The 21-cm PDF (i.e., distribution of pixel brightness temperatures) is expected to be highly non-Gaussian during reionization and to provide important information on the distribution of density and ionization. We measure the 21-cm PDF as a function of redshift in a large simulation of cosmic reionization and propose a simple empirical fit. Guided by the simulated PDF, we then carry out a maximum likelihood analysis of the ability of upcoming experiments to measure the shape of the 21-cm PDF and derive from it the cosmic reionization history. Under the strongest assumptions, we find that upcoming experiments can measure the reionization history in the mid to late stages of reionization to 1-10% accuracy. Under a more flexible approach that allows for four free parameters at each redshift, a similar accuracy requires the lower noise levels of second-generation 21-cm experiments.
In the coming decade, low-frequency radio arrays will begin to probe the epoch of reionization via the redshifted 21-cm hydrogen line. Successful interpretation of these observations will require effective statistical techniques for analyzing the data. Due to the difficulty of these measurements, it is important to develop techniques beyond the standard power spectrum analysis in order to offer independent confirmation of the reionization history, probe different aspects of the topology of reionization, and have different systematic errors. In order to assess the promise of probability distrib…
During cosmic reionization, the 21-cm brightness fluctuations were highly non-Gaussian, and complementary statistics can be extracted from the distribution of pixel brightness temperatures that are not derivable from the 21-cm power spectrum. One such statistic is the 21-cm difference PDF, the probability distribution function of the difference in the 21-cm brightness temperatures between two points, as a function of the distance between the points. Guided by 21-cm difference PDFs extracted from simulations, we perform a maximum likelihood analysis on mock observational data, and analyze the a…
We study how uncertainty in the reionization history of the universe affects estimates of other cosmological parameters from the Cosmic Microwave Background. We analyze WMAP7 data and synthetic Planck-quality data generated using a realistic scenario for the reionization history of the universe obtained from high-resolution numerical simulation. We perform parameter estimation using a simple sudden reionization approximation, and using the Principal Component Analysis (PCA) technique proposed by Mortonson and Hu. We reach two main conclusions: (1) Adopting a simple sudden reionization model do…
In La Plante et al. (2017), we presented a new suite of hydrodynamic simulations with the aim of accurately capturing the process of helium II reionization. In this paper, we discuss the observational signatures present in the He II Ly$α$ forest. We show that the effective optical depth of the volume $τ_\mathrm{eff}$ is not sufficient for capturing the ionization state of helium II, due to the large variance inherent in sightlines. However, the He II flux PDF can be used to determine the timing of helium II reionization. The amplitude of the one-dimensional flux power spectrum can also determi…
We investigate the constraints on the reionization history of the Universe from a joint analysis of the cosmic microwave background and neutral hydrogen fraction data. The $\tanh$ parametrization and principal component analysis methods are applied to the reionization history respectively. The commonly used $\tanh$ parametrization is oversimplistic when the neutral hydrogen fraction data are taken into account. Using the principal component analysis method, the reconstructed reionization history is consistent with the neutral hydrogen fraction data. With the principal component analysis method…
A new parametrization of the reionization history is presented to facilitate robust comparisons between different observations and with theory. The evolution of the ionization fraction with redshift can be effectively captured by specifying the midpoint, duration, and asymmetry parameters. Lagrange interpolating functions are then used to construct analytical curves that exactly fit corresponding ionization points. The shape parametrizations are excellent matches to theoretical results from radiation-hydrodynamic simulations. The comparative differences for reionization observables are: ioniza…
Cosmic Reionization imprints its signature on the temperature and polarization anisotropies of the cosmic microwave background (CMB). Advances in CMB telescopes have already placed a significant constraint on the history of reionization. As near-future CMB telescopes target the maximum sensitivity, or observations limited only by the cosmic variance (CV), we hereby forecast the potential of future CMB observations in constraining the history of reionization. In this study, we perform Markov Chain Monte Carlo analysis for CV-limited E-mode polarization observations such as the LiteBIRD (Light s…
We compare the results from several sets of cosmological simulations of cosmic reionization, produced under Cosmic Reionization On Computers (CROC) project, with existing observational data on the high-redshift Ly-alpha forest and the abundance of Ly-alpha emitters. We find good consistency with the observational measurements and the previous simulation work. By virtue of having several independent realizations for each set of numerical parameters, we are able to explore the effect of cosmic variance on observable quantities. One unexpected conclusion we are forced into is that cosmic variance…
