Bright point sources associated with extragalactic AGN and radio galaxies are an important foreground for low frequency radio experiments aimed at detecting the redshifted 21cm emission from neutral hydrogen during the epoch of reionization. The frequency dependence of the synthesized beam implies that the sidelobes of these sources will move across the field of view as a function of observing frequency, hence frustrating line-of-sight foreground subtraction techniques. We describe a method for subtracting these point sources from dirty maps produced by an instrument such as the MWA. This technique combines matched filters with an iterative centroiding scheme to locate and characterize point sources in the presence of a diffuse background. Simulations show that this technique can improve the dynamic range of EOR maps by 2-3 orders of magnitude.
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…
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…
Measuring the small primordial nonGaussianity (PNG) predicted by cosmic inflation theories may help diagnose them. The detectability of PNG by its imprint on the 21cm power spectrum from the epoch of reionization is reassessed here in terms of $f_{NL}$, the local nonlinearity parameter. We find that an optimum, multi-frequency observation by SKA can achieve $Δf_{NL} \sim 3$ (comparable to recent Planck CMB limits), while a cosmic-variance-limited array of this size like Omniscope can even detect $Δf_{NL} \sim 0.2$. This substantially revises the methods and results of previous work.
It has been several years since the first detection of Gunn-Peterson troughs in the z > 6 Ly-alpha forest and since the first measurement of the Thomson scattering optical depth through reionization from the cosmic microwave background (CMB). Present day CMB measurements provide a significant constraint on the mean redshift of reionization, and the Ly-alpha forest provides a lower bound on the redshift at which reionization ended. However, no observation has provided definitive information on the duration and morphology of this process. This article is intended as a short review on the most pr…
We present the algorithm and main results of our semi-numerical simulation, islandFAST, which is developed from the 21cmFAST and designed for the late stage of reionization. The islandFAST predicts the evolution and size distribution of the large scale under-dense neutral regions (neutral islands), and we find that the late Epoch of Reionization (EoR) proceeds very fast, showing a characteristic scale of the neutral islands at each redshift. Using islandFAST, we compare the impact of two types of absorption systems, i.e. the large scale under-dense neutral islands versus small scale over-dense…
Observations of the 21cm Epoch of Reionization (EoR) signal are dominated by Galactic and extragalactic foregrounds. The need for foreground removal has led to the development of two main techniques, often referred to as "foreground avoidance" and "foreground subtraction." Avoidance is associated with filtering foregrounds in Fourier space, while subtraction uses an explicit foreground model that is removed. Using 1088 hours of data from the 64-element PAPER array, we demonstrate that subtraction of a foreground model prior to delay-space foreground filtering results in a modest but measurable…
Primordial non-Gaussianity is a crucial test of inflationary cosmology. We consider the impact of non-Gaussianity on the ionization power spectrum from 21 cm emission during the epoch of reionization. We focus on the power spectrum on large scales at redshifts of 7 to 8 and explore the expected constraint on the local non-Gaussianity parameter f_NL for current and next-generation 21 cm experiments. We show that experiments such as SKA and MWA could measure f_NL values of order 10. This can be improved by an order of magnitude with a fast-Fourier transform telescope like Omniscope.
