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DOE OSTI2022

Gravitationally lensed quasars in Gaia – IV. 150 new lenses, quasar pairs, and projected quasars

ABSTRACT We report the spectroscopic follow-up of 175 lensed quasar candidates selected using Gaia Data Release 2 observations following Paper III of this series. Systems include 86 confirmed lensed quasars and a further 17 likely lensed quasars based on imaging and/or similar spectra. We also confirm 11 projected quasar pairs and 11 physical quasar pairs, while 25 systems are left as unclassified quasar pairs – pairs of quasars at the same redshift, which could be either distinct quasars or potential lensed quasars. Especially interesting objects include eight quadruply imaged quasars of which two have BAL sources, an apparent triple, and a doubly lensed LoBaL quasar. The source redshifts and image separations of these new lenses range between 0.65–3.59 and 0.78–6.23 arcsec, respectively. We compare the known population of lensed quasars to an updated mock catalogue at image separations between 1 and 4 arcsec, showing a very good match at z < 1.5. At z > 1.5, only 47 per cent of the predicted number are known, with 56 per cent of these missing lenses at image separations below 1.5 arcsec. The missing higher redshift, small-separation systems will have fainter lensing galaxies, and are partially explained by the unclassified quasar pairs and likely lenses presented in this work, which require deeper imaging. Of the 11 new reported projected quasar pairs, 5 have impact parameters below 10 kpc, almost tripling the number of such systems, which can probe the innermost regions of quasar host galaxies through absorption studies. We also report four new lensed galaxies discovered through our searches, with source redshifts ranging from 0.62 to 2.79.

Astronomy & Astrophysics↗
DOE OSTI2022

The relation between quasars’ optical spectra and variability

Abstract Brightness variation is an essential feature of quasars, but its mechanism and relationship to other physical quantities are not understood well. We aimed to find the relationship between the optical variability and spectral features to reveal the regularity behind the random variation. It is known that a quasar’s Fe ii/Hβ flux ratio and equivalent width of [O iii]5007 are negatively correlated; this is called Eigenvector 1. In this work, we visualized the relationship between the position on this Eigenvector 1 (EV1) plane and how the brightness of the quasars had changed after ∼10 yr. We conducted three analyses, using a different quasar sample in each. The first analysis showed the relation between the quasars’ distributions on the EV1 plane and how much they had changed brightness, using 13438 Sloan Digital Sky Survey quasars. This result shows how brightness changes later are clearly related to the position on the EV1 plane. In the second analysis, we plotted the sources reported as “changing-look quasars” (or “changing-state quasars”) on the EV1 plane. This result shows that the position on the EV1 plane corresponds to the activity level of each source, and the bright or dim states of them are distributed on the opposite sides divided by the typical quasar distribution. In the third analysis, we examined the transition vectors on the EV1 plane using sources with multiple-epoch spectra. This result shows that the brightening and dimming sources move on a similar path and they reach a position corresponding to the opposite activity level. We also found this trend is opposite to the empirical rule that $R_{\rm {Fe\, \small {II}}}$ positively correlated with the Eddington ratio, which has been proposed based on the trends of a large number of quasars. From all these analyses, it is indicated that quasars tend to oscillate between both sides of the distribution ridge on the EV1 plane; each of them corresponds to a dim state and a bright state. This trend in optical variation suggests that significant brightness changes, such as changing-look quasars, are expected to repeat.

Astronomy & Astrophysics↗
DOE OSTI2023

The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) Quasar Survey: Quasar Properties from Data Releases 6 to 9