We reconstruct the history of reionization using Gaussian process regression. Using the UV luminosity data compilation from Hubble Frontiers Fields we reconstruct the redshift evolution of UV luminosity density and thereby the evolution of the source term in the ionization equation. This model-independent reconstruction rules out single power-law evolution of the luminosity density but supports the logarithmic double power-law parametrization. We obtain reionization history by integrating ionization equations with the reconstructed source term. Using optical depth constraint from Planck Cosmic…
We extend the existing analytical model of reionization by Furlanetto et al. (2004) to include the biasing of reionization sources and additional absorption by Lyman Limit systems. Our model is, by construction, consistent with the observed evolution of the galaxy luminosity function at z<8 and with the observed evolution of Ly-α forest at z<6. We also find that, for a wide range of values for the relative escape fraction that we consider reasonable, and which are consistent with the observational constraints on the relative escape fraction from lower redshifts, our reionization model is consi…
We present a comparison between several observational tests of the post-reionization IGM and the numerical simulations of reionization completed under the Cosmic Reionization On Computers (CROC) project. The CROC simulations match the gap distribution reasonably well, and also provide a good match for the distribution of peak heights, but there is a notable lack of wide peaks in the simulated spectra and the flux PDFs are poorly matched in the narrow redshift interval 5.5<z<5.7, with the match at other redshifts being significantly better, albeit not exact. Both discrepancies are related: simu…
At cosmic recombination, there was supersonic relative motion between baryons and dark matter, which originated from the baryonic acoustic oscillations in the early universe. This motion has been considered to have a negligible impact on the late stage of cosmic reionization because the relative velocity quickly decreases. However, recent studies have suggested that the recombination in gas clouds smaller than the local Jeans mass ($\lesssim$ $10^8~M_\odot$) can affect the reionization history by boosting the number of ultraviolet photons required for ionizing the intergalactic medium. Motivat…
We explore the constraints on the history of reionization from Planck 2015 Cosmic Microwave Background (CMB) data and we derive the forecasts for future CMB observations. We consider a class of monotonic histories of reionization as parametrized by two additional extra parameters with respect to the average optical depth used in the instantaneous reionization modeling. We investigate the degeneracies between the history of reionization and selected extensions of the standard cosmological model. In particular, we consider the degeneracies with the total mass of the neutrino sector and we discus…
An emulator for the ionizing photon mean free path in ultra-high resolution simulations: the implications of mean free path measurements for the reionization history
University of Washington（华盛顿大学）；Arizona State University（亚利桑那州立大学）；University of California Riverside（加州大学河滨分校）
We explore the ability of observations of the global brightness temperature of the 21-cm signal to constrain the reionization history and the properties of the ionizing sources. In order to describe the reionization signal, we employ either a commonly-used toy model or a structure formation model that parameterizes the properties of the ionizing sources. If the structure formation model captures the actual evolution of the reionization signal, then detecting the signal is somewhat easier than it would be for the toy model; using the toy model in this case also leads to systematic errors in rec…
We present a new investigation of the thermal history of the intergalactic medium (IGM) during and after reionization using the Lyman-$α$ forest flux power spectrum at $4.0\lesssim z\lesssim5.2$. Using a sample of 15 high-resolution spectra, we measure the flux power down to the smallest scales ever probed at these redshifts ($-1\lesssim \log(k/$km$^{-1}$s)$\lesssim -0.7$). These scales are highly sensitive to both the instantaneous temperature of the IGM and the total energy injected per unit mass during and after reionization. We measure temperatures at the mean density of $T_{0}\sim7000$-80…
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）
In this paper, we study the evolution of the ionization fraction $x_e(z)$ during the epoch of reionization by using the dispersion measurements (DMs) of fast radio bursts (FRBs). Different from the previous studies, here we turn to consider the large-scale clustering information of observed DMs of FRB catalog, which only needs the rough redshift distribution, instead of the exact redshift information of each FRB. Firstly, we consider the instantaneous ``\texttt{tanh}'' model for $x_e(z)$ and find that including the auto-correlation information of the mock catalog, about $10^4$ FRBs with the in…
We perform an analysis of the cosmic reionization in the standard cold dark matter (CDM) paradigm and in alternative dark matter scenarios. Building upon the work of Corasaniti et al. (2017), we predict the reionization history for CDM, for warm dark matter (WDM), late-forming dark matter (LFDM) and ultra-light axion dark matter (ADM) models which reproduce state-of-art measurements of the galaxy luminosity function at very high-redshifts $6\le z\le 10$. To this purpose we adopt a reionization model parametrized in terms of the limiting UV-magnitude of galaxies contributing to the reionization…
