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Minju Lee

Publications and source records attributed to Minju Lee.

At least 19 recordsLinked to original sources

Constraints on the host galaxy and AGN properties of three z > 6 JWST AGN from NOEMA observations

We targeted with deep NOEMA observations the [CII]158$\mu$m emission of three JWST-discovered AGN at z>6. Two of them have the typical features of Little Red Dots (LRDs), while the third one is a blue, extended, Type I AGN. We do not significantly detect [CII] emission or dust continuum in any of the targets, even after stacking. The resulting [CII] luminosity upper limits, $\log (L_{[CII]}/L_{\odot})<7.77-8.1$, lie $\sim2\sigma$ below the values expected from the [CII]-SFR relation, and we explore different scenarios to explain the lack of [CII]. We obtained upper limits on the gas masses of $\log (M_{gas}/M_{\odot})<9.26-9.59$ corresponding to $\log( M_{dust}/M_{\odot})<5.68-6.55$ assuming a metallicity dependent dust to gas ratio. Using the continuum non-detections (rms $\sim 16-25 ~\mu Jy$) together with JWST/MIRI constraints, we performed a revised SED-fitting decomposition, resulting in stellar masses up to $\sim 2$ dex lower than previously reported, and implying $0.03\lesssim M_{BH}/M_{*}\lesssim0.7$. For the two LRDs, the SED is well reproduced by stellar emission in the rest-frame UV, while the rising rest-frame optical slope, flattening toward the near-infrared, is consistent with emission from a Type I AGN partially obscured along the polar direction with $E(B-V)_{\rm polar}\simeq 1$, in agreement with attenuation derived from the broad lines Balmer decrement. This decomposition demonstrates that a relatively standard AGN configuration can reproduce the SEDs of the two LRDs, without invoking more exotic scenarios. Finally, we investigate the positions of the three sources in the $IRX-\beta_{UV}$ plane, finding that they lie in a parameter space where galaxies are typically characterized by patchy dust distributions. Our analysis highlights the importance of millimeter constraints to characterize the different physical properties of high-z AGN.

astro-ph.GA

The ionised interstellar medium of DSFGs revealed by JWST/NIRSpec and ALMA: Super-solar metallicity, low ionisation parameters and, typical electron densities

We present a detailed study of near-infrared (2-4$\rm \mu$m) JWST/NIRSpec spectra of 48 high-redshift ($z=2.53^{+1.32}_{-0.70}$) galaxies detected with ALMA at $>3\sigma$. From a multi-wavelength SED analysis we establish the sample has a a median stellar mass of $\rm\log_{10}(M_\ast/M_\odot)=10.8\pm0.1$ and dust mass of $\rm\log_{10}(M_{\rm d}/M_\odot)=8.7\pm0.1$, covering a broad range of far-infrared luminosity $\rm (\log_{10}(L_{FIR}/L_\odot)=10.9-12.7)$. The majority of sources show no signs of AGN activity, with 40% having either X-ray counterparts $(\rm L_{Xc}>10^{42}erg/s)$, elevated optical line ratios, or broad (FWHM>800 km/s) H$\alpha$ profiles, although we note this is a lower limit due to the stochastic placement of NIRSpec slits. We establish the sample has a median gas-phase metallicity of $12+\log({\rm O/H})=8.71\pm0.02$, as derived from the [NII]/H$\alpha$ ratio, with the most FIR-luminous galaxies ($\rm\log_{10}(L_{\rm FIR}/L_\odot)>12$) falling $0.15\pm0.03$dex above the fundamental metallicity relation. From the [SII] emission-line doublet ratio, we measure a median electron density of $\log_{10}(n_{\rm e}/{\rm cm}^{-3})=2.53\pm0.07$ consistent with less-massive, star-forming, galaxies at the same epoch. For nine galaxies with [OII] and H$\beta$ detections (median $\rm\log_{10}(L_{\rm FIR}/L_\odot)=11.81\pm0.15$), we derive a median observed (dust-uncorrected) ionisation parameter of $\rm\log_{10}(U)=-2.84\pm0.06$. Our results indicate that luminous far-infrared galaxies are massive, chemically evolved systems that appear to deviate from the standard dust and metal production equilibrium observed in less obscured galaxies. This study demonstrates the synergy of JWST and ALMA in unveiling the nature of DSFGs, and highlights the need for a NIRSpec survey of uniformly selected, massive, dust-obscured, galaxies to fully characterise their interstellar medium.

