in figure A2; the left and
right panels show (𝑁C/𝑉) along with the uncertainty as functions of
¯𝑥Hiand filling factor respectively. Again we note that the results from
our primary large simulation volume is consistent with that from the
smaller boxes. The uncertainty is larger for inside-out models.
Figure A3 shows the LCS as functions of ¯𝑥Hiand filling factor in
the left and right panels respectively together with the uncertainties
estimated using the standard deviation of the results from the smaller
boxes. The solid/dashed curves represent the results for the primary
box, and hence same as in figure 7. The uncertainty, shown by the
shaded regions with respective colours, is found to be larger before
and near the percolation transitions especially for the inside-out sce-
narios. Particularly for PL (𝑛=3) model, we get largest error in
the pre-percolation phase since this model produces largest ionized
bubbles among all the scenarios tested in this article, hence exhibits
more variation in LCS with different realizations.
In summary, we crudely estimate the uncertainty in various quan-
tities studied in the main text by separately analysing the simulation
volume divided into 8 equal pieces. As we expected, the results from
the primary simulation volume and the smaller ones are statistically
consistent with each other. The uncertainties in the quantities like
filling factor, number of ionized region per unit volume and most
importantly in LCS is larger for inside-out scenarios where larger
ionized regions typically forms as compared to the outside-in scenar-
ios. Nevertheless, the estimated errors in LCS for all the scenarios are
small enough so that one can easily distinguish the source models;
especially between the inside-out and outside-in scenarios.
APPENDIX B: BUBBLE SIZE DISTRIBUTION
In this appendix, we have used Bubble Size Distribution (BSD)
method to check the consistency of our Largest Cluster Statistics
(LCS) which have been discussed in section 4.1. This particular
method extensively gives information about the distribution of num-
ber of ionized bubbles of given size or volume in our simulated data.
This BSD can be achieved via various routes like the Mean Free
MNRAS 000,1–16 (2022)
14 Pathak et al.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8
Filling Factor
Fiducial
Clumping
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
Figure A2. The number of ionized regions per unit volume for different reionization models has been shown against the neutral fraction ( ¯𝑥Hi) and the ionized
filling factor (FF) in the left and right panel respectively. Note in the right panel that different models go up to different filling factor by the end of the redshift
range we consider in this work. The shaded regions represent the standard deviations of the results for the smaller boxes studied separately.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Largest Cluster Statistics
0.0 0.2 0.4 0.6 0.8
Filling Factor
____ : Inside-Out
----- : Outside-In
Fiducial
Clumping
PL (n=3)
UIB Dom
SXR Dom
UV+SXR+UIB
Figure A3. Same as figure 7but with error on LCS, shown by the shaded regions, estimated using the standard deviation of the results from the smaller boxes.
Path (MFP) method in which one can basically choose any random
location and cast the rays in all possible direction which stop when
meet the stopping criteria and after repeating this process several
times, one can plot a histogram of different ray values obtained. Sec-
ond, one can use the Spherical Averaged (SPA) method in which the
largest possible spheres of different radii around each ionized regions
are constructed on the basis of given threshold and then a histogram
is plotted corresponding to the different radii of spheres. We have
adopted the third method in our case which is the Friends of Friend
(FoF) method in which one can plot the histogram of the volumes
of different ionized region formed when the different data points sat-
isfying the condition of given linking length group together to form
a cluster. One can find more information about this BSD method in
the literature (Giri et al. 2018;Friedrich et al. 2011).
Once we apply this FoF technique on our simulated data, we
have a number of ionized bubbles whose volumes are known to us.
Later, these volumes can be plotted in the form of histogram to see
the distribution of ionized bubbles for specific volume range. The
quantity which we expect to be plotted against V is :
𝑝(𝑉)=𝑉𝑑𝑛
𝑑𝑉 (B1)
But considering the largest ionized region’s contribution into effect
we can modify the equation B1 as 𝑉/𝑉𝑖 𝑜𝑛𝑖𝑧 𝑒𝑑 𝑝(𝑉)which can be
MNRAS 000,1–16 (2022)
Probing EoR with 21-cm LCS 15
Figure B1. From top to bottom, the BSD at different neutral fraction (in each
row where from left to right we have different stages of reionization as : at
percolation, just after percolation and at the end of reionization respectively)
is shown for all six reionization scenarios which are fiducial, Clumping, UIB
Dom., SXR Dom., UV+SXR+UIB, PL (n=3).
further written as 𝑉2𝑑 𝑝
𝑑𝑉 . This is the quantity which we have plotted
against the volume (V) in our BSD plot for all reionization scenarios
as shown in Figure B1.