The first major star-forming galaxies and Active Galactic Nuclei will produce Balmer and higher order extended haloes during the Epoch of Reionization through the scattering of Lyman resonance line photons off the surrounding neutral intergalactic gas. The optical depth dependence of the scattering rates will produce a signal sensitive to both the density and velocity fluctuations of the gas, offering the possibility of probing the ionization region and flow field surrounding young star-forming galaxies. The requirements for detecting the haloes in the infra-red using a space-based telescope a…
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…
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
Observations of the 21-centimetre line of atomic hydrogen in the early Universe directly probe the history of the reionization of the gas between galaxies. The observations are challenging, though, because of the low expected signal strength (~10 mK), and contamination by strong (>100 K) foreground synchrotron emission in the Milky Way and extragalactic continuum sources2. If reionization happened rapidly, there should be a characteristic signature visible against the smooth foreground in an all-sky spectrum. Here we report an all-sky spectrum between 100 and 200 MHz, corresponding to the reds…
The Universe's dark ages end with the formation of the first generation of galaxies. These objects start emitting ultraviolet radiation that carves out ionized regions around them. After a sufficient number of ionizing sources have formed, the ionized fraction of the gas in the Universe rapidly increases until hydrogen becomes fully ionized. This period, during which the cosmic gas went from neutral to ionized, is known as the Universe's Epoch of Reionization . The Epoch of Reionization is related to many fundamental questions in cosmology, such as properties of the first galaxies, physics of …
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…
The motion of the solar system with respect to the cosmic rest frame modulates the monopole of the Epoch of Reionization 21-cm signal into a dipole. This dipole has a characteristic frequency dependence that is dominated by the frequency derivative of the monopole signal. We argue that although the signal is weaker by a factor of $\sim100$, there are significant benefits in measuring the dipole. Most importantly, the direction of the cosmic velocity vector is known exquisitely well from the cosmic microwave background and is not aligned with the galaxy velocity vector that modulates the foregr…
We review the observable consequences of the epoch of reionization (EoR) on the cosmic microwave background (CMB), and the resulting constraints on the EoR. We discuss how Thomson scattering with the free electrons produced during EoR equates to an optical depth for CMB photons. The optical depth measurements from the WMAP and Planck satellites, using large-scale CMB polarization power spectra, is one of the few current constraints on the timing of cosmic reionization. We also present forecasts for the precision with which the optical depth will be measured by future satellite missions. Second…
Simulations estimating the differential brightness temperature of the redshifted 21-cm from the epoch of reionization (EoR) often assume that the spin temperature is decoupled from the background CMB temperature and is much larger than it. Although a valid assumption towards the latter stages of the reionization process, it does not necessarily hold at the earlier epochs. Violation of this assumption will lead to fluctuations in differential brightness temperature that are neither driven by density fluctuations nor by HII regions. Therefore, it is vital to calculate the spin temperature self-c…
We investigate self-shielding of intergalactic hydrogen against ionizing radiation in radiative transfer simulations of cosmic reionization carefully calibrated with Lyman alpha forest data. While self-shielded regions manifest as Lyman-limit systems in the post-reionization Universe, here we focus on their evolution during reionization (redshifts z=6-10). At these redshifts, the spatial distribution of hydrogen-ionizing radiation is highly inhomogeneous, and some regions of the Universe are still neutral. After masking the neutral regions and ionizing sources in the simulation, we find that t…
Due to the large cross section of Ly$α$ photons with hydrogen, Lyman Alpha Emitters (LAEs) are sensitive to the presence of neutral hydrogen in the intergalactic medium (IGM) during the Epoch of Reionization (EoR): the period in the Universe's history where neutral hydrogen in the IGM is ionized. The correlation of the ionized regions in the IGM with respect to the underlying intrinsic LAEs has a pronounced effect on the number of observed LAEs and their apparent clustering. As a result, observations of LAEs during the EoR can be used as a probe of the EoR morphology. Here we build on previous…
The 21cm background from the epoch of reionization is a promising cosmological probe: line-of-sight velocity fluctuations distort redshift, so brightness fluctuations in Fourier space depend upon angle, which linear theory shows can separate cosmological from astrophysical information. Nonlinear fluctuations in ionization, density and velocity change this, however. The validity and accuracy of the separation scheme are tested here for the first time, by detailed reionization simulations. The scheme works reasonably well early in reionization (< 40% ionized), but not late (> 80% ionized).
