Predicting the 21cm signal from the epoch of reionization and cosmic dawn is a complex and challenging task. Various simplifying assumptions have been applied over the last decades to make the modeling more affordable. In this paper, we investigate the validity of several such assumptions, using a simulation suite consisting of three different astrophysical source models that agree with the current constraints on the reionization history and the UV luminosity function. We first show that the common assumption of a saturated spin temperature may lead to significant errors in the 21cm clustering signal over the full reionization period. The same is true for the assumption of a neutral universe during the cosmic dawn which may lead to significant deviation from the correct signal during the heating and the Lyman-$α$ coupling period. Another popular simplifying assumption consists of predicting the global differential brightness temperature ($dT_b$) based on the average quantities of the reionization fraction, gas temperature, and Lyman-$α$ coupling. We show that such an approach leads to a 10 percent deeper absorption signal compared to the results obtained by averaging the final $dT_b$-map. Finally, we investigate the simplifying method of breaking the 21cm clustering signal into different auto and cross components that are then solved assuming linearity. We show that even though the individual fields have a variance well below unity, they often cannot be treated perturbatively as the perturbations are strongly non-Gaussian. As a consequence, predictions based on the perturbative solution of individual auto and cross power spectra may lead to strongly biased results, even if higher-order terms are taken into account.
The Cosmic Dawn and Reionization epochs remain a fundamental but challenging frontier of astrophysics and cosmology. We advocate a large-scale, multi-tracer approach to develop a comprehensive understanding of the physics that led to the formation and evolution of the first stars and galaxies. We highlight the line intensity mapping technique to trace the multi-phase reionization topology on large scales, and measure reionization history in detail. Besides 21cm, we advocate for Lya tomography mapping during the epoch of Wouthuysen-Field coupling as an additional probe of the cosmic dawn era.
We provide an overview of 21cm tomography of the Cosmic Dawn and Epoch of Reionization as possible with SKA-Low. We show why tomography is essential for studying CD/EoR and present the scales which can be imaged at different frequencies for the different phases of SKA- Low. Next we discuss the different ways in which tomographic data can be analyzed. We end with an overview of science questions which can only be answered by tomography, ranging from the characterization of individual objects to understanding the global processes shaping the Universe during the CD/EoR
Studying the cosmic dawn and the epoch of reionization through the redshifted 21 cm line are among the major science goals of the SKA1. Their significance lies in the fact that they are closely related to the very first stars in the universe. Interpreting the upcoming data would require detailed modelling of the relevant physical processes. In this article, we focus on the theoretical models of reionization that have been worked out by various groups working in India with the upcoming SKA in mind. These models include purely analytical and semi-numerical calculations as well as fully numerical…
Conventional wisdom was that thermal relics from the epoch of reionization (EOR) would vanish swiftly. Recently, however, it was shown that these relics can survive to lower redshifts ($z \sim 2$) than previously thought, due to gas at mean density being heated to $T \sim 3 \times 10^4$ K by reionization, which is inhomogeneous, and shocks. Given the high sensitivities of upcoming Ly$α$ forest surveys, this effect will be a novel broadband systematic for cosmological application. From the astrophysical point of view, however, the imprint of inhomogeneous reionization can shed light on the EOR …
I review recent progress and challenges in studies of the earliest galaxies, seen when the Universe was less than 1 billion years old. Can they be used as reliable tracers of the physics of cosmic reionization thereby complementing other, more direct, probes of the evolving neutrality of the intergalactic medium? Were star-forming galaxies the primary agent in the reionization process and what are the future prospects for identifying the earliest systems devoid of chemical enrichment? Ambitious future facilities are under construction for exploring galaxies and the intergalactic medium in the …