The Epoch of Reionization (EoR) features a rich interplay between the first luminous sources and the low-density gas of the intergalactic medium (IGM), where photons from these sources ionize the IGM. There are currently few observational constraints on key observables related to the EoR, such as the midpoint and duration of reionization. Although upcoming observations of the 21 cm power spectrum with next-generation radio interferometers such as the Hydrogen Epoch of Reionization Array (HERA) and the Square Kilometre Array (SKA) are expected to provide information about the midpoint of reioni…
A key physical quantity during reionization is the size of HII regions. Previous studies found a characteristic bubble size which increases rapidly during reionization, with apparent agreement between simulations and analytic excursion set theory. Using four different methods, we critically examine this claim. In particular, we introduce the use of the watershed algorithm -- widely used for void finding in galaxy surveys -- which we show to be an unbiased method with the lowest dispersion and best performance on Monte-Carlo realizations of a known bubble size PDF. We find that a friends-of-fri…
We constrain the history of reionization using the data from Planck 2015 Cosmic Microwave Background (CMB) temperature and polarization anisotropy observations. We also use prior constraints on the reionization history at redshifts $\sim7-8$ obtained from Lyman-$α$ emission observations. Using the free electron fractions at different redshifts as free parameters, we construct the complete reionization history using polynomials. Our construction provides an extremely flexible framework to search for the history of reionization as a function of redshifts. We present a conservative and an optimis…
Based on the work by Mitra, Choudhury & Ferrara (2010), we obtain model-independent constraints on reionization from cosmic microwave background (CMB) and QSO absorption line data by decomposing the function N_{ion}(z) (the number of photons entering the IGM per baryon in collapsed objects) into its principal components. The main addition in this work is that for the CMB data set, we explicitly include the angular power spectra C_l for TT, TE and EE modes in our analysis which seem to contain somewhat more information than taking the electron scattering optical depth tau_{el} as a single data …
In this paper, we explore the constraints on the reionization history that are provided by current observations of the Lyman-alpha forest and the CMB. Rather than using a particular semi-analytic model, we take the novel approach of parametrizing the ionizing sources with arbitrary functions, and perform likelihood analyses to constrain possible reionization histories. We find model independent conclusions that reionization is likely to be mostly complete by z=8 and that the IGM was 50% ionized at z=9-10. Upcoming low-frequency observations of the redshifted 21 cm line of neutral hydrogen are …
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…
We consider the prospects of probing features in the primordial power spectrum with future Cosmic Microwave Background (CMB) polarization measurements. In the scope of the inflationary scenario, such features in the spectrum can be produced by local non-smooth pieces in an inflaton potential (smooth and quasi-flat in general) which in turn may originate from fast phase transitions during inflation in other quantum fields interacting with the inflaton. They can fit some outliers in the CMB temperature power spectrum which are unaddressed within the standard inflationary ${\mathrmΛ}$CDM model. W…
We present the first complete calculation of the history of the inhomogeneous 21-cm signal from neutral hydrogen during the era of the first stars. We use hybrid computational methods to capture the large-scale distribution of the first stars, whose radiation couples to the neutral hydrogen emission, and to evaluate the 21-cm signal from z ~ 15-35. In our realistic picture large-scale fluctuations in the 21-cm signal are sourced by the inhomogeneous density field and by the Ly-alpha and X-ray radiative backgrounds. The star formation is suppressed by two spatially varying effects: negative fee…
Small-scale density fluctuations can significantly affect reionization but are typically modelled quite crudely. Unresolved fluctuations in numerical simulations and analytical calculations are included using a gas clumping factor, typically assumed to be independent of the local environment. In Paper I, we presented an improved, local density-dependent model for the sub-grid gas clumping. Here we extend this using an empirical stochastic model based on the results from high-resolution numerical simulations which fully resolve all relevant fluctuations. Our model reproduces well both the mean …
Using hydrodynamical simulations coupled to a radiative transfer code, we study the additional heating effects in the intergalactic medium (IGM) produced by $z\sim 6$ quasars in their near-zones. If helium is predominantly in HeII to begin with, both normalization ($T_0$) and slope ($γ$) of the IGM effective equation-of-state get modified by the excess ionization from the quasars. Using the available constraints on $T_0$ at $z\sim6$, we discuss implications for the nature and epoch of HI and HeII reionization. We study the extent of the HeIII region as a function of quasar age and show, for a …