astro-ph.GA

There and back again? Neutral outflows in z~3.5 quiescent galaxies

Neutral gas outflows play a crucial role in the baryon cycle of galaxies, their properties provide key insights into the transition from star formation to quiescence. In this work, we investigate the neutral gas outflow of 23 massive ($M_\star = 10^{10.1-11.6}\,\rm M_\odot$) quiescent galaxies (QGs) at z=2.82--4.61, selected from the JWST NIRSpec (R~1000) and NIRCam program DeepDive. We trace the neutral gas outflows using the NaI Doublet absorption lines and detect excess NaI D in 13/23 (57%) targets, of which 7/23 (30%) show blueshifted absorption with velocity offsets $|{\Delta}v|$ >~ 150 km/s. The z ~ 3.5 targets have ${\Delta}v$ similar to those of their local counterparts; they are also equivalent when compared in SFR--${\Delta}v$ space. We derive mass outflow rates and identify the most extreme neutral gas outflow rate $\log(\dot M_{\rm out} / \mathrm{M_\odot \, yr}^{-1})=2.68\pm0.27$ beyond the local Universe, coincident with an X-ray AGN. For all NaI D detected systems, the inferred mass outflow rate can, in principle, suppress ongoing star formation; however, the outflows are unlikely to escape their hosts, suggesting recycling on relatively short timescales (~3--180 Myr), depending on the assumed potential and launching radius. All NaI D detected targets occupy the LI(N)ER region of the BPT diagram and/or are X-ray detected, but we find no strong correlation between ongoing AGN and the neutral outflow: 2/4 broad-line/X-ray AGNs are NaI D undetected -- yet, the outflows can be powered by fossil/episodic AGNs, and one broad-line target shows a possible P-Cygni profile that indicates strong outflows. As neutral outflows alone are not able to permanently quench star formation by removing gas in our sample at z ~ 3.5, the presence of gas cycling in and out of massive passive systems may instead be the signature of feedback-regulated quenching-maintenance processes.

astro-ph.GA

A high-dynamic-range view of the growth of structure and the warm/hot Universe

Baryons heat to temperatures above $>\!\!10^5\,\mathrm{K}$ as they accrete onto massive overdensities -- galaxies, groups, clusters, and filaments -- where they ionize and become optically transparent. Deep mm-wave observations such as those with ALMA have begun to probe a handful ($\sim\,$4) of massive systems at $z\!\sim\!2-4$, while low-resolution mm-wave surveys have detected thousands of objects at arcminute resolution out to $z\!\approx\!2$. To truly advance the field of the evolution of large-scale structures, mapping the warm/hot distribution of ionized gas out to the redshift of their formation, the ESO community requires a large-aperture single-dish (sub-)mm telescope. This will need to provide several orders of magnitude higher mapping speeds than currently available while preserving the few arcsecond resolution required for imaging the gas and removing contaminating radio and dusty thermal signals across the full (sub-)mm wavelength range.

astro-ph.CO

Properness and finiteness of totally geodesic submanifolds in the convex core

We study totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds. Although the infinite-volume setting can exhibit highly complicated behavior, including geodesic planes with fractal closures, we show that a strong rigidity persists inside the convex core. This rigidity has striking consequences in the infinite volume setting: every maximal totally geodesic submanifold of dimension at least two contained in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur. These results stand in sharp contrast to the behavior in the finite-volume setting. Moreover, combining this finiteness result with the work of Bader-Fisher-Miller-Stover and of Gromov-Schoen, we deduce that any geometrically finite rank-one manifold with infinitely many maximal totally geodesic submanifolds of dimension at least two and of finite volume must be arithmetic.