The process of heating and reionization of the Universe at high redshift links small scale structures/galaxy formation and large scale inter-galactic medium properties. Even if the first one is difficult to observe, an observation window is opening on the second one, with the promising development of current and future radio telescopes. They will permit to observe the 21cm brightness temperature global signal and fluctuations. The need of large scale simulations is therefore strong to understand the properties of the IGM that will be observed. But at the same time the urge to resolve the struc…
The dark age of the universe, when no luminous object had existed, ended with the birth of the first stars, galaxies, and blackholes. This epoch is called cosmic dawn. Cosmic reionization is the major transition of the intergalactic medium (IGM) in the universe driven by ionizing photons emitted from luminous objects. Although the epoch through the dark age to reionization is a milestone in the universe, our knowledge of this epoch has not been sufficient yet. Cosmic 21cm signal, which is emitted from neutral hydrogen, is expected to open a new window for this epoch. In this review paper, we f…
The Square Kilometre Array (SKA) will offer an unprecedented view onto the early Universe, using interferometric observations of the redshifted 21cm line. The 21cm line probes the thermal and ionization state of the cosmic gas, which is governed by the birth and evolution of the first structures in our Universe. Here we show how the evolution of the 21cm signal will allow us to study when the first generations of galaxies appeared, what were their properties, and what was the structure of the intergalactic medium. We highlight qualitative trends which will offer robust insights into the early …
Cosmic baryon evolution during the Cosmic Dawn and Reionization results in redshifted 21-cm spectral distortions in the cosmic microwave background (CMB). These encode information about the nature and timing of first sources over redshifts 30-6 and appear at meter wavelengths as a tiny CMB distortion along with the Galactic and extragalactic radio sky, which is orders of magnitude brighter. Therefore, detection requires precise methods to model foregrounds. We present a method of foreground fitting using maximally smooth (MS) functions. We demonstrate the usefulness of MS functions over tradit…
The 21cm neutral hydrogen line is likely to be a key probe for studying the epoch of reionization and comic dawn in the forthcoming decades. This prospect stimulates the development of the theoretical basis for simulating the power spectrum of this line. Because of the beam size of the upcoming radio telescopes at high redshifts, most of the theoretical models are focused on the inhomogeneities on scales above few comoving megaparsecs. Therefore, smaller scales are often neglected and modeled with approximated sub-grid models. In this study we explore whether the perturbations on small scales …
The cosmic microwave background (CMB) serves as a backlight to large-scale structure during the epoch of reionization, where Thomson scattering gives rise to temperature anisotropies on small angular scales from the kinetic Sunyaev Zel'dovich (kSZ) effect. In this paper, we demonstrate that the technique of kSZ tomography (velocity reconstruction), based on cross correlations between CMB temperature and 21cm surveys, can significantly improve constraints on models of inhomogeneous reionization and provide information about large-scale modes that are poorly characterized by 21cm measurements th…
Redshifted 21cm signal is a promising tool to investigate the state of intergalactic medium (IGM) in the Cosmic Dawn (CD) and Epoch of Reionization(EoR). In our previous work (Shimabukuro et al 2015), we studied the variance and skewness to give a clear interpretation of 21cm power spectrum and found that skewness is a good indicator of the epoch when X-ray heating becomes effective. Thus, the non-Gaussian feature of the spatial distribution of the 21cm signal is expected to be useful to investigate the astrophysical effects in the CD and EoR. In this paper, in order to investigate such a non-…
The 21-cm power spectrum (PS) has been shown to be a powerful discriminant of reionization and cosmic dawn astrophysical parameters. However, the 21-cm tomographic signal is highly non-Gaussian. Therefore there is additional information which is wasted if only the PS is used for parameter recovery. Here we showcase astrophysical parameter recovery directly from 21-cm images, using deep learning with convolutional neural networks (CNN). Using a database of 2D images taken from 10,000 21-cm lightcones (each generated from different cosmological initial conditions), we show that a CNN is able to …
Improved measurement of the Cosmic Microwave Background polarization from Planck allows a detailed study of reionization beyond the average optical depth. The lower value of the optical depth disfavours an early onset and an early completion of reionization in favour of a redsfhit range where different astrophysical probes provide sensible information on the sources of reionization and the status of the intergalactic medium. In this work we extend our previous study in which we constrained reionization by combining three different probes - CMB, UV luminosity density and neutral hydrogen fracti…