We use the Aurora radiation-hydrodynamical simulations to study the mean free path (MFP) for hydrogen ionizing photons during the epoch of reionization. We directly measure the MFP by averaging the distance 1 Ry photons travel before reaching an optical depth of unity along random lines-of-sight. During reionization the free paths tend to end in neutral gas with densities near the cosmic mean, while after reionizaton the end points tend to be overdense but highly ionized. Despite the increasing importance of discrete, over-dense systems, the cumulative contribution of systems with $N_{\rm{HI}}…
Fractal dimensions can be used to characterize the clustering and lacunarities in density distributions. We use generalized fractal dimensions to study the neutral hydrogen distribution (HI) during the epoch of reionization. Using a semi-numeric model of ionized bubbles to generate the HI field, we calculate the fractal dimensions for length scales $\sim 10 h^{-1}$ cMpc. We find that the HI field displays significant multifractal behaviour and is not consistent with homogeneity at these scales when the mass averaged neutral fraction $\bar{x}_{\rm HI}^M \gtrsim 0.5$. This multifractal nature is…
We study the visibility of the Lyman Alpha (Lya) emission line during the Epoch of Reionization (EoR). Combining galactic outflow models with large-scale semi-numeric simulations of reionization, we quantify the probability distribution function (PDF) of the fraction of Lya photons transmitted through the intergalactic medium (IGM), T_IGM. Our study focusses on galaxies populating dark matter halos with masses of M_halo=1e10 M_sun at z=8.6, which is inspired by the recent reported discovery of a galaxy at z=8.6 with strong Lya line emission. For reasonable model assumptions, we find that winds…
Precise subtraction of foreground sources is crucial for detecting and estimating 21cm HI signals from the Epoch of Reionization (EoR). We quantify how imperfect point source subtraction due to limitations of the measurement dataset yields structured residual signal in the dataset. We use the Cramer-Rao lower bound, as a metric for quantifying the precision with which a parameter may be measured, to estimate the residual signal in a visibility dataset due to imperfect point source subtraction. We then propagate these residuals into two metrics of interest for 21cm EoR experiments - the angular…
The reionization epoch of singly ionized helium (He II) is believed to start at redshifts z~3.5--4 and be nearly complete by z~2.7. We explore the post-reionization epoch with far-ultraviolet spectra of the bright, high-redshift quasar HS1700+6416 taken by the Cosmic Origins Spectrograph (COS) on the Hubble Space Telescope, which show strong He II (λ303.78) absorption shortward of the QSO redshift, z_QSO=2.75. We discuss these data as they probe the post-reionization history of He II and the local ionization environment around the quasar and transverse to the line of sight. We compare previous…
The GMRT Epoch of Reionization (EoR) experiment is an ongoing effort to measure the power spectrum from neutral hydrogen at high redshift. We have previously reported an upper limit of (70 mK)^2 at wavenumbers of k=0.65 h/Mpc using a basic piecewise-linear foreground subtraction. In this paper we explore the use of a singular value decomposition to remove foregrounds with fewer assumptions about the foreground structure. Using this method we also quantify, for the first time, the signal loss due to the foreground filter and present new power spectra adjusted for this loss, providing a revised …
We introduce a versatile and spatially resolved morphological characterisation of binary fields, rooted in the opening transform of mathematical morphology. We subsequently apply it to the thresholded ionization field in simulations of cosmic reionization and study the morphology of ionized regions. We find that an ionized volume element typically resides in an ionized region with radius $\sim8\,h^{-1}\mathrm{cMpc}$ at the midpoint of reionization ($z\approx7.5$) and follow the bubble size distribution even beyond the overlap phase. We find that percolation of the fully ionized component sets …
A number of experiments are currently working towards a measurement of the 21 cm signal from the Epoch of Reionization. Whether or not these experiments deliver a detection of cosmological emission, their limited sensitivity will prevent them from providing detailed information about the astrophysics of reionization. In this work, we consider what types of measurements will be enabled by a next-generation of larger 21 cm EoR telescopes. To calculate the type of constraints that will be possible with such arrays, we use simple models for the instrument, foreground emission, and the reionization…
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 …
We apply the Correlated Component Analysis (CCA) method on simulated data of the Square Kilometre Array, with the aim of accurately cleaning the 21 cm reionization signal from diffuse foreground contamination. The CCA has been developed for the Cosmic Microwave Background, but the application of the Fourier-domain implementation of this method to the reionization signal is straightforward. The CCA is a parametric method to estimate the frequency behaviour of the foregrounds from the data by using second-order statistics. We test its performance on foreground simulations of increasing complexit…
Baryons both increase halo concentration through adiabatic contraction and expel mass through feedback processes. However, it is not well understood how the radiation fields prevalent during the epoch of reionization affect the evolution of concentration in dark matter halos. We investigate how baryonic physics during the epoch of reionization modify the structure of dark matter halos in the Cosmic Reionization On Computers (CROC) simulations. We use two different measures of halo concentration to quantify the effects. We compare concentrations of halos matched between full physics simulations…