math.GT

ALMA Deep Field in SSA22: Reconstructed [CII] Luminosity Function at z = 6

The ADF22 line survey reported detections of two high-$z$ line-emitting source candidates above 6-$\sigma$, both of which were shown to be spurious after follow-up observations. We investigate the detectability of far-infrared emitters in ALMA deep fields using mock observations by injecting artificial line-emitting sources into the visibility planes. We also discuss our investigation, conducted together with the ALMA operations team, of a possible technical problem in the original observations. Finally, we devise a method to estimate the [CII] luminosity function (LF) at $z \sim 6$, including a full analysis of signal contamination and sample completeness. The comparison of pixel distributions between the real and mock datacubes does not show significant differences, confirming that the effect of non-Gaussian noise is negligible for the ADF22 datacube. Using 100 blank mock-mosaic datasets, we show 0.43 $\pm$ 0.67 false detections per datacube with the previous source-finding method. We argue that the underestimation of the contamination rate in the previous work is caused by the smaller number of datacubes, using only 4 real ADF22 datacubes. We compare the results of clump-finding between the time division mode and frequency division mode correlator datacubes and confirm that the velocity widths of the clumps in the TDM case are up to 3 times wider than in the FDM case. The LF estimation using our model shows that a correction for the number count is required, up to one order of magnitude, in the luminosity range of $\geq 5 \times 10^8 L_\odot$. Our reconstruction method for the line LF can be applied to future blind line surveys.

astro-ph.GA

Galaxy morphologies at cosmic noon with JWST: A foundation for exploring gas transport with bars and spiral arms

The way in which radial flows shape galaxy structure and evolution remains an open question. Internal drivers of such flows, such as bars and spiral arms, known to mediate gas flows in the local Universe, are now observable at high redshift thanks to JWST's unobscured view. We investigated the morphology of massive star-forming galaxies at 0.8 3$ and median values of $v/\sigma > 7$ for spirals and $v/\sigma > 5$ for barred galaxies. This study establishes a population-wide framework for linking galaxy morphology and dynamics at cosmic noon, providing a key reference for future studies on the role of detailed structural features in galaxy evolution.

astro-ph.GA

Horocycles in hyperbolic 3-manifolds with round Sierpi\'nski limit sets

Let M be a geometrically finite hyperbolic 3-manifold whose limit set is a round Sierpi\'nski gasket, i.e. M is geometrically finite and acylindrical with a compact, totally geodesic convex core boundary. In this paper, we classify orbit closures of the 1-dimensional horocycle flow on the frame bundle of M. As a result, the closure of a horocycle in M is a properly immersed submanifold. This extends the work of McMullen-Mohammadi-Oh where M is further assumed to be convex cocompact.

math.DS

Revealing the hidden cosmic feast: A z=4.3 galaxy group hosting two optically dark, efficiently star-forming galaxies

We present the confirmation of a compact galaxy group candidate, CGG-z4, at $z=4.3$ in the COSMOS field. This structure was identified by two spectroscopically confirmed $z=4.3$ $K_s$-dropout galaxies with ALMA $870\rm\, \mu m$ and 3 mm continuum detections, surrounded by an overdensity of NIR-detected galaxies with consistent photometric redshifts of $4.0 4$. Their high CO(5-4)/CO(4-3) ratios indicate that the inter-stellar mediums (ISMs) are close to thermalization, suggesting either high gas temperatures, densities, and/or pressure, while the low [CI](1-0)/CO(4-3) line ratios indicate high star formation efficiencies. With [CI]-derived gas masses we found the two galaxies have extremely short gas depletion times of $99$ Myr and $<63$ Myr respectively, suggesting the onset of quenching. With an estimated halo mass of $\rm log (M_{\rm halo}[M_{\odot}])\sim12.8$, we suggest that this structure is likely in the process of forming a massive galaxy cluster.