The study of reionization history plays an important role in understanding the evolution of our universe. It is commonly believed that the intergalactic medium (IGM) in our universe are fully ionized today, however the reionizing process remains to be mysterious. A simple instantaneous reionization process is usually adopted in modern cosmology without direct observational evidence. However, the history of ionization fraction, $x_e(z)$ will influence cosmic microwave background (CMB) observables and constraints on optical depth $τ$. With the mocked future data sets based on featured reionizati…
We show how cross-correlating a high redshift external tracer field, such as the 21cm neutral hydrogen distribution and product maps involving Cosmic Microwave Background (CMB) temperature and polarisation fields, that probe mixed bispectrum involving these fields, can help to determine the reionization history of the Universe, beyond what can be achieved from cross-spectrum analysis. Taking clues from recent studies for the detection of primordial non-Gaussianity (Munshi & Heavens 2010), we develop a set of estimators that can study reionization using a power spectrum associated with the bisp…
In cosmology, the fine-structure constant can affect the whole ionization history. However, the previous works confine themselves to the recombination epoch and give various strong constraints on the fine-structure constant. In this paper, we also take the reionization epoch into consideration and do a consistency test of the fine-structure constant from the whole ionization history. From the data combination of Planck 2018, BAO data, SNIa samples, SFR density from UV and IR measurements, and the $Q_\text{HII}$ constraints, we find the constraint on the fine-structure constant during the recom…
The simple assumption of an instantaneous reionization of the Universe may bias estimates of cosmological parameters. In this paper a model-independent principal component method for the reionization history is applied to give constraints on the cosmological parameters from recent Planck 2015 data. We find that the Universe are not completely reionized at redshifts $z \ge 8.5$ at 95% CL. Both the reionization optical depth and the matter fluctuation amplitude are higher than but consistent with those obtained in the standard instantaneous reionization scheme. The high estimated value of the ma…
We study the impact of the extra density fluctuations induced by primordial magnetic fields on the reionization history in the redshift range: $6 < z < 10$. We perform a comprehensive MCMC physical analysis allowing the variation of parameters related to primordial magnetic fields (strength, $B_0$, and power-spectrum index $n_{\scriptscriptstyle \rm B}$), reionization, and $Λ$CDM cosmological model. We find that magnetic field strengths in the range: $B_0 \simeq 0.05{-}0.3$ nG (for nearly scale-free power spectra) can significantly alter the reionization history in the above redshift range and…
One of the most promising probes to constrain the reionization history of the universe is the power spectrum of neutral hydrogen 21 cm emission fluctuations. The corresponding analyses require computationally efficient modelling of reionization, usually achieved through semi-numerical simulations. We investigate the capability of one such semi-numerical code, SCRIPT, to constrain the reionization parameters. Our study involves creating a mock data set corresponding to the upcoming SKA-Low, followed by a Bayesian inference method to constrain the model parameters. In particular, we explore in d…
One of the milestones in the cosmic history is the formation of the first luminous objects and Hydrogen reionization. The standard theory of cosmic structure formation predicts that the first generation of stars were born about a few hundred million years after the Big Bang. The dark Universe was then lit up once again, and eventually filled with ultraviolet photons emitted from stars, galaxies, and quasars. The exact epoch of the cosmic reionization and the details of the process, even the dominant sources, are not known except the fact that the universe was reionized early on. Signatures of …
We explore the fundamental limits to which reionization histories can be constrained using only large-scale cosmic microwave background (CMB) anisotropy measurements. The redshift distribution of the fractional ionization $x_e(z)$ affects the angular distribution of CMB polarization. We project constraints on the reionization history of the universe using low-noise full-sky temperature and E-mode measurements of the CMB. We show that the measured TE power spectrum, $\hat C_\ell^\mathrm{TE}$, has roughly one quarter of the constraining power of $\hat C_\ell^\mathrm{EE}$ on the reionization opti…
Star forming galaxies represent a valuable tracer of cosmic history. Recent observational progress with Hubble Space Telescope has led to the discovery and study of the earliest-known galaxies corresponding to a period when the Universe was only ~800 million years old. Intense ultraviolet radiation from these early galaxies probably induced a major event in cosmic history: the reionization of intergalactic hydrogen. New techniques are being developed to understand the properties of these most distant galaxies and determine their influence on the evolution of the universe.
