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The Physical Properties of Intracluster Gas at z > 1

We have used XMM-Newton, Chandra and HST/ACS data on one of the most distant clusters known to date, RDCS1252-29 at z= 1.24, to measure the mass of its baryonic and dark components for the first time at these large redshifts. By comparing physical properties of cluster galaxies and of the X-ray emitting intra-cluster medium (including the iron abundance) with those in low-redshift clusters, we have found that little evolution has taken place over 60% of the lifetime of the Universe. This suggests that most of the stars formed at z>approx.3 and metal enrichment processes took place early in the evolutionary history of galaxy clusters. These findings have a strong bearing on galaxy and cluster evolution models.

Rosati, Piero↗

Venus Flagship Mission Concept: A Decadal Survey Study

More than any other known planet, Venus is essential to our understanding of the evolution and habitability of Earth-size planets throughout the galaxy. We address two critical questions for planetary science: 1) How, if at all, did Venus evolve through a habitable phase? 2) What circumstances affect how volatiles shape habitable worlds? More than any other group, volatile elements have a strong influence on the evolutionary paths of rocky bodies and are critical to understanding solar system evolution. It is clear that Venus experienced a different volatile element history from the Earth and provides the only accessible example of one end-state of habitable Earth-size planets. Venus will allow us to identify the mechanisms that operate together to produce and maintain habitable worlds like our own.The (VFM) concept architecture relies on five collaborative platforms: an Orbiter, Lander, variable-altitude Aerobot and two Small Satellites (SmallSats) delivered via a single launch on a Falcon 9 heavy expendable. The platforms would use multiple instruments to measure the exosphere, atmosphere and surface at multiple scales with high precision and over time. VFM would provide the first measurements of mineralogy and geochemistry of tessera terrain to examine rocks considered to be among the most likely to have formed in a habitable climate regime. Landed, descent, aerial and orbital platforms would work synergistically to measure the chemical composition of the atmosphere including the Aerobot operating for 60 days in the Venus clouds. Loss mechanisms would be constrained by the SmallSats in two key orbits. The baseline payload for VFM includes instruments to make the first measurements of seismicity and remanent magnetism, the first long-lived (60 day) surface platform and the first life detection instrument at Venus to interrogate what could be an inhabited world.The VFM concept directly addresses each of the three Venus Exploration Analysis Group (VEXAG) goals as well as several of the strategic objectives of the 2020 NASA Science Plan, Planetary Science Division, Heliophysics and Astrophysics. The simultaneous, synergistic measurements of the solid body, surface, atmosphere and space environment provided by the VFM would allow us to target the most accessible Earth-size planet in our galaxy, and gain a profound new understanding of the evolution of our solar system and habitable worlds.

Segura, Marcia↗

Deep near-infrared survey of the Southern Sky (DENIS)

DENIS (Deep Near-Infrared Survey of the Southern Sky) will be the first complete census of astronomical sources in the near-infrared spectral range. The challenges of this novel survey are both scientific and technical. Phenomena radiating in the near-infrared range from brown dwarfs to galaxies in the early stages of cosmological evolution, the scientific exploitation of data relevant over such a wide range requires pooling expertise from several of the leading European astronomical centers. The technical challenges of a project which will provide an order of magnitude more sources than given by the IRAS space mission, and which will involve advanced data-handling and image-processing techniques, likewise require pooling of hardware and software resources, as well as of human expertise. The DENIS project team is composed of some 40 scientists, computer specialists, and engineers located in 5 European Community countries (France, Germany, Italy, The Netherlands, and Spain), with important contributions from specialists in Australia, Brazil, Chile, and Hungary. DENIS will survey the entire southern sky in 3 colors, namely in the I band at a wavelength of 0.8 micron, in the 1.25 micron J band, and in the 2.15 micron K' band. The sensitivity limits will be 18th magnitude in the I band, 16th in the J band, and 14.5th in the K' band. The angular resolution achieved will be 1 arcsecond in the I band, and 3.0 arcseconds in the J and K' bands. The European Southern Observatory 1 m telescope on La Silla will be dedicated to survey use during operations expected to last four years, commencing in late 1993. DENIS aims to provide the astronomical community with complete digitized infrared images of the full southern sky and a catalogue of extracted objects, both of the best quality and in readily accessible form. This will be achieved through dedicated software packages and specialized catalogues, and with assistance from the Leiden and Paris Data Analysis Centers. The data will be mailed on DAT tapes from La Silla to the two Data Analysis Centers for further processing. Two centers are necessary because of the shear quantity of data and because of the complementary roles the Centers will develop, each exploiting its own particular expertise. The Leiden Data Analysis Center (LDAC) will extract objects, establish their parameters, and archive them into a source catalogue. The LDAC will collaborate with the Groningen Space Research group that has gained experience in infrared image handling from the IRAS satellite. The Paris Data Analysis Center (PDAC) will be responsible for archiving and preprocessing the raw data to provide a homogeneous set of data suitable for further reduction in both the Leiden and Paris data analysis streams. The PDAC will also extract and archive images for the sources flagged by the LDAC as extended, and create a catalogue of galaxies. In exploiting the DENIS data we foresee the collaboration with other data analysis centers, such as the Observatoire de Lyon where the relevant DENIS catalogue of galaxies can be incorporated into their extragalactic database. The Point Sources and the Small Extended Sources catalogues could be incorporated in the Late Type Star database at Montpellier, and in the SIMBAD database as CDS. At Groningen the IRAS Point Source catalogue and/or image data can be merged with the DENIS catalogues. At Meudon algorithms and software will be developed with main goal assessing the limits reachable for the homogeneity and intrinsic consistency between the ensemble of the images in the data base (flat-fielding, relative positioning of the fields, bootstrapped flux calibration) but also for the data analysis.