astro-ph.GA

Deep kiloparsec view of the molecular gas in a massive star-forming galaxy at cosmic noon

We present deep ($\sim$ 20 hr), high-angular resolution Atacama Large Millimeter/submillimeter Array (ALMA) observations of the $\rm CO ~ (4-3)$ and $\rm [CI] ~ (1-0)$ transitions, along with the rest-frame 630 $\mu$m dust continuum, in BX610 --a massive, main-sequence galaxy at the peak epoch of cosmic star formation $(z = 2.21)$. Combined with deep Very Large Telescope (VLT) SINFONI observations of the H$\alpha$ line, we characterize the molecular gas and star formation activity on kiloparsec scales. Our analysis reveals that the excitation of the molecular gas, as traced by the $L'_{\rm CO ~ (4-3)} / L'_{\rm [CI] ~ (1-0)}$ line luminosity ratio, decreases with increasing galactocentric radius. While the line luminosity ratios in the outskirts are similar to those typically found in main-sequence galaxies at $z \sim 1$, the ratios in the central regions of BX610 are comparable to those observed in local starbursts. There is also a giant extra-nuclear star-forming clump in the southwest of BX610 that exhibits high star formation activity, molecular gas abundance, and molecular gas excitation. Furthermore, the central region of BX610 is rich in molecular gas $(M_{\rm mol} / M_{\rm \star} \approx 1)$; however, at the current level of star formation activity, such molecular gas is expected to be depleted in $\sim$ 450 Myr. This, along with recent evidence for rapid inflow toward the center, suggests that BX610 may be experiencing an evolutionary phase often referred to as wet compaction, which is expected to lead to central gas depletion and subsequent inside-out quenching of star formation activity.

astro-ph.GA

The Radio Galaxy Environment Reference Survey (RAGERS): a submillimetre study of the environments of massive radio-quiet galaxies at $z = 1{\rm -}3$

Measuring the environments of massive galaxies at high redshift is crucial to understanding galaxy evolution and the conditions that gave rise to the distribution of matter we see in the Universe today. While high-$z$ radio galaxies (H$z$RGs) and quasars tend to reside in protocluster-like systems, the environments of their radio-quiet counterparts are relatively unexplored, particularly in the submillimetre, which traces dust-obscured star formation. In this study we search for 850 $\mu$m-selected submillimetre galaxies in the environments of massive ($M_{\star} > 10^{11} M_{\odot}$), radio-quiet ($L_{500 {\rm MHz}} \lesssim 10^{25}$ W Hz$^{-1}$) galaxies at $z \sim 1\text{--}3$ using S2COSMOS data. By constructing number counts in circular regions of radius 1--6 arcmin and comparing with blank-field measurements, we find no significant overdensities of SMGs around massive radio-quiet galaxies at any of these scales, despite being sensitive down to overdensities of $\delta \sim 0.4$. To probe deeper than the catalogue we also examine the distribution of peaks in the SCUBA-2 SNR map, which reveals only tentative signs of any difference in the SMG densities of the radio-quiet galaxy environments compared to the blank field, and only on smaller scales (1$^{\prime}$ radii, corresponding to $\sim0.5$ Mpc) and higher SNR thresholds. We conclude that massive, radio-quiet galaxies at cosmic noon are typically in environments with $\delta\lesssim0.4$, which are either consistent with the blank field or contain only weak overdensities spanning sub-Mpc scales. The contrast between our results and studies of H$z$RGs with similar stellar masses and redshifts implies an intrinsic link between the wide-field environment and radio AGN luminosity at high redshift.