We report the fourth installment in the series of the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) quasar survey, which includes quasars observed between 2017 September and 2021 June. There are in total 13,066 quasars reliably identified, of which 6685 are newly discovered that are not reported in the Sloan Digital Sky Survey (SDSS) DR14 quasar catalog or Million Quasars catalog. Because LAMOST does not provide accurate absolute flux calibration, we recalibrate the spectra with the SDSS/Pan-STARRS1 multiband photometric data. The emission-line properties of Hα, Hβ, Mg II , and C Iv and the continuum luminosities are measured by fitting the recalibrated spectra. We also estimate the single-epoch virial black hole masses (M BH ) using the derived emission-line and continuum parameters. This is the first time that the emission-line and continuum fluxes were estimated based on LAMOST recalibrated quasar spectra. The catalog and spectra for these quasars are available online. After the 9 yr LAMOST quasar survey, there are in total 56,175 identified quasars, of which 24,127 are newly discovered. The LAMOST quasar survey not only discovers a great number of new quasars but also provides a database for investigating the spectral variability of the quasars observed by both LAMOST and SDSS and finding rare quasars, including changing-look quasars and broad absorption line quasars.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Measurements of Quasar Proximity Zones with the Lyα Forest of DESI Y1 Quasars

Abstract The intergalactic medium (IGM) around a quasar is shaped by its dense environment and by its excess ionizing radiation, which form a “quasar proximity zone” whose size and anisotropy depend on the quasar’s halo mass, luminosity, age, and radiation geometry. Using over 10,000 quasar pairs from the Dark Energy Spectroscopic Instrument (DESI) Year 1 data, with projected comoving separations r ⊥ < 2 h −1 Mpc, we investigate how the proximity zone of foreground quasars at z ∼ 2–3.5 affects Lyα absorption in their background quasars. The large DESI sample enables unprecedented precision in measuring this “transverse proximity” effect, allowing a detailed investigation of the signal’s dependence on the projected separation of quasar pairs and the luminosity of the foreground quasar. We find that enhanced gas clustering near quasars dominates over their ionizing effect, leading to stronger absorption on neighboring sightlines. Under the assumption that quasar ionizing luminosity is isotropic and steady, we infer the IGM overdensity profile in the vicinity of quasars, finding overdensities as high as Δ ∼ 10 at comoving distance ∼1 h −1 Mpc from the most luminous systems. Surprisingly, however, we find no significant dependence of the proximity profile on the luminosity of the foreground quasar. This lack of luminosity dependence could reflect a cancellation between higher ionizing flux and higher gas overdensity, or it could indicate that quasar emission is highly time-variable or anisotropic, so that the observed luminosity does not trace the ionizing flux on nearby sightlines.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

DESI z ≳ 5 Quasar Survey. I. A First Sample of 400 New Quasars at z ~ 4.7–6.6

We report the first results of a high-redshift (z ≳ 5) quasar survey using the Dark Energy Spectroscopic Instrument (DESI). As a DESI secondary target program, this survey is designed to carry out a systematic search and investigation of quasars at 4.8 < z < 6.8. The target selection is based on the DESI Legacy Imaging Surveys (the Legacy Surveys) DR9 photometry, combined with the Pan-STARRS1 data and J-band photometry from public surveys. A first quasar sample has been constructed from the DESI Survey Validation 3 (SV3) and first-year observations until 2022 May. This sample includes more than 400 new quasars at redshift 4.7 ≤ z < 6.6, down to 21.5 magnitude (AB) in the z band, discovered from 35% of the entire target sample. Remarkably, there are 220 new quasars identified at z ≥ 5, more than one-third of existing quasars previously published at this redshift. The observations so far result in an average success rate of 23% at z > 4.7. The current spectral data set has already allowed analysis of interesting individual objects (e.g., quasars with damped Lyα absorbers and broad absorption line features), and statistical analysis will follow the survey's completion. A set of science projects will be carried out leveraging this program, including quasar luminosity function, quasar clustering, intergalactic medium, quasar spectral properties, intervening absorbers, and properties of early supermassive black holes. Additionally, a sample of 38 new quasars at z ~ 3.8–5.7 discovered from a pilot survey in the DESI SV1 is also published in this paper.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

A novel Bayesian approach for decomposing the radio emission of quasars: I. Modelling the radio excess in red quasars