Deul, E.↗

Effect of local environment on Ly$α$ line profile in DESI/ODIN LAEs

Lyman-Alpha Emitters (LAEs) are star-forming galaxies with significant Ly$α$ emission and are often used as tracers of large-scale structure at high redshift. We explore the relationship between the Ly$α$ line profile and environmental density with spectroscopy from the Dark Energy Spectroscopic Instrument (DESI) of LAEs selected with narrow-band photometry through the One-hundred-deg$^2$ DECam Imaging in Narrowbands (ODIN) survey. We use LAE surface density maps in the N419 (z $\sim$ 2.45) and N501 (z $\sim$ 3.12) narrow bands to probe the relationship between local environmental density and the Ly$α$ line profile. In both narrow bands, we stack the LAE spectra in bins of environmental density and inside and outside of protocluster regions. The N501 data shows $\sim$15% higher Ly$α$ line luminosity for galaxies in protoclusters, suggesting increased star formation in these regions. However, the line luminosity is not appreciably greater in protocluster galaxies in the N419 band, suggesting a potential redshift evolution of this effect. The shape of the line profile itself does not vary with environmental density, suggesting that line shape changes are caused by local effects independent of a galaxy's environment. These data indicate a potential relationship between LAE local environmental density, ionized gas distribution, and Ly$α$ line luminosity.

Uzsoy, Ana M. [Harvard-Smithsonian Ctr. Astrophys.↗

SDSS-IV MaNGA: when is morphology imprinted on galaxies?

ABSTRACT It remains an open question as to how long ago the morphology that we see in a present-day galaxy was typically imprinted. Studies of galaxy populations at different redshifts reveal that the balance of morphologies has changed over time, but such snapshots cannot uncover the typical time-scales over which individual galaxies undergo morphological transformation, nor which are the progenitors of today’s galaxies of different types. However, these studies also show a strong link between morphology and star formation rate (SFR) over a large range in redshift, which offers an alternative probe of morphological transformation. We therefore derive the evolution in SFR and stellar mass of a sample of 4342 galaxies in the SDSS-IV MaNGA survey through a stellar population ‘fossil record’ approach, and show that the average evolution of the population shows good agreement with known behaviour from previous studies. Although the correlation between a galaxy’s contemporaneous morphology and SFR is strong over a large range of lookback times, we find that a galaxy’s present-day morphology only correlates with its relatively recent ($\sim \! 2\, \textrm {Gyr}$) star formation history. We therefore find strong evidence that morphological transitions to galaxies’ current appearance occurred on time-scales as short as a few billion years.