astro-ph.GA

A photo-z cautionary tale: Redshift confirmation of COSBO-7 at z=2.625

Photometric redshifts are widely used in studies of dusty star-forming galaxies (DSFGs), but catastrophic photo-$z$ failure can undermine all redshift-dependent results. Here we report the spectroscopic redshift confirmation of COSBO-7, a strongly lensed DSFG in the COSMOS-PRIMER field. Recently, a photometric redshift solution of $z\gtrsim7.0$ was reported for COSBO-7 based on ten bands of {\it James Webb} Space Telescope (JWST) NIRCam and MIRI imaging data. This $z$ value was favored by four independent spectral energy distribution (SED) fitting codes, and the result provided an appealing candidate for the most distant massive DSFG known to date. This photo-$z$ solution was also supported by a single line detection in Atacama Large Millimeter Array (ALMA) Band 3 consistent with CO(7-6) at $z=7.46$. However, our new ALMA observations robustly detect two lines in Band 6 identified as CO(7-6) and [CI](2-1) at $z_{\rm spec}=2.625$, and thus the Band 3 line as CO(3-2). These three robust line detections decidedly place COSBO-7 at $z=2.625$, refuting the photo-$z$ solution. We derive physical parameters by fitting near-infrared(NIR)-to-millimeter(mm) photometry and lens modeling, revealing that COSBO-7 is a main sequence galaxy. We examine possible reasons for this photo-$z$ failure and attribute it to (1) the likely underestimation of photometric uncertainties at 0.9\,$\mu$m and 1.15 \,$\mu$m; and (2) the lack of photometry at wavelengths beyond 20\,$\mu$m. Notably, we recover a bona fide $z_{\rm phot}\sim 2.3$ by including the existing MIPS $24\,\mu$m photometry, demonstrating the critical importance of mid-infrared (MIR) data in bolstering photo-$z$ measurements. This work highlights a common challenge in modeling the SEDs of DSFGs, and provides a cautionary tale regarding the reliability of photometric redshifts as well as pseudo-spectroscopic redshifts based on single line detection.

astro-ph.GA

The Structure of Massive Star-Forming Galaxies from JWST and ALMA: Dusty, High Redshift Disk Galaxies

We present an analysis of the JWST NIRCam and MIRI morphological properties of 80 massive ($\log_{10}(M_\ast[M_{\odot}])$=11.2$\pm$0.1) dusty star-forming galaxies at $z$$=$2.7$^{+1.2}_{-0.7}$, identified as sub-millimetre galaxies (SMGs) by ALMA, that have been observed as part of the JWST PRIMER project. To compare the structure of these massive, active galaxies to more typical less actively star-forming galaxies, we define two comparison samples. The first of 850 field galaxies matched in specific star-formation rate and redshift and the second of 80 field galaxies matched in stellar mass. We identify 20$\pm$5% of the SMGs as candidate late-stage major mergers, a further 40$\pm$10% as potential minor mergers and 40$\pm$10% which have comparatively undisturbed disk-like morphologies, with no obvious massive neighbours. These rates are comparable to those for the field samples and indicate that the majority of the sub-millimetre-detected galaxies are not late-stage major mergers, but have interaction rates similar to the less-active population at $z$$\sim$2-3. We establish that SMGs have comparable near-infrared sizes to the less active populations, but exhibit lower S\'ersic indices, consistent with bulge-less disks and have more structured morphologies at 2$\mu$m relative to 4$\mu$m. We find evidence for dust reddening as the origin of the morphological differences between the populations, identifying a strong correlation between the F200W$-$F444W pixel colour and the 870$\mu$m surface brightness. We conclude that SMGs and less active galaxies at the same epochs share a common disk-like structure, but the weaker bulge components of the SMGs results in a lower dynamical stability. Consequently, instabilities triggered either secularly or by minor external perturbations result in higher levels of activity (and dust content) in SMGs compared to typical star-forming galaxies. [Abridged]

astro-ph.GA

Atacama Large Aperture Submillimeter Telescope (AtLAST) Science: Surveying the distant Universe