ABSTRACT Studies show that both radio jets from the active galactic nuclei (AGNs) and the star formation (SF) activity in quasar host galaxies contribute to the quasar radio emission; yet their relative contributions across the population remain unclear. Here, we present an improved parametric model that allows us to statistically separate the SF and AGN components in observed quasar radio flux density distributions, and investigate how their relative contributions evolve with AGN bolometric luminosity ($L_\mathrm{bol}$) and redshift (z) using a fully Bayesian method. Based on the newest data from LOw-Frequency ARray Two-metre Sky Survey data release 2, our model gives robust fitting results out to $z\sim 4$, showing a quasar host galaxy SF rate (SFR) evolution that increases with bolometric luminosity and with redshift out to $z\sim 4$. This differs from the global cosmic SFR density, perhaps due to the importance of galaxy mergers. The prevalence of radio AGN emissions increases with quasar luminosity, but has little dependence on redshift. Furthermore, our new methodology and large sample size allow us to subdivide our data set to investigate the role of other parameters. Specifically, in this paper, we explore quasar colour and demonstrate that the radio excess in red quasars is due to an enhancement in AGN-related emission, since the host galaxy SF contribution to the total radio emission is independent of quasar colour. We also find evidence that this radio enhancement occurs mostly in quasars with weak or intermediate radio power.

Yue, B. -H (ORCID:0009000989352929)↗
DOE OSTI2026

Quantifying biases in stellar masses of JWST high- z quasar host galaxies caused by quasar subtraction

The James Webb Space Telescope (JWST) has enabled dozens of high-z quasar host galaxy detections. Many of these observations imply galaxies with black holes that are overmassive compared to their low-z counterparts. However, the bright quasar point source removal can cause significant biases in recovered host magnitudes and stellar mass measurements due to the degeneracy in host galaxy and quasar light. We develop a statistical method to disentangle the quasar host galaxy stellar mass measurements from observational biases during the point source removal assuming the PSF is modelled perfectly. We use the BlueTides simulation to generate mock images and perform point source removal on thousands of simulated high-z quasar host galaxies, constructing corrected host magnitude posteriors. We find that removing a bright quasar in JWST photometry tends to either correctly recover or modestly misestimate host magnitudes, with a maximum magnitude underestimate of 0.2 mag. With our corrected magnitude posteriors, we perform SED fitting on each quasar host galaxy and compare the stellar mass measurement before and after the correction. We find that stellar mass estimates are generally robust, or misestimated by $<$ 0.3 dex. We also find that the stellar masses of a subset of hosts (J0844−0132, J0911+0152, and J1146−0005) remain unconstrained, as key photometric bands provide only flux upper limits. Accounting for observational biases does not resolve the apparent mismatch between black hole and host galaxy growth at high-z, where some quasars appear to host overmassive black holes while others reside in relatively massive galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Nine lensed quasars and quasar pairs discovered through spatially extended variability in Pan-STARRS

We present the proof of concept of a method for finding strongly lensed quasars using their spatially extended photometric variability through difference imaging in cadenced imaging survey data. We applied the method to Pan-STARRS, starting with an initial selection of 14 107Gaiamultiplets with quasar-like infrared colours from WISE. We identified 229 candidates showing notable spatially extended variability during the Pan-STARRS survey period. These include 20 known lenses and an additional 12 promising candidates for which we obtained long-slit spectroscopy follow-up. This process resulted in the confirmation of four doubly lensed quasars, four unclassified quasar pairs, and one projected quasar pair. Only three are pairs of stars or quasar+star projections. The false-positive rate accordingly is 25%. The lens separations are between 0.81″ and 1.24″, and the source redshifts lie betweenz = 1.47 andz = 2.46. Three of the unclassified quasar pairs are promising dual-quasar candidates with separations ranging from 6.6 to 9.3 kpc. We expect that this technique is a particularly efficient way to select lensed variables in the upcomingRubin-LSST, which will be crucial given the expected limitations for spectroscopic follow-up.