Peterken, Thomas↗

Einstein imaging observations of clusters of galaxies

X-ray observations suggest a broad division between clusters having a centrally-located, X-ray dominant galaxy characterized by a centrally-peaked X-ray emission, and clusters with a broader X-ray profile. Within these categories, such evolutionary indicators as X-ray temperature, vorticity dispersion, spiral fraction and central galaxy density may indicate the relative stages of evolution. X-ray imaging observations are presented for (1) the Virgo cluster, (2) other dynamically unevolved clusters, (3) double clusters which may be in an intermediate evolutionary stage, and (4) dynamically evolved clusters. The data are analyzed under the assumption of an isothermal X-ray gas which is gravitationally bound by galaxies and clusters.

Forman, W.↗

The 4 Ms CHANDRA Deep Field-South Number Counts Apportioned by Source Class: Pervasive Active Galactic Nuclei and the Ascent of Normal Galaxies

We present 0.5-2 keV, 2-8 keV, 4-8 keV, and 0.5-8 keV (hereafter soft, hard, ultra-hard, and full bands, respectively) cumulative and differential number-count (log N-log S ) measurements for the recently completed approx. equal to 4 Ms Chandra Deep Field-South (CDF-S) survey, the deepest X-ray survey to date. We implement a new Bayesian approach, which allows reliable calculation of number counts down to flux limits that are factors of approx. equal to 1.9-4.3 times fainter than the previously deepest number-count investigations. In the soft band (SB), the most sensitive bandpass in our analysis, the approx. equal to 4 Ms CDF-S reaches a maximum source density of approx. equal to 27,800 deg(sup -2). By virtue of the exquisite X-ray and multiwavelength data available in the CDF-S, we are able to measure the number counts from a variety of source populations (active galactic nuclei (AGNs), normal galaxies, and Galactic stars) and subpopulations (as a function of redshift, AGN absorption, luminosity, and galaxy morphology) and test models that describe their evolution. We find that AGNs still dominate the X-ray number counts down to the faintest flux levels for all bands and reach a limiting SB source density of approx. equal to 14,900 deg(sup -2), the highest reliable AGN source density measured at any wavelength. We find that the normal-galaxy counts rise rapidly near the flux limits and, at the limiting SB flux, reach source densities of approx. equal to 12,700 deg(sup -2) and make up 46% plus or minus 5% of the total number counts. The rapid rise of the galaxy counts toward faint fluxes, as well as significant normal-galaxy contributions to the overall number counts, indicates that normal galaxies will overtake AGNs just below the approx. equal to 4 Ms SB flux limit and will provide a numerically significant new X-ray source population in future surveys that reach below the approx. equal to 4 Ms sensitivity limit. We show that a future approx. equal to 10 Ms CDF-S would allow for a significant increase in X-ray-detected sources, with many of the new sources being cosmologically distant (z greater than or approx. equal to 0.6) normal galaxies.

Field-South↗

Deficiency of ''Thin'' Stellar Bars in Seyfert Host Galaxies

Using all available major samples of Seyfert galaxies and their corresponding control samples of closely matched non-active galaxies, we find that the bar ellipticities (or axial ratios) in Seyfert galaxies are systematically different from those in non-active galaxies. Overall, there is a deficiency of bars with large ellipticities (i.e., 'fat' or 'weak' bars) in Seyferts, compared to non-active galaxies. Accompanied with a large dispersion due to small number statistics, this effect is strictly speaking at the 2 sigma level. To obtain this result, the active galaxy samples of near-infrared surface photometry were matched to those of normal galaxies in type, host galaxy ellipticity, absolute magnitude, and, to some extent, in redshift. We discuss possible theoretical explanations of this phenomenon within the framework of galactic evolution, and, in particular, of radial gas redistribution in barred galaxies. Our conclusions provide further evidence that Seyfert hosts differ systematically from their non-active counterparts on scales of a few kpc.

Shlosman, Isaac↗

The DESI DR1 peculiar velocity survey: Growth rate measurements from the galaxy power spectrum