Detection of individual luminous sources during the reionization epoch and cosmic dawn through their signatures in the HI 21-cm signal is one of the direct approaches to probe the epoch. Here, we summarize our previous works on this and present preliminary results on the prospects of detecting such sources using the SKA1-low experiment. We first discuss the expected HI 21-cm signal around luminous sources at different stages of reionization and cosmic dawn. We then introduce two visibility based estimators for detecting such signal: one based on the matched filtering technique and the other re…
Recent determinations of the mean free path of ionising photons (mfp) in the intergalactic medium (IGM) at $\rm z=6$ are lower than many theoretical predictions. To gain insight into this issue, we investigate the evolution of the mfp in our new massive fully coupled radiation hydrodynamics cosmological simulation of reionization: Cosmic Dawn III (CoDaIII). CoDaIII's scale ($\rm 94^3 \, cMpc^3$) and resolution ($\rm 8192^3$ grid) make it particularly suitable to study the evolution of the IGM during Reionization. The simulation was performed with RAMSES-CUDATON on Summit, and used 131072 proce…
The 21-cm signal of neutral hydrogen is a sensitive probe of the Epoch of Reionization (EoR) and Cosmic Dawn. Currently operating radio telescopes have ushered in a data-driven era of 21-cm cosmology, providing the first constraints on the astrophysical properties of sources that drive this signal. However, extracting astrophysical information from the data is highly non-trivial and requires the rapid generation of theoretical templates over a wide range of astrophysical parameters. To this end emulators are often employed, with previous efforts focused on predicting the power spectrum. In thi…
We highlight recent progress in the sophistication and diversification of cosmic dawn and reionization simulations. The application of these modeling tools to current observations has allowed us narrow down the timing of reionization, which we now know to within dz ~ 1 for the bulk of reionization. The strongest constraints come from the optical depth to the CMB measured with the {\it Planck} satellite and the first detection of ongoing reionization from the spectra of the z=7.1 QSOs ULASJ1120+0641. However, we still know virtually nothing about the astrophysical sources during the first billi…
Cosmic dawn and the Epoch of Reionization (EoR) are among the least explored observational eras in cosmology: a time at which the first galaxies and supermassive black holes formed and reionized the cold, neutral Universe of the post-recombination era. With current instruments, only a handful of the brightest galaxies and quasars from that time are detectable as individual objects, due to their extreme distances. Fortunately, a multitude of multi-wavelength intensity mapping measurements, ranging from the redshifted 21 cm background in the radio to the unresolved X-ray background, contain a pl…
Cosmic reionization is known to be a major phase transition of the gas in the Universe. Since astronomical objects formed in the early Universe, such as the first stars, galaxies and black holes, are expected to have caused cosmic reionization, the formation history and properties of such objects are closely related to the reionization process. In spite of the importance of exploring reionization, our understandings regarding reionization is not sufficient yet. Square Kilometre Array (SKA) is a next-generation large telescope that will be operated in the next decade. Although several programs …
Imaging the 21-cm Signal from the Cosmic Dawn & Epoch of Reionization and the Connection with the Global Signal
Technical University of Munich（慕尼黑工业大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；ETH Zurich（苏黎世联邦理工学院）；Indian Institute of Technology Madras（马德拉斯理工学院）；University of Cambridge（剑桥大学）；Indian Institute of Technology Indore（印多尔理工学院）；Jadavpur University（贾达普大学）；The Open University of Israel（以色列开放大学）；University of Groningen（格罗宁根大学）；Université PSL（巴黎文理研究大学）；Sorbonne Université（索邦大学）；Université Paris Cité（巴黎西岱大学）；Shanghai Astronomical Observatory, Chinese Academy of Sciences（中国科学院上海天文台）
Line Intensity Mapping Prediction from the Cosmic Dawn (CoDa) III Simulation for H$α$ from Galaxies and the Intergalactic Medium during the Epoch of Reionization
The Square Kilometre Array (SKA) will have a low frequency component (SKA-low) which has as one of its main science goals the study of the redshifted 21cm line from the earliest phases of star and galaxy formation in the Universe. This 21cm signal provides a new and unique window on both the formation of the first stars and accreting black holes and the later period of substantial ionization of the intergalactic medium. The signal will teach us fundamental new things about the earliest phases of structure formation, cosmology and even has the potential to lead to the discovery of new physical …