Patchy reionization leaves a number of imprints on the small-scale cosmic microwave background (CMB) temperature fluctuations, the largest of which is the kinematic Sunyaev-Zel'dovich (kSZ), the Doppler shift of CMB photons scattering off moving electrons in ionized bubbles. It has long been known that in the CMB power spectrum, this imprint of reionization is largely degenerate with the kSZ signal produced by late-time galaxies and clusters, thus limiting our ability to constrain reionization. Following Smith & Ferraro (2017), it is possible to isolate the reionization contribution in a model…
Tomographic intensity mapping of the HI using the redshifted 21 cm observations opens up a new window towards our understanding of cosmological background evolution and structure formation. This is a key science goal of several upcoming radio telescopes including the Square Kilometer Array (SKA). In this article we focus on the post-reionization signal and investigate the of cross correlating the 21 cm signal with other tracers of the large scale structure. We consider the cross-correlation of the post-reionization 21 cm signal with the Lyman-alpha forest, Lyman-break galaxies and late time an…
A major goal of observational and theoretical cosmology is to observe the largely unexplored time period in the history of our universe when the first galaxies form, and to interpret these measurements. Early galaxies dramatically impacted the gas around them in the surrounding intergalactic medium (IGM) by photoionzing the gas during the Epoch of Reionization (EoR). This epoch likely spanned an extended stretch in cosmic time: ionized regions formed and grew around early generations of galaxies, gradually filling a larger and larger fraction of the volume of the universe. At some time -- thus…
The cosmological coincidences between the matter and radiation energy densities at recombination as well as between the densities of matter and the cosmological constant at present time are well known. We point out that moreover the third intersection between the energy densities of radiation and the cosmological constant coincides with the reionization epoch. To quantify the statistical relevance of this concurrence, we compute the Bayes factor between the concordance cosmology with free Thomson scattering optical depth and a model for which this parameter is inferred from imposing a match be…
The epoch of reionization (EoR) marks the end of the Cosmic Dawn and the beginning of large-scale structure formation in the universe. The impulsive ionization fronts (I-fronts) heat and ionize the gas within the reionization bubbles in the intergalactic medium (IGM). The temperature during this process is a key yet uncertain ingredient in current models. Typically, reionization simulations assume that all baryonic species are in instantaneous thermal equilibrium with each other during the passage of an I-front. Here we present a new model of the temperature evolution for the ionization front …
Redshifted 21 cm signal from neutral hydrogen is one of the most competitive probes of the epoch of reionization (EoR). Unpolarized 21 cm radiation acquires a certain level of linear polarization during the EoR due to Thompson scattering. This linear polarization, if measured, could probe important information about the EoR. We study the effect of Galactic Faraday rotation (FR) in the detection of polarization of 21 cm radiation. Since the effect of FR is strong due to the large wavelength nature of redshifted 21 cm signal, the initial E-mode signal carried by 21 cm photons is modified in its …
The reionization process is expected to be prolonged by the small-scale absorbers (SSAs) of ionizing photons, which have been seen as Lyman-limit systems in quasar absorption line observations. We use a set of semi-numerical simulations to investigate the effects of absorption systems on the reionization process, especially their impacts on the neutral islands during the late Epoch of Reionization (EoR). Three models are studied, i.e. the extreme case of no-SSA model with a high level of ionizing background, the moderate-SSA model with a relatively high level of ionizing background, and the de…
The fraction of ionizing photons (fesc) that escape from z>6 galaxies is an important parameter when assessing the role of these objects in the reionization of the Universe, but the opacity of the intergalactic medium precludes a direct measurement of fesc for individual galaxies at these epochs. We argue, that since fesc regulates the impact of nebular emission on the spectra of galaxies, it should nonetheless be possible to indirectly probe fesc well into the reionization epoch. As a first step, we demonstrate that by combining measurements of the rest-frame UV slope beta with the equivalent…
The epoch of first star formation and the state of the intergalactic medium (IGM) at that time are not directly observable with current telescopes. The radiation from those early sources is now part of the Cosmic Infrared Background (CIB) and, as these sources ionize the gas around them, the IGM plasma would produce faint temperature anisotropies in the Cosmic Microwave Background (CMB) via the thermal Sunyaev-Zeldovich (TSZ) effect. While these TSZ anisotropies are too faint to be detected, we show that the cross-correlation of maps of source-subtracted CIB fluctuations from {\it Euclid}, wit…
Current surveys are underway to utilize gravitational lensing by galaxy clusters with Einstein radii >35" in the search for the highest redshift galaxies. Associated supernova from the epoch of reionization would have their fluxes boosted above the detection threshold, extending their duration of visibility. We predict that the James Webb Space Telescope (JWST) will be able to discover lensed core-collapse supernovae at redshifts exceeding z=7-8.