The large-scale structure of the Universe and its evolution encapsulate a wealth of cosmological information. A powerful means of unlocking this knowledge lies in measuring the auto-power spectrum and/or the cross-power spectrum of the galaxy density and momentum fields, followed by the estimation of cosmological parameters based on these spectrum measurements. In this study, we generalize the cross-power spectrum model to accommodate scenarios in which the density and momentum fields are derived from distinct galaxy surveys. The growth rate of the large-scale structures of the Universe, commonly represented as fσ 8 , was extracted by jointly fitting the monopole and quadrupole moments of the auto-density power spectrum, the monopole of the auto-momentum power spectrum, and the dipole of the cross-power spectrum. Our estimators, theoretical models, and parameter-fitting framework were tested using mocks, confirming their robustness and accuracy in retrieving the fiducial growth rate from simulation. These techniques were then applied to analyse the power spectrum of the DESI Bright Galaxy Survey and Peculiar Velocity Survey. The fit result of the growth rate is fσ8 = 0.440$^{+0.080}_{−0.096}$ at effective redshift zeff = 0.07. By synthesizing the fitting outcomes from correlation functions, maximum likelihood estimation, and the power spectrum, a consensus value is yielded of fσ 8 (z eff = 0.07) = 0.450$^{+0.055}_{−0.055}$, and correspondingly we obtain γ = 0.580$^{+0.110}_{−0.110}$, Ω m = 0.301$^{+0.011}_{−0.011}$, and σ 8 = 0.834$^{+0.032}_{−0.032}$. The measured fσ8 and γ are consistent with the prediction of the Λ cold dark matter model and general relativity.

79 ASTRONOMY AND ASTROPHYSICS↗

The DESI DR1 Peculiar Velocity Survey: Growth Rate Measurements from the Galaxy Power Spectrum

The large-scale structure of the Universe and its evolution encapsulate a wealth of cosmological information. A powerful means of unlocking this knowledge lies in measuring the auto-power spectrum and/or the cross-power spectrum of the galaxy density and momentum fields, followed by the estimation of cosmological parameters based on these spectrum measurements. In this study, we generalize the cross-power spectrum model to accommodate scenarios where the density and momentum fields are derived from distinct galaxy surveys. The growth rate of the large-scale structures of the Universe, commonly represented as $fσ_8$, is extracted by jointly fitting the monopole and quadrupole moments of the auto-density power spectrum, the monopole of the auto-momentum power spectrum, and the dipole of the cross-power spectrum. Our estimators, theoretical models and parameter-fitting framework have been tested using mocks, confirming their robustness and accuracy in retrieving the fiducial growth rate from simulation. These techniques are then applied to analyze the power spectrum of the DESI Bright Galaxy Survey and Peculiar Velocity Survey, and the fit result of the growth rate is $fσ_8=0.440^{+0.080}_{-0.096}$ at effective redshift $z_{\rm eff}=0.07$. By synthesizing the fitting outcomes from correlation functions, maximum likelihood estimation and power spectrum, yields a consensus value of $fσ_8(z_{\rm eff}=0.07) = 0.450 ^{+0.055}_{-0.055}$, and correspondingly we obtain $γ=0.580^{+0.110}_{-0.110}$, $Ω_\mathrm{m}=0.301^{+0.011}_{-0.011}$ and $σ_8=0.834^{+0.032}_{-0.032}$. The measured $fσ_8$ and $γ$ are consistent with the prediction of the $Λ$ Cold Dark Matter Model and General Relativity.

Qin, F. [Marseille, CPPM] (ORCID:0000000179507864)↗

Illuminating White Dwarf Spectra through Laboratory Experiments at Cosmic Conditions

We establish the work on white dwarf stars in the larger context of the experiments of the Wootton Center for Astrophysical Plasma Properties (WCAPP); these experiments are conducted at the plasma conditions found in the cosmos, without the need for scaling. We briefly summarize the results of these experiments to-date and their astrophysical and physical import, before focusing on the white dwarf experiments. Over 97% of stars either are, or will become, white dwarf stars, giving them broad relevance. We describe the astrophysical and physical problems associated with white dwarf photospheres, the plasma where the observed light arises. Here, the astrophysical questions include the age of the universe, the age and history of star formation in our Galaxy's varied morphological components and the evolution of stars. The compact nature of these ubiquitous stars means that the atomic physics is not well constrained even in the outermost layers. Further, it suggests that many important processes, including crystallization in dense Coulomb plasma, occur and have a significant effect on the evolution and structure of these stars and thus all their many applications.