The detection of Lyman-$α$ emitting galaxies (LAEs) puts severe constraints on the reionization history. In this Paper we derive the properties of very high-$z$ LAEs predicted in the only two reionization scenarios shown in a previous Paper to be consistent with current data on 15 independent evolving global (or averaged) cosmic properties regarding luminous objects and the intergalactic medium and the optical depth to electron scattering of ionized hydrogen to CMB photons: one with a monotonic behavior, which is completed by $z=6$, as commonly considered, and another one with a non-monotonic …
The line of sight direction in the redshifted 21-cm signal coming from the cosmic dawn and the epoch of reionization is quite unique in many ways compared to any other cosmological signal. Different unique effects, such as the evolution history of the signal, non-linear peculiar velocities of the matter etc will imprint their signature along the line of sight axis of the observed signal. One of the major goals of the future SKA-LOW radio interferometer is to observe the cosmic dawn and the epoch of reionization through this 21-cm signal. It is thus important to understand how these various eff…
Cosmic Dawn Intensity Mapper is a "Probe Class" mission concept for reionization studies of the universe. It will be capable of spectroscopic imaging observations between 0.7 to 6-7 microns in the near-Infrared. The primary observational objective is pioneering observations of spectral emission lines of interest throughout the cosmic history, but especially from the first generation of distant, faint galaxies when the universe was less than 800 million years old. With spectro-imaging capabilities, using a set of linear variable filters (LVFs), CDIM will produce a three-dimensional tomographic …
Direct detection of regions of ionized hydrogen (HII) has been suggested as a promising probe of cosmic reionization. Observing the redshifted 21-cm signal of hydrogen from the epoch of reionization (EoR) is a key scientific driver behind new-generation, low-frequency radio interferometers. We investigate the feasibility of combining low-frequency observations with the Square Kilometre Array and near infra-red survey data of the Wide-Field Infrared Survey Telescope to detect cosmic reionization by imaging HII bubbles surrounding massive galaxies during the cosmic dawn. While individual bubbles…
Cosmic Dawn II (CoDa II) is a new, fully-coupled radiation-hydrodynamics simulation of cosmic reionization and galaxy formation and their mutual impact, to redshift $z < 6$. With $4096^3$ particles and cells in a 94 Mpc box, it is large enough to model global reionization and its feedback on galaxy formation while resolving all haloes above $10^8$ M$_{\odot}$. Using the same hybrid CPU-GPU code RAMSES-CUDATON as CoDa I in Ocvirk et al. (2016), CoDa II modified and re-calibrated the subgrid star-formation algorithm, making reionization end earlier, at $z \gtrsim 6$, thereby better matching the …
Cosmic reionization by starlight from early galaxies affected their evolution, thereby impacting reionization, itself. Star formation suppression, for example, may explain the observed underabundance of Local Group dwarfs relative to N-body predictions for Cold Dark Matter. Reionization modelling requires simulating volumes large enough [~(100Mpc)^3] to sample reionization "patchiness", while resolving millions of galaxy sources above ~10^8 Msun , combining gravitational and gas dynamics with radiative transfer. Modelling the Local Group requires initial cosmological density fluctuations pre-s…
The effect of photon re-scattering due to cosmic reionization on 21-cm images and power spectra
School of Physics and Astronomy, Tel-Aviv University（特拉维夫大学物理与天文学院）；Institute of Astronomy, University of Cambridge（剑桥大学天体研究所）；Kavli Institute for Cosmology（卡弗里宇宙学研究所）
The 21 cm brightness temperature $δT_{\rm b}$ fluctuations from reionization promise to provide information on the physical processes during that epoch. We present a formalism for generating the $δT_{\rm b}$ distribution using dark matter simulations and an one-dimensional radiative transfer code. Our analysis is able to account for the spin temperature $T_{\rm S}$ fluctuations arising from inhomogeneous X-ray heating and Ly$α$ coupling during cosmic dawn. The $δT_{\rm b}$ power spectrum amplitude at large scales ($k \sim 0.1$ Mpc$^{-1}$) is maximum when $\sim 10\%$ of the gas (by volume) is h…