During reionization, neutral hydrogen in the intergalactic medium (IGM) imprints a damping wing absorption feature on the spectrum of high-redshift quasars. A detection of this signature provides compelling evidence for a significantly neutral Universe, and enables measurements of the hydrogen neutral fraction $x_{\rm HI}(z)$ at that epoch. Obtaining reliable quantitative constraints from this technique, however, is challenging due to stochasticity induced by the patchy inside-out topology of reionization, degeneracies with quasar lifetime, and the unknown unabsorbed quasar spectrum close to r…
Previously we identified a new class of early galaxy that we estimate contributes up to 30\% of the ionizing photons responsible for reionization. These are low mass halos in the range $M_h =10^{6.5}-10^{8} M_{\odot}$ that have been chemically enriched by supernova ejecta from prior Pop III star formation. Despite their low star formation rates, these Metal Cooling halos (MCs) are significant sources of ionizing radiation, especially at the onset of reionization, due to their high number density and ionizing escape fractions. Here we present a fully-coupled radiation hydrodynamic simulation of…
Given that the reionization history of cosmic hydrogen is yet to be stringently constrained, it is worth checking the prospects of doing so using physically motivated models and available observational data. For this purpose, we use an extended version of the explicitly photon-conserving semi-numerical model of reionization, $\texttt{SCRIPT}$, which also includes thermal evolution of the intergalactic medium (IGM). The model incorporates the effects of inhomogeneous recombination and radiative feedback self-consistently and is characterized by five free parameters (two for the redshift-depende…
Despite decades of effort, the timing and duration of He II reionization, as well as its morphology and the properties of the quasars believed to drive it, are still not well constrained. In this paper we present a new method to study both He II reionization and quasars via the thermal proximity effect -- the photoelectric heating of the intergalactic medium around quasars when their hard radiation doubly ionizes helium. We post-process a SPH simulation with 1D radiative calculations, and study how the thermal proximity effect depends on the amount of singly ionized helium, $x_{\rm HeII,0}$, w…
We present an analysis of the star formation history (SFH) of the transition-type (dIrr/dSph) Local Group galaxy \objectname[]{LGS-3} (Pisces) based on deep photometry obtained with the {\it Advanced Camera for Surveys} onboard the {\it Hubble Space Telescope}. Our analysis shows that the SFH of \objectname[]{LGS-3} is dominated by a main episode $\sim 11.7$ Gyr ago with a duration of $\sim$ 1.4 Gyr which formed $\sim 90%$ of the stars. Subsequently, \objectname[]{LGS-3} continued forming stars until the present, although at a much lower rate. The lack of early chemical enrichment is in contra…
We present a detailed comparison of three different simulations of the epoch of reionization (EoR). The radiative transfer simulation (${\rm C}^2$-RAY) among them is our benchmark. Radiative transfer codes can produce realistic results, but are computationally expensive. We compare it with two semi-numerical techniques: one using the same halos as ${\rm C}^2$-RAY as its sources (Sem-Num), and one using a conditional Press-Schechter scheme (CPS+GS). These are vastly more computationally efficient than ${\rm C}^2$-RAY, but use more simplistic physical assumptions. We evaluate these simulations i…
We describe a new methodology to analyze the reionization process in numerical simulations: the chronology and the geometry of reionization is investigated by means of merger histories of individual HII regions. From the merger tree of ionized patches, one can track the individual evolution of the regions properties such as e.g. their size, or the intensity of the percolation process by looking at the formation rate, the frequency of mergers and the number of individual HII regions involved in the mergers. We apply the merger tree technique to simulations of reionization with three different k…