79 ASTRONOMY AND ASTROPHYSICS↗

Convective shells in the interior of Cepheid variable stars: Overshooting models based on hydrodynamic simulations

Context. Because Cepheid variable stars have long been used as a cosmic benchmark for scaling distances in our Galaxy and beyond, the accuracy of stellar evolution models for Cepheids have wide-reaching effects. However, our understanding of the dynamics in the interiors of these physically complex stars is limited. Aims. Our goal is to provide a detailed multi-dimensional picture of hydrodynamic convection and convective boundary mixing in the interior of Cepheids. Methods. Using the Modules for Experiments in Stellar Astrophysics (MESA), we studied the structure of intermediate-mass stars that cross the instability strip. Then, we performed two-dimensional hydrodynamic simulations of six stars with the fully compressible Multidimensional Stellar Implicit Code (MUSIC). Our simulations did not model the radial pulsations but focused on the interior structure of this family of stars. We developed and applied a new statistical analysis to examine convection and convective boundary mixing in the interior of these stellar simulations. Results. Based on a grid of MESA models, we demonstrated that a common structure for intermediate mass Cepheids includes an interior convective shell as well as a thin outer convective envelope. Using the extreme value theory approach to analyze our MUSIC simulation data, we found that overshooting above the convective shell fills the space between these convectively unstable layers. We developed a new statistical analysis that provides a clearer picture of how overshooting fills this layer; it also allowed us to formulate a detailed comparison between overshooting above and below the convective shell. Our analysis effectively decomposes the overshooting layer into two layers: a weak overshooting layer and a strong overshooting layer. Statistically, this is accomplished by decomposing the strongly non-Gaussian probability density function into a mixture of gamma distributions. Using our mixture model, we showed that the ratio of overshooting lengths above and below the convective shell depends directly on the radial extent of the convective shell as well as its depth in the star. We proposed a new form for the diffusion coefficient that addresses the need for overlapping overshooting layers between convective shells. We introduced the idea of a “super-mixing layer” where overshooting from both the convective shell and the convective envelope results in efficient mixing and could be viewed as merging the two adjacent convective zones.

79 ASTRONOMY AND ASTROPHYSICS↗

The First Identification of Lyα Changing-look Quasars at High Redshift in DESI

We present two cases of Lyα changing-look (CL) quasars (J1306 and J1512) along with two additional candidates (J1511 and J1602), all discovered serendipitously at z > 2 through the Dark Energy Spectroscopic Instrument and the Sloan Digital Sky Survey. It is the first time to capture CL events in Lyα at high redshift, which is crucial for understanding the underlying mechanisms driving the CL phenomenon and the evolution of high-redshift quasars and galaxies. We find that the accretion rate in the dim state for these CL objects corresponds to a relatively low value ($\mathop{\,}\limits^{{\mathscr{M}}}\,\approx 2\times 1{0}^{-3}$), which suggests that the inner region of the accretion disk might be in transition between the advection dominated accretion flow ($\dot{\,{\mathscr{M}}}\,\lt 1{0}^{-3}\sim 1{0}^{-2}$) and the canonical accretion disk (optically thick, geometrically thin). However, unlike in C IV CL quasars in which broad Lyα remained, the broad C IV may still persist after a CL event occurs in Lyα, making the physical origin of the CL and ionization mechanism event more puzzling and interesting.

accretion↗

Distortion of the Microwave Blackbody Background Radiation Implied by the Baryon-symmetric Cosmology of Omnes and the Galaxy Formation Theory of Stecker and Puget

Theories on the evolution of the universe are evaluated. Particular attention was given to Omnes and Stecker and Puget theories. Data cover distortion of the microwave black body background energy distribution at red shifts between 10,000 and 1.000, and black body distortion due to antimatter and annihilation reactions.

Stecker, F. W.↗

The High Resolution Spectrograph for the Space Telescope

The High Resoltuion Spectrograph (HRS) in conjunction with the Space Telescope (ST), extends ultraviolet astronomical spectroscopy to higher spectra, spatial, and time resolutions than previously achieved, as well as to fainter and more distant celestial objects. Other significant advances inherent in the instrument are high photometric accuracy and efficient operation via exposure meter control and real time rejection of bad data. These capabilities are provided to accomplish the scientific programs of the HRS investigation definition team, which concern the interstellar medium, stellar winds, and evolutionary aspects of stellar atmosphere studies; the determination of chemical abundances relevant to stellar evolution; the investigation of quasars and Seyfert galaxy nuclei; and the analysis of the atmospheres of solar system objects, including comets.