In the left most panel, BSD at the point of percolation is shown for
all reionization scenarios, the middle panel shows the BSD just after
percolation and the right most panel shows the BSD at the end of
reionization. It can be observed from the plots that the BSD mainly at
the later stage of reionization i.e for lower neutral fractions shows the
bimodal distribution . That means large collection of small ionized
bubbles are basically separated from the single large ionized bubble.
This further strengthen the fact that the largest ionized bubble upto
percolation cannot be distinguished from the other ionized bubbles,
so we can essentially observe almost same size of ionized regions
including our largest ionized region. But after percolation, this largest
ionized region becomes the dominant contributor in our ionized
volume which is true for all scenarios. This behaviour of Bubble Size
Distribution is consistent with our findings based on Largest Cluster
Statistics presented in section 4.1. Thus, it shows the robustness of
our LCS statistics.
APPENDIX C: SHAPES OF IONIZED REGIONS IN THE
CLUMPING MODEL
We have seen in section 4.2 that clumping model in general gives rise
to large genus values in the ionized regions because of the fact that the
non-uniform recombination produces large number of small neutral
pockets those remain neutral till pretty late reionization stage. Since
they form many negatively curved tunnels, the IMC often becomes
very low or even negative sometimes. Interestingly this translates
into higher values of planarity which may not be entirely physical.
However, one can use this behaviour to easily distinguish this type
of model with non-uniform IGM from the rests. We demonstrate this
in this appendix in more details.
The evolution of the shape of the largest ionized region (LIR) in
the clumping model has been compared with that for the rest of the
reionization models in the left panels of figure C1 (the results for
the rest 5 scenarios have been already shown in the main text, in
figures 9and 10). The top panel shows the morphological evolution
of the LIR in terms of its planarity (plotted by thin curves along left
y-axis) and filamentarity (thick curves plotted along right y-axis).
Like in the other scenarios, the filamentarity of the LIR in clumping
model (thick blue curve) also increases with reionization at the onset
of percolation. But, the planarity of the LIR is significantly higher
in the clumping model (thin blue curve) because of large genus
values, i.e. due to the presence of large number of neutral tunnels
that bring down the IMC. Therefore, the morphology of the LIR in
clumping model is not strikingly filamentary as opposed to the rest
of models. The middle and bottom panels of figure C1 compare the
evolution of the cross-section and the length of the LIR in clumping
model with that in the other scenarios. In contrast to other scenarios,
the cross-section of the LIR in the clumping model does not remain
stable during the percolation transition due to the enhanced planarity.
However, the abrupt increase in the length (third Shapefinder) is at
par with other scenarios.
The right panel of figure C1 compares the morphology distribution
of the individual ionized regions in clumping model with that of the
rest of models (these results for the rest 5 scenarios have been already
shown in the main text, in figure 11). The clumping model behaves
entirely differently from the rest of the source models in all three
panels. Because of the high recombination rate in this model, there
exist many pockets of neutral hydrogen which gives rise to a very
high genus values for the ionized regions, as vividly evident from the
top panel. The middle and the bottom panel illustrate the difference
in the filamentarity and planarity distribution in the clumping model
from the rests. We find overall decreased filamentarity and increased
planarity in the ionized regions for the clumping model as compared
to the other scenarios. Moreover, we notice large dispersion in the fil-
amentarity and planarity distributions that set clumping model apart
from the rest of the scenarios considered in this work. Therefore, in
view of the enhanced planarity of the ionized regions in the clumping
model, due to the extreme inhomogeneity in the modelling of IGM,
one can easily distinguish this type of reionization source mode using
the Shapefinders analyses.
MNRAS 000,1–16 (2022)
16 Pathak et al.
0
200
400
600
800
1000
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure C1. The evolution of the shape of the largest ionized region in the clumping model has been compared with that in the rest of the scenarios in the left
panels in terms of planarity, filamentarity (top), cross-section (middle) and length (bottom). Note that in the top left panel we plot planarity (thin curves) along
the left y-axis whereas filamentarity (thick curves) is plotted along the right y-axis. The right panels illustrate how the shape distributions of the ionized regions
in the clumping model are different from the other scenarios considered in this article.
This paper has been typeset from a T
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X/L
A
T
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X file prepared by the author.
MNRAS 000,1–16 (2022)