Observations of galaxy luminosity function at high redshifts typically focus on fields of view of limited sizes preferentially containing bright sources. These regions possibly are overdense and hence biased with respect to the globally averaged regions. Using a semi-analytic model based on Choudhury & Ferrara (2006) which is calibrated to match a wide range of observations, we study the reionization and thermal history of the universe in overdense regions. The main results of our calculation are: (i) Reionization and thermal histories in the biased regions are markedly different from the aver…
We calculate the angular power spectrum of the Cosmic Microwave Background (CMB) temperature fluctuations induced by the kinetic Sunyaev-Zel'dovich (kSZ) effect from the epoch of reionization (EOR). We use detailed N-body+radiative transfer simulations to follow inhomogeneous reionization of the intergalactic medium (IGM). For the first time we take into account the "self-regulation" of reionization: star formation in low-mass dwarf galaxies (10^8 M_\sun \lesssim M \lesssim 10^9 M_\sun) or minihalos (10^5 M_\sun \lesssim M \lesssim 10^8 M_\sun) is suppressed if these halos form in the regions …
[abridged] With their long mean free paths and efficient heating of the intergalactic medium (IGM), X-rays could have a dramatic impact on the thermal and ionization history of the Universe. We explore this in various signals: (i) Reionization history: including X-rays results in an earlier, more extended reionization. Efficient thermal feedback from X-ray heating could yield an extended, ~10% ionized epoch. (ii) Reionization morphology: a sizable (~10%) contribution of X-rays to reionization results in a more uniform morphology, though the impact is modest when compared at the same global neu…
The observed 21-cm signal from the epoch of reionization will be distorted along the line-of-sight by the peculiar velocities of matter particles. These redshift-space distortions will affect the contrast in the signal and will also make it anisotropic. This anisotropy contains information about the cross-correlation between the matter density field and the neutral hydrogen field, and could thus potentially be used to extract information about the sources of reionization. In this paper, we study a collection of simulated reionization scenarios assuming different models for the sources of reion…
Cosmic Microwave Background temperature and polarization anisotropies from Planck have estimated a lower value of the optical depth to reionization ($τ$) compared to WMAP. A significant period in the reionization history would then fall within $6<z< 10$, where detection of galaxies with Hubble Frontier Fields program and independent estimation of neutral hydrogen in the inter galactic medium by Lyman-$α$ observations are also available. This overlap allows an analysis of cosmic reionization which utilizes a direct combination of CMB and these astrophysical measurements and potentially breaks d…
The relative velocity between baryons and dark matter in the early Universe can suppress the formation of small-scale baryonic structure and leave an imprint on the baryon acoustic oscillation (BAO) scale at low redshifts after reionization. This "streaming velocity" affects the post-reionization gas distribution by directly reducing the abundance of pre-existing mini-halos ($\lesssim 10^7 M_{\bigodot}$) that could be destroyed by reionization and indirectly modulating reionization history via photoionization within these mini-halos. In this work, we investigate the effect of streaming velocit…
We present simulations of the 21-cm signal during the epoch of reionization. We focus on properly modeling the absorption regime in the presence of inhomogeneous Wouthuysen-Field effect and X-ray heating. We ran radiative transfer simulations for three bands in the source spectrum (Lyman, UV, and X-ray) to fully account for these processes. We find that the brightness temperature fluctuation of the 21 cm signal has an amplitude greater than 100 mK during the early reionization, up to 10 times greater than the typical amplitude of a few 10 mK obtained during the later emission phase. More impor…