Astronomy & Astrophysics↗
DOE OSTI2022

The 2175 Å bump features in FeLoBAL quasars: One indicator of MW-like dust in the nuclear region of quasars

To investigate the properties of dust in the nuclear region of quasars, we explored the extinction curves of the iron low-ionisation broad absorption line (FeLoBAL) quasar SDSS J163004.29+311957.6 and its two analogues. The parametrised extinction curves indicated the Milky Way-like 2175 Å bump features in underlying extinction, which are similar to those seen in the Local Group and in a subset of high-redshift star-forming galaxies. Compared to the bump features in the Large Magellanic Cloud (LMC), the detections in this work are much closer to those in the Milky Way (MW). These bump features, as well as those in the high- and low-ionisation broad absorption line (BAL) quasars with 2175 Å bumps, are probably the counterpart of the 2175 Å bump features in the quasar environment. This type of dust grain is generally small, easily disrupted by high-energy photons, and has difficulty surviving in the radiation field of the active galactic nucleus. However, due to the presence of absorption-line outflows, the 2175 Å bump feature in quasars, which should be rare, is seen many times in BAL quasars. The shielding effect of outflow clouds allows the MW-like dust grains to be assembled or extends the survival period in the quasar nuclear region. The process and physical and chemical conditions deserve further observational study and investigation.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

The VST ATLAS quasar survey I: Catalogue of photometrically selected quasar candidates

We present the VST ATLAS Quasar Survey, consisting of ~1229 000 quasar (QSO) candidates with 16 < g < 22.5 over ~4700 deg 2 . The catalogue is based on VST ATLAS+NEOWISE imaging surveys and aims to reach a QSO sky density of 130 deg –2 for $z$ < 2.2 and ~30 deg –2 for $z$ > 2.2. To guide our selection, we use X-ray/UV/optical/MIR data in the extended William Herschel Deep Field (WHDF) where we find a g < 22.5 broad-line QSO density of 269 ± 67 deg –2 , roughly consistent with the expected ~196 deg –2 . We find that ~25 per cent of our QSOs are morphologically classed as optically extended. Overall, we find that in these deep data, MIR, UV, and X-ray selections are ~70–90 per cent complete while X-ray suffers less contamination than MIR and UV. MIR is however more sensitive than X-ray or UV to $z$ > 2.2 QSOs at g < 22.5 and the S X (0.5 – 10 keV) > 1 X 10 -14 ergs cm –2 s –1 limit of eROSITA. We adjust the selection criteria from our previous 2QDES pilot survey and prioritize VST ATLAS candidates that show both UV and MIR excess, also selecting candidates initially classified as extended. We test our selections using data from DESI (which will be released in DR1) and 2dF to estimate the efficiency and completeness, and we use ANNz2 to determine photometric redshifts. Applying over the ~4700 deg 2 ATLAS area gives us ~ 917 000 $z$ < 2.2 QSO candidates of which 472 000 are likely to be $z$ < 2.2 QSOs, implying a sky density of ~100 deg –2 , which our WHDF analysis suggests will rise to at least 130 deg –2 when eROSITA X-ray candidates are included. At $z$ > 2.2, we find ~310() 000 candidates, of which 169 000 are likely to be QSOs for a sky density of ~36 deg –2 .

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

The WISSH quasars project: X. Discovery of a multi-component and highly variable UV ultra-fast outflow in a z = 3.6 quasar

We report on the variability of a multi-component broad absorption line (BAL) system observed in the hyper-luminous quasar J1538+0855 at z = 3.6. Observations from the Sloan Digital Sky Survey (SDSS), Very Large Telescope (VLT), Large Binocular Telescope (LBT), and Subaru telescope taken at five different epochs, spanning 17 yr in the observed frame, are presented. We detect three (A, B, and C) CIV variable troughs exhibiting extreme velocities (~40 000–54 000 km s –1 ) similar to the ultra-fast outflows (UFOs) typically observed in the X-ray spectra. The A component of the BAL UFO (v ufo ~ 0.17c) shows strength variations, while B (v ufo ~ 0.15c) and C (v ufo ~ 0.13c) components show changes both in shape and strength, appearing and disappearing at different epochs. In addition, during the last observation on June 2021, the entire BAL system disappeared. The variability trends observed during the first two epochs (1.30 yr rest frame) in the CIV, SiIV, OVI, and NV absorption spectral regions are the same for B and C troughs, while the A component of the BAL varies independently.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

CMB lensing and Ly⁢ α forest cross bispectrum from DESI’s first-year quasar sample