Current and upcoming radio interferometric experiments are aiming to make a statistical characterization of the high-redshift 21cm fluctuation signal spanning the hydrogen reionization and X-ray heating epochs of the universe. However, connecting 21cm statistics to underlying physical parameters is complicated by the theoretical challenge of modeling the relevant physics at computational speeds quick enough to enable exploration of the high dimensional and weakly constrained parameter space. In this work, we use machine learning algorithms to build a fast emulator that mimics expensive simulat…
Galaxy Proximate Damped Lyman-Alpha Systems and HI Reionization Topology in TECHNICOLOR DAWN
New Mexico State University（新墨西哥州立大学）；Cosmic Dawn Center (DAWN)（宇宙黎明中心（DAWN））；Niels Bohr Institute, University of Copenhagen / DTU-Space Technical University of Denmark（哥本哈根大学尼尔斯·玻尔研究所/丹麦技术大学空间研究所）
The nature of dark matter remains unknown, but upcoming measurements probing the high-redshift Universe may provide invaluable insight. In the presence of dark matter-baryon scattering, the suppression in the matter power spectrum and the colder mean gas temperature are expected to modify the evolution of cosmic dawn and reionization. However, the contributions from such interactions to the baryon and dark matter temperature perturbations have been neglected thus far. In this work, we derive these contributions, evolve the cosmological perturbations until the end of the dark ages and show that…
We present a detailed analysis of an astrophysical mechanism that generates cosmological magnetic fields during the Epoch of Reionization. It is based on the photoionization of the Intergalactic Medium by the first sources formed in the Universe. First the induction equation is derived, then the characteristic length and time scales of the mechanism are identified, and finally numerical applications are carried out for first stars, primordial galaxies and distant powerful quasars. In these simple examples, the strength of the generated magnetic fields varies between the order of $10^{-23}$ G o…
Machine Learning and the SKA for Cosmic Dawn and the Epoch of Reionization
Indian Institute of Technology Indore（印度理工学院印多尔分校）；Stockholm University（斯德哥尔摩大学）；University of Groningen（格罗宁根大学）；University of Michigan（密歇根大学）；University of Nevada Las Vegas（内华达大学拉斯维加斯分校）；The Johns Hopkins University（约翰斯·霍普金斯大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；ETH Zurich（苏黎世联邦理工学院）；University of California, Berkeley（加州大学伯克利分校）；Tsinghua University（清华大学）；Yunnan University（云南大学）；Nagoya University（名古屋大学）；Zhejiang Laboratory（浙江实验室）；McGill University（麦吉尔大学）
The formation of the first galaxies during cosmic dawn and reionization (at redshifts $z=5-30$), triggered the last major phase transition of our universe, as hydrogen evolved from cold and neutral to hot and ionized. The 21-cm line of neutral hydrogen will soon allow us to map these cosmic milestones and study the galaxies that drove them. To aid in interpreting these observations, we upgrade the publicly available code {\tt 21cmFAST}. We introduce a new, flexible parametrization of the additive feedback from: an inhomogeneous, $H_2$-dissociating (Lyman-Werner; LW) background; and dark matter…
The properties of the first galaxies, expected to drive the Cosmic Dawn (CD) and the Epoch of Reionization (EoR), are encoded in the 3D structure of the cosmic 21-cm signal. Parameter inference from upcoming 21-cm observations promises to revolutionize our understanding of these unseen galaxies. However, prior inference was done using models with several simplifying assumptions. Here we introduce a flexible, physically-motivated parametrization for high-$z$ galaxy properties, implementing it in the public code 21cmFAST. In particular, we allow their star formation rates and ionizing escape fra…
Concerted effort is currently ongoing to open up the Epoch of Reionization (EoR) ($z\sim$15-6) for studies with IR and radio telescopes. Whereas IR detections have been made of sources (Lyman-$α$ emitters, quasars and drop-outs) in this redshift regime in relatively small fields of view, no direct detection of neutral hydrogen, via the redshifted 21-cm line, has yet been established. Such a direct detection is expected in the coming years, with ongoing surveys, and could open up the entire universe from $z\sim$6-200 for astrophysical and cosmological studies, opening not only the EoR, but also…
An overview on the current status of the census of the early universe population is given. Observational surveys of high redshift objects provide direct opportunities to study the early epoch of the Universe. The target population included are Lyman Alpha Emitters (LAE), Lyman Break Galaxies (LBG), gravitationally lensed galaxies, quasars and gamma-ray bursts (GRB). The basic properties of these objects and the methods used to study them are reviewed. The present paper highlights the fact that the Subaru Telescope group made significant contributions in this field of science to elucidate the e…
Square Kilometer Array Synergies for the Epoch of Reionization and Cosmic Dawn