During the most active period of star formation in galaxies, which occurs in the redshift range 1<z<3, strong bursts of star formation result in significant quantities of dust, which obscures new stars being formed as their UV/optical light is absorbed and then re-emitted in the infrared, which redshifts into the mm/sub-mm bands for these early times. To get a complete picture of the high-z galaxy population, we need to survey a large patch of the sky in the sub-mm with sufficient angular resolution to resolve all galaxies, but we also need the depth to fully sample their cosmic evolution, and therefore obtain their redshifts using direct mm spectroscopy with a very wide frequency coverage. This requires a large single-dish sub-mm telescope with fast mapping speeds at high sensitivity and angular resolution, a large bandwidth with good spectral resolution and multiplex spectroscopic capabilities. The proposed 50-m Atacama Large Aperture Submillimeter Telescope (AtLAST) will deliver these specifications. We discuss how AtLAST allows us to study the whole population of high-z galaxies, including the dusty star-forming ones which can only be detected and studied in the sub-mm, and obtain a wealth of information for each of these up to z~7: gas content, cooling budget, star formation rate, dust mass, and dust temperature. We present worked examples of surveys that AtLAST can perform, both deep and wide, and also focused on galaxies in proto-clusters. In addition we show how such surveys with AtLAST can measure the growth rate and the Hubble constant with high accuracy, and demonstrate the power of the line-intensity mapping method in the mm/sub-mm wavebands to constrain the cosmic expansion history at high redshifts, as good examples of what can uniquely be done by AtLAST in this research field.

astro-ph.CO

Atacama Large Aperture Submillimeter Telescope (AtLAST) Science: Solar and stellar observations

Observations at (sub-)millimeter wavelengths offer a complementary perspective on our Sun and other stars, offering significant insights into both the thermal and magnetic composition of their chromospheres. Despite the fundamental progress in (sub-)millimeter observations of the Sun, some important aspects require diagnostic capabilities that are not offered by existing observatories. In particular, simultaneous observations of the radiation continuum across an extended frequency range would facilitate the mapping of different layers and thus ultimately the 3D structure of the solar atmosphere. Mapping large regions on the Sun or even the whole solar disk at a very high temporal cadence would be crucial for systematically detecting and following the temporal evolution of flares, while synoptic observations, i.e., daily maps, over periods of years would provide an unprecedented view of the solar activity cycle in this wavelength regime. As our Sun is a fundamental reference for studying the atmospheres of active main sequence stars, observing the Sun and other stars with the same instrument would unlock the enormous diagnostic potential for understanding stellar activity and its impact on exoplanets. The Atacama Large Aperture Submillimeter Telescope (AtLAST), a single-dish telescope with 50\,m aperture proposed to be built in the Atacama desert in Chile, would be able to provide these observational capabilities. Equipped with a large number of detector elements for probing the radiation continuum across a wide frequency range, AtLAST would address a wide range of scientific topics including the thermal structure and heating of the solar chromosphere, flares and prominences, and the solar activity cycle. In this white paper, the key science cases and their technical requirements for AtLAST are discussed.

astro-ph.SR

Atacama Large Aperture Submillimeter Telescope (AtLAST) Science: Resolving the Hot and Ionized Universe through the Sunyaev-Zeldovich effect