An obstacle to the detection of redshifted 21cm emission from the epoch of reionization (EoR) is the presence of foregrounds which exceed the cosmological signal in intensity by orders of magnitude. We argue that in principle it would be better to fit the foregrounds non-parametrically - allowing the data to determine their shape - rather than selecting some functional form in advance and then fitting its parameters. Non-parametric fits often suffer from other problems, however. We discuss these before suggesting a non-parametric method, Wp smoothing, which seems to avoid some of them. After o…
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 …
Using Neural Emulators and Hamiltonian Monte Carlo to constrain the Epoch of Reionization's History with the Ly$α$ Forest Power Spectrum
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理学研究所）；The Ohio State University（俄亥俄州立大学）；Leiden University（莱顿大学）；University of Washington（华盛顿大学）；University of California, Berkeley（加州大学伯克利分校）
We carry out a new suite of cosmological radiation hydrodynamic simulations and explore the relative impacts on reionization-epoch star formation of galactic outflows and photoionization heating. By itself, an extragalactic ultraviolet background (EUVB) suppresses the luminosity function by less than 50% at z=6, overproducing the observed galaxy abundance by a factor of 3-5. Galactic outflows restore agreement with observations without preventing Population II star formation from reionizing the Universe by z=6. The resulting EUVB suppresses star formation in halos with virial temperatures belo…
In this chapter we provide an overview of the current status of the simulations and modelling of the Cosmic Dawn and Epoch of Reionization. We discuss the modelling requirements as dictated by the characteristic scales of the problem and the SKA instrumental properties and the planned survey parameters. Current simulations include most of the relevant physical processes. They can follow the full nonlinear dynamics and are now reaching the required scale and dynamic range, although small-scale physics still needs to be included at sub-grid level. However, despite a significant progress in devel…
We propose a deep learning analyzing technique with convolutional neural network (CNN) to predict the evolutionary track of the Epoch of Reionization (EoR) from the 21-cm differential brightness temperature tomography images. We use 21cmFAST, a fast semi-numerical cosmological 21-cm signal simulator, to produce mock 21-cm maps between $z=6 \sim 13$. We then apply two observational effects into those 21-cm maps, such as instrumental noise and limit of (spatial and depth) resolution somewhat suitable for realistic choices of the Square Kilometre Array (SKA). We design our deep learning model wit…
Density inhomogeneity in the intergalactic medium (IGM) can boost the recombination rate of ionized gas substantially, affecting the growth of HII regions during reionization. Previous attempts to quantify this effect typically failed to resolve down to the Jeans scale in the pre-ionization IGM, which is important in establishing this effect, along with the hydrodynamical back-reaction of reionization on it. Towards that end, we perform a set of fully-coupled, radiation-hydrodynamics simulations from cosmological initial conditions, extending the mass resolution of previous work to the scale o…
It is seen that foregrounds of 21cm Epoch of Reionization experiments, which are expected to have smooth spectral dependence, are dominant in a wedge shaped region of the Fourier space called as Foreground Wedge. A possible way forward to isolate the 21cm Epoch of Reionization (EoR) signal from the much larger foreground component is to focus on the remaining portion of the Fourier space called as the EoR window. There are in-fact three distinct regions in the Fourier space, (i) the Foreground wedge portion, (ii) the EoR window region which lies beyond Horizon wedge and (iii) the region in bet…
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-…
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…