National Centre for Radio Astrophysics, Tata Institute of Fundamental Research（塔塔基础研究院射电天体物理国家中心）；Jadavpur University（贾达普大学）；University of Toronto（多伦多大学）；University of Cambridge（剑桥大学）；Kavli Institute for Cosmology（卡弗里宇宙学研究所）；Stockholm University（斯德哥尔摩大学）；University of Groningen（格罗宁根大学）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS, Institut d'Astrophysique Spatiale（法国国家科学研究中心空间天体物理研究所）；University of Vienna（维也纳大学）；Indian Institute of Technology Indore（印度理工学院印多尔分校）；University of Catania（卡塔尼亚大学）；The University of Tokyo（东京大学）；The Open University of Israel（以色列开放大学）；Australian National University（澳大利亚国立大学）；National Astronomical Observatory of Japan（日本国立天文台）；Uppsala University（乌普萨拉大学）
The neutral hydrogen (HI) and its 21 cm line are promising probes to the reionization process of the intergalactic medium (IGM). To use this probe effectively, it is imperative to have a good understanding on how the neutral hydrogen traces the underlying matter distribution. Here we study this problem using semi-numerical modeling by combining the HI in the IGM and the HI from halos during the epoch of reionization (EoR), and investigate the evolution and the scale-dependence of the neutral fraction bias as well as the 21 cm line bias. We find that the neutral fraction bias on large scales is…
Magnetic fields are ubiquitous in the Universe. They seem to be present at virtually all scales and all epochs. Yet, whether the fields on cosmological scales are of astrophysical or cosmological origin remains an open major problem. Here we focus on an astrophysical mechanism based on the photoionization of the intergalactic medium during the Epoch of Reionization. Building upon previous studies that depicted the physical mechanism around isolated sources of ionization, we present here an analytic model to estimate the level at which this mechanism contributed to the magnetization of the whol…
The global 21 cm signal from Cosmic Dawn (CD) and the Epoch of Reionization (EoR), at redshifts $z \sim 6-30$, probes the nature of first sources of radiation as well as physics of the Inter-Galactic Medium (IGM). Given that the signal is predicted to be extremely weak, of wide fractional bandwidth, and lies in a frequency range that is dominated by Galactic and Extragalactic foregrounds as well as Radio Frequency Interference, detection of the signal is a daunting task. Critical to the experiment is the manner in which the sky signal is represented through the instrument. It is of utmost impo…
In the most distant reaches of the Universe, the 21-cm hyperfine transition in neutral hydrogen provides one of the only available tracers of large-scale structure. A number of instruments have been working and planned to measure the 21-cm line signals, and in particular, Experiment to Detect the Global EoR Signature (EDGES) recently has reported the first detection of an absorption signal, which corresponds to the 21-cm line global signal at the epoch of reionization (EoR). The future large radio telescope, Square Kilometre Array (SKA) will be able to deliver the high-precision measurement of…
The shape of the faint-end of the high-z galaxy luminosity function (LF) informs early star formation and reionization physics during the Cosmic Dawn and Epoch of Reionization. Until recently, based on the strong gravitational lensing cluster deep surveys, the Hubble Frontier Fields (HFF) has found a potential turnover in the ultraviolet (UV) LF at z$\sim$6. In this paper, we analyze the contribution of extremely faint galaxies with the magnitude larger than the turnover magnitude in LF to cosmic reionization. We apply the measurement from HFF to our suppressed star formation efficiency model,…
[abridged] We propose a tiered, UV--near-IR, cosmological broad- and medium-band imaging and grism survey that covers ~10 deg^2 in two epochs to m_AB=28, ~3 deg^2 in seven epochs to m_AB=28, and ~1 deg^2 in 20 epochs to m_AB=30 mag. Such a survey is an essential complement to JWST surveys (<~0.1 deg^2 to m_AB<~31 mag at lambda>1100nm and z>~8). We aim to: (1) understand in how galaxies formed from perturbations in the primordial density field by studying faint Lyα-emitting and Lyman-break galaxies at 5.5<~z<~8 and trace the metal-enrichment of the IGM; (2) measure the evolution of the faint en…
We employ the rank-2 {\em contour} Minkowski Tensor in two dimensions to probe length and time scales of ionized bubbles during the epoch of reionization. We demonstrate that the eigenvalues of this tensor provide excellent probes of the distribution of the sizes of ionized bubbles, and from it the characteristic bubble sizes, at different redshifts. We show that ionized bubbles are not circular, and hence not spherical in three dimensions, as is often assumed for simplified analytic arguments. We quantify their shape anisotropy by using the ratio of the two eigenvalues. The shape parameter pr…