An omnipresent feature of the multi-phase ``cosmic web'' is that warm/hot (>$10^5$ K) ionized gas pervades it. This gas constitutes a relevant contribution to the overall universal matter budget across multiple scales, from the several tens of Mpc-scale IGM filaments, to the Mpc ICM, all the way down to the CGM surrounding individual galaxies from ~1 kpc up to their respective virial radii (~100 kpc). The study of the hot baryonic component of cosmic matter density represents a powerful means for constraining the intertwined evolution of galactic populations and large-scale cosmological structures, for tracing the matter assembly in the Universe and its thermal history. To this end, the SZ effect provides the ideal observational tool for measurements out to the beginnings of structure formation. The SZ effect is caused by the scattering of the photons from the cosmic microwave background off the hot electrons embedded within cosmic structures, and provides a redshift-independent perspective on the thermal and kinematic properties of the warm/hot gas. Still, current and future (sub)mm facilities have been providing only a partial view of the SZ Universe due to any combination of: limited angular resolution, spectral coverage, field of view, spatial dynamic range, sensitivity. In this paper, we motivate the development of a wide-field, broad-band, multi-chroic continuum instrument for the Atacama Large Aperture Submillimeter Telescope (AtLAST) by identifying the scientific drivers that will deepen our understanding of the complex thermal evolution of cosmic structures. On a technical side, this will necessarily require efficient multi-wavelength mapping of the SZ signal with an unprecedented spatial dynamic range (from arcsecond to tens of arcminutes) and we employ theoretical forecasts to determine the key instrumental constraints for achieving our goals. [abridged]

astro-ph.CO

Cosmic Vine: A z=3.44 large-scale structure hosting massive quiescent galaxies

We report the discovery of a large-scale structure at z=3.44 revealed by JWST data in the Extended Groth Strip (EGS) field. This structure, called the Cosmic Vine, consists of 20 galaxies with spectroscopic redshifts at 3.43 70\%$). Comparisons with simulations suggest that the Cosmic Vine would form a cluster with halo mass $M_{\rm halo}>10^{14}M_\odot$ at z=0, and the two massive galaxies are likely forming the brightest cluster galaxies (BCGs). The results unambiguously reveal that massive quiescent galaxies can form in growing large-scale structures at z>3, thus disfavoring the environmental quenching mechanisms that require a virialized cluster core. Instead, as suggested by the interacting and bulge-dominated morphologies, the two galaxies are likely quenched by merger-triggered starburst or active galactic nucleus (AGN) feedback before falling into a cluster core. Moreover, we found that the observed specific star formation rates of massive quiescent galaxies in z>3 dense environments are one to two orders of magnitude lower than that of the BCGs in the TNG300 simulation. This discrepancy potentially poses a challenge to the models of massive cluster galaxy formation. Future studies comparing a large sample with dedicated cluster simulations are required to solve the problem.

astro-ph.GA

Torus counting and self-joinings of Kleinian groups

For any $d\geq 1$, we obtain counting and equidistribution results for tori with small volume for a class of $d$-dimensional torus packings, invariant under a self-joining $Γ_ρ<\prod_{i=1}^d\mathrm{PSL}_2(\mathbb{C})$ of a Kleinian group $Γ$ formed by a $d$-tuple of convex cocompact representations $ρ=(ρ_1, \cdots, ρ_d)$. More precisely, if $\mathcal P$ is a $Γ_ρ$-admissible $d$-dimensional torus packing, then for any bounded subset $E\subset \mathbb{C}^d$ with $\partial E$ contained in a proper real algebraic subvariety, we have $$\lim_{s\to 0} { s^{δ_{L^1}(ρ) }} \cdot \#\{T\in \mathcal{P}: \mathrm{Vol} (T)> s,\, T\cap E\neq \emptyset \}= c_{\mathcal P}\cdot ω_ρ (E\cap Λ_ρ).$$ Here $0<δ_{L^1}(ρ)\le 2/\sqrt d$ is the critical exponent of $Γ_ρ$ with respect to the $L^1$-metric on the product $\prod_{i=1}^d \mathbb{H}^3$, $Λ_ρ\subset (\mathbb{C}\cup\{\infty\})^d$ is the limit set of $Γ_ρ$, and $ω_ρ$ is a locally finite Borel measure on $\mathbb{C}^d\cap Λ_ρ$ which can be explicitly described. The class of admissible torus packings we consider arises naturally from the Teichmüller theory of Kleinian groups. Our work extends previous results of Oh-Shah on circle packings (i.e. one-dimensional torus packings) to $d$-torus packings.

math.DS