Brandt, J. C.↗

The extended medium sensitivity survey distant cluster sample - X-ray data and interpretation of the luminosity evolution

The X-ray properties of a cluster of galaxies subsample of the Einstein Extended Medium Sensitivity Survey is described. A summary of this sample and its implication has been presented previously; this paper gives the full details. The cluster subsample is 98.4 percent identified and contains 93 X-ray-selected clusters to a redshift of 0.58. The cluster X-ray luminosity function at three cosmic epochs is derived. While the present luminosity function agrees with previous determinations at the lowest redshifts, it is found that the volume density of high-luminosity clusters is greater now than it was in the past. The normalization, shape, and time dependence of the luminosity function can be described by a simple hierarchical formation model with parameters which also describe the temperature function of an independent sample of low-redshift clusters. In this model the comoving hot gas density remains constant with time at least to redshifts of order 0.35.

Henry, J. P.↗

Gamma ray bursts from extragalactic sources

The properties of gamma ray bursts of classical type are found to be explicable in terms of high speed collisions between stars. A model is proposed in which the frequency of such collisions can be calculated. The model is then applied to the nuclei of galaxies in general on the basis that galaxies, or at least some fraction of them, originate in the expulsion of stars from creation centers. Evidence that low level activity of this kind is also taking place at the center of our own Galaxy is discussed. The implications for galactic evolution are discussed and a negative view of black holes is taken.

Hoyle, Fred↗

Reionization in a cold dark matter universe: The feedback of galaxy formation on the intergalactic medium

We study the coupled evolution of the intergalactic medium (IGM) and the emerging structure in the universe in the context of the cold dark matter (CDM) model, with a special focus on the consequences of imposing reionization and the Gunn-Peterson constraint as a boundary condition on the model. We have calculated the time-varying density of the IGM by coupling our detailed, numerical calculations of the thermal and ionization balance and radiative transfer in a uniform, spatially averaged IGM of H and He, including the mean opacity of an evolving distribution of gas clumps which correspond to quasar absorption line clouds, to the linearized equations for the growth of density fluctuations in both the gaseous and dark matter components in a CDM universe. We use the linear growth equations to identify the fraction of the gas which must have collapsed out at each epoch, an approach similar in spirit to the so-called Press-Schechter formalism. We identify the IGM density with the uncollapsed baryon fraction. The collapsed fraction is postulated to be a source of energy injection into the IGM, by radiation or bulk hydrodynamical heating (e.g., via shocks) or both, at a rate which is marginally enough to satisfy the Gunn-Peterson constraint at z less than 5. Our results include the following: (1) We find that the IGM in a CDM model must have contained a substantial fraction of the total baryon density of the universe both during and after its reionization epoch. (2) As a result, our previous conclusion that the observed Quasi-Stellar Objects (QSOs) at high redshift are not sufficient to ionize the IGM enough to satisfy the Gunn-Peterson constraint is confirmed. (3) We predict a detectable He II Gunn-Peterson effect at 304(1 + z) A in the spectra of quasars at a range of redshift z greater than or approx. 3, depending on the nature of the sources of IGM reionization. (4) We find, moreover, that a CDM model with high bias parameter b (i.e., b greater than or approx. 2) cannot account for the baryon content of the universe at z approximately 3 observed in quasar absorption line gas unless Omega (sub B) significantly exceeds the maximum value allowed by big bang nucleocynthesis. (5) For a CDM model with bias parameter within the allowed range of (lower) values, the lower limit to Omega(sub B) imposed by big bang nucleosynthesis (Omega(sub B) h(sup 2) greater than or equal to 0.01) combines with our results to yield the minimum IGM density for the CDM fodel. For CDM with b = 1 (Cosmic Background Explorer (COBE) normalization), we find Omega(sub IGM)(sup min) (z approximately 4) approx. equal 0.02-0.03, and Omega(sub IGM)(sup min)(z approximately 0) approx. equal 0.005-0.03, depending upon the nature of the sources of IGM reionization. (6) In general, we find that self-consistent reionization of the IGM by the collapsed baryon fraction has a strong effect on the rate of collapse. (7) As a further example, we show that the feedback effect on the IGM of energy release by the collapsed baryon fraction may explain the slow evolution of the observed comoving QSO number density between z = 5 and z = 2, followed by the sharp decline after z = 2.

Shapiro, Paul R.↗