The squeezed cross-bispectrum B κ,Ly α between the gravitational lensing in the cosmic microwave background and the 1D Ly α forest power spectrum can constrain bias parameters and break degeneracies between σ8 and other cosmological parameters. We detect B κ,Ly ⁢α with 4.8⁢σ significance at an effective redshift z eff =2.4 using Planck PR3 lensing map and over 280,000 quasar spectra from the Dark Energy Spectroscopic Instrument’s first-year data. We test our measurement against metal contamination and foregrounds such as Galactic extinction and clusters of galaxies by deprojecting the thermal Sunyaev-Zeldovich effect. Finally, we compare our results to a tree-level perturbation theory calculation and find reasonable agreement between the model and measurement.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Quasar Properties from the Sloan Digital Sky Survey. III. The Quasars Obtained by the SDSS-IV

This paper provides the compilations of properties of the quasars newly obtained by the Sloan Digital Sky Survey (SDSS) during the fourth stage (SDSS-IV). The catalog is available on the journal's website. We have measured the main properties of emission lines around C $\tiny{IV}$ Mg $\tiny{II}$, Hβ, and Hα spectral regions. We estimate the quasar redshifts from our fits of narrow [O $\tiny{III}$], broad Mg $\tiny{II}$, and broad C $\tiny{IV}$ emission lines and find that the best redshifts included in DR16Q are robust with respect to our measurements. Based on the broad C $\tiny{IV}$, Mg $\tiny{II}$, Hβ, and/or Hα emission lines, we calculate the virial black hole mass with empirical relationships provided by previous works. Systematic biases are found among the different line-based mass estimators.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Standardizing reverberation-measured C iv time-lag quasars, and using them with standardized Mg ii quasars to constrain cosmological parameters

ABSTRACT We use 38 C iv quasar (QSO) reverberation-mapped observations, which span eight orders of magnitude in luminosity and the redshift range 0.001064 ≤ z ≤ 3.368, to simultaneously constrain cosmological-model and QSO radius–luminosity (R–L) relation parameters in six cosmological models, using an improved technique that more correctly accounts for the asymmetric errors bars of the time-lag measurements. We find that R–L relation parameters are independent of the cosmological models used in the analysis and so the R–L relation can be used to standardize the C iv QSOs. The C iv QSO cosmological constraints are consistent with those from Mg ii QSOs, allowing us to derive joint C iv + Mg ii QSO cosmological constraints which are consistent with currently accelerated cosmological expansion, as well as consistent with cosmological constraints derived using better-established baryon acoustic oscillation (BAO) and Hubble parameter [H(z)] measurements. When jointly analysed with H(z) + BAO data, current C iv + Mg ii QSO data mildly tighten current H(z) + BAO data cosmological constraints.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Performance of the Quasar Spectral Templates for the Dark Energy Spectroscopic Instrument

Abstract Millions of quasar spectra will be collected by the Dark Energy Spectroscopic Instrument (DESI), leading to a fourfold increase in the number of known quasars. High-accuracy quasar classification is essential to tighten constraints on cosmological parameters measured at the highest redshifts DESI observes ( z > 2.0). We present spectral templates for identification and redshift estimation of quasars in the DESI Year 1 data release. The quasar templates are comprised of two quasar eigenspectra sets, trained on spectra from the Sloan Digital Sky Survey. The sets are specialized to reconstruct quasar spectral variation observed over separate yet overlapping redshift ranges and, together, are capable of identifying DESI quasars from 0.05 < z < 7.0. The new quasar templates show significant improvement over the previous DESI quasar templates regarding catastrophic failure rates, redshift precision and accuracy, quasar completeness, and the contamination fraction in the final quasar sample.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Clustering of high-redshift quasars with DESI DR2

We present clustering measurements for high-redshift quasars using data from the Dark Energy Spectroscopic Instrument Data Release 2. Our sample consists of quasars with $2.0 < z < 3.5$ in the luminosity range $M_{1450} \leq -19.94$ mag. We measure the mean quasar bias $b_Q(\bar{z} = 2.48) = 3.61 \pm 0.01$ for the full sample of $\sim 715,000$ quasars and quantify the redshift evolution of quasar bias by dividing the sample into four equal redshift bins. There is strong evolution of the quasar bias with redshift that is well fit by the function $b_Q(z) = a [(1 + z)^2 - 6.565] + b$ with $a=0.230 \pm 0.007$ and $b=2.394 \pm 0.035$, and this fit is also a good match to lower redshift measurements in the literature. This bias evolution is consistent with a characteristic halo mass of $\bar{M}_{\mathrm{h}} \sim 10^{12}\,\mathrm{M_\odot}$ that does not vary significantly with redshift. The inferred duty cycles for quasars in our sample are $f_{\mathrm{duty}} \sim 10^{-2}$, staying mostly constant over redshifts. We investigate the luminosity dependence of quasar clustering by dividing each of our four redshift bins into three luminosity bins. The size of our quasar sample permits the first statistically significant measurement of the luminosity dependence of quasar bias at these redshifts. We measure weak dependence of quasar bias on luminosity at fixed redshift, inconsistent with no dependence, but weaker than predicted by a model in which quasar luminosity is tightly correlated with halo mass. These clustering measurements provide a stringent test for models of active black hole light curves and the black hole-halo connection at high redshift.

Charles, M. [Ohio State U.; Chicago U., KICP] (ORC↗
DOE OSTI2023

Target Selection and Validation of DESI Quasars

The Dark Energy Spectroscopic Instrument (DESI) survey will measure large-scale structures using quasars as direct tracers of dark matter in the redshift range 0.9 < z < 2.1 and using Lyα forests in quasar spectra at z > 2.1. We present several methods to select candidate quasars for DESI, using input photometric imaging in three optical bands (g, r, z) from the DESI Legacy Imaging Surveys and two infrared bands (W1, W2) from the Wide-field Infrared Survey Explorer. These methods were extensively tested during the Survey Validation of DESI. In this paper, we report on the results obtained with the different methods and present the selection we optimized for the DESI main survey. The final quasar target selection is based on a random forest algorithm and selects quasars in the magnitude range of 16.5 < r < 23. Visual selection of ultra-deep observations indicates that the main selection consists of 71% quasars, 16% galaxies, 6% stars, and 7% inconclusive spectra. Using the spectra based on this selection, we build an automated quasar catalog that achieves a fraction of true QSOs higher than 99% for a nominal effective exposure time of ~1000 s. With a 310 deg -2 target density, the main selection allows DESI to select more than 200 deg -2 quasars (including 60 deg -2 quasars with z > 2.1), exceeding the project requirements by 20%. The redshift distribution of the selected quasars is in excellent agreement with quasar luminosity function predictions.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Luminosity Function of Quasars at 1.0 < z < 3.5 from SDSS and DESI

We present a study of the evolution of type 1 quasars at 1.0 < z < 3.5, covering the peak epoch of quasar activity. The quasar evolution has been extensively explored by a variety of previous works and the derived quasar luminosity functions (QLFs) are not very consistent with each other, presumably due to the complexities introduced by different quasar selection techniques and associated completeness corrections. We use a new strategy to construct QLFs based on a library of all known quasars. We focus on a wide region of ∼1700 deg 2 and a deep field of ∼265 deg 2 that have rich spectroscopic data primarily from the Sloan Digital Sky Survey and DESI. We then apply traditional color cuts in the rest-frame UV–optical to select quasar candidates and use the quasar library to identify them. Our final sample consists of 62,426 quasars at 1.0 < z < 3.5, with a high completeness (∼96%) and a high purity (∼93%) in the color selection. Simple color cuts can potentially minimize selection biases for the study of quasar evolution. We derive binned QLFs and characterize them using a double power-law model. Sample incompleteness and contamination are considered as part of the uncertainties in the calculation. Compared to previous results, our QLFs are slightly higher at the faint end, and also higher at the bright end at 2.5 < z < 3.5. The QLFs suggest that the quasar evolution at 1.0 < z < 2.5 can be well described by the pure luminosity evolution model, while at 2.5 < z < 3.5, it can be described by either the pure luminosity evolution or the pure density evolution model.

Luminosity function↗