arXiv subjects

Lars Hernquist

Publications and source records attributed to Lars Hernquist.

At least 343 records · Page 19Linked to original sources

arXiv2017

First results from the IllustrisTNG simulations: matter and galaxy clustering

Hydrodynamical simulations of galaxy formation have now reached sufficient volume to make precision predictions for clustering on cosmologically relevant scales. Here we use our new IllustrisTNG simulations to study the non-linear correlation functions and power spectra of baryons, dark matter, galaxies and haloes over an exceptionally large range of scales. We find that baryonic effects increase the clustering of dark matter on small scales and damp the total matter power spectrum on scales up to k ~ 10 h/Mpc by 20%. The non-linear two-point correlation function of the stellar mass is close to a power-law over a wide range of scales and approximately invariant in time from very high redshift to the present. The two-point correlation function of the simulated galaxies agrees well with SDSS at its mean redshift z ~ 0.1, both as a function of stellar mass and when split according to galaxy colour, apart from a mild excess in the clustering of red galaxies in the stellar mass range 10^9-10^10 Msun/h^2. Given this agreement, the TNG simulations can make valuable theoretical predictions for the clustering bias of different galaxy samples. We find that the clustering length of the galaxy auto-correlation function depends strongly on stellar mass and redshift. Its power-law slope gamma is nearly invariant with stellar mass, but declines from gamma ~ 1.8 at redshift z=0 to gamma ~ 1.6 at redshift z ~ 1, beyond which the slope steepens again. We detect significant scale-dependencies in the bias of different observational tracers of large-scale structure, extending well into the range of the baryonic acoustic oscillations and causing nominal (yet fortunately correctable) shifts of the acoustic peaks of around ~5%.

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arXiv2017

Similar star formation rate and metallicity evolution timescales drive the fundamental metallicity relation

The fundamental metallicity relation (FMR) is a postulated correlation between galaxy stellar mass, star formation rate (SFR), and gas-phase metallicity. At its core, this relation posits that offsets from the mass-metallicity relation (MZR) at a fixed stellar mass are correlated with galactic SFR. In this Letter, we quantify the timescale with which galactic SFRs and metallicities evolve using hydrodynamical simulations. We find that Illustris and IllustrisTNG predict that galaxy offsets from the star formation main sequence and MZR evolve over similar timescales, are often anti-correlated in their evolution, evolve with the halo dynamical time, and produce a pronounced FMR. In fact, for a FMR to exist, the metallicity and SFR must evolve in an anti-correlated sense which requires that they evolve with similar time variability. In contrast to Illustris and IllustrisTNG, we speculate that the SFR and metallicity evolution tracks may become decoupled in galaxy formation models dominated by globally-bursty SFR histories, which could weaken the FMR residual correlation strength. This opens the possibility of discriminating between bursty and non-bursty feedback models based on the strength and persistence of the FMR -- especially at high redshift.

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arXiv2017

The Size Evolution of Star-forming and Quenched Galaxies in the IllustrisTNG simulation

We analyze scaling relations and evolution histories of galaxy sizes in TNG100, part of the IllustrisTNG simulation suite. Observational qualitative trends of size with stellar mass, star-formation rate and redshift are reproduced, and a quantitative comparison of projected r-band sizes at 0~ ~10^{9.5}Msun, the evolution of the median main progenitor differs, with quenched galaxies hardly growing in median size before quenching, whereas main-sequence galaxies grow their median size continuously, thus opening a gap from the progenitors of quenched galaxies. This is partly because the main-sequence high-redshift progenitors of quenched z=0 galaxies are drawn from the lower end of the size distribution of the overall population of main-sequence high-redshift galaxies. (ii) Quenched galaxies with M_{*,z=0}>~10^{9.5}Msun experience a steep size growth on the size-mass plane after their quenching time, but with the exception of galaxies with M_{*,z=0}>~10^{11}Msun, the size growth after quenching is small in absolute terms, such that most of the size (and mass) growth of quenched galaxies (and its variation among them) occurs while they are still on the main-sequence. After they become quenched, the size growth rate of quenched galaxies as a function of time, as opposed to versus mass, is similar to that of main-sequence galaxies. Hence, the size gap is retained down to z=0.

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arXiv2017

Probing the Hot X-ray Corona around the Massive Spiral Galaxy, NGC 6753, Using Deep XMM-Newton observations

X-ray emitting gaseous coronae around massive galaxies are a basic prediction of galaxy formation models. Although the coronae around spiral galaxies offer a fundamental test of these models, observational constraints on their characteristics are still scarce. While the presence of extended hot coronae has been established around a handful of massive spiral galaxies, the short X-ray observations only allowed for measurements of the basic characteristics of the coronae. In this work, we utilize deep XMM-Newton observations of NGC 6753 to explore its extended X-ray corona in unprecedented detail. Specifically, we establish the isotropic morphology of the hot gas, suggesting that it resides in hydrostatic equilibrium. The temperature profile of the gas shows a decrease with increasing radius: it drops from $kT\approx0.7$ keV in the innermost parts to $kT\approx0.4$ keV at 50 kpc radius. The temperature map reveals the complex temperature structure of the gas. We study the metallicity distribution of the gas, which is uniform at $Z\approx0.1$ Solar. This value is about an order of magnitude lower than that obtained for elliptical galaxies with similar dark matter halo mass, hinting that the hot gas in spiral galaxies predominantly originates from external gas inflows rather than from internal sources. By extrapolating the density profile of the hot gas out to the virial radius, we estimate the total gas mass and derive the total baryon mass of NGC 6753. We conclude that the baryon mass fraction is $f_{\rm b} \approx 0.06$, implying that about half of the baryons are missing.

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arXiv2017

The evolution of the mass-metallicity relation in IllustrisTNG

The coevolution of galaxies and their metal content serves as an important test for galaxy feedback models. We analyze the distribution and evolution of metals within the IllustrisTNG simulation suite with a focus on the gas-phase mass-metallicity relation (MZR). We find that the IllustrisTNG model broadly reproduces the slope and normalization evolution of the MZR across the redshift range $0 2$) evolution fit by $\mathrm{d\;log(Z)}/\mathrm{dz} \approx - 0.064$. Our simulations indicate that the metal retention efficiency of the interstellar medium (ISM) is low: a majority of gas-phase metals ($\sim$ 85 per cent at $z=0$) live outside of the ISM, either in an extended gas disk, the circumgalactic medium, or outside the halo. Nevertheless, the redshift evolution in the simulated MZR normalization is driven by the higher gas fractions of high redshift galaxies, not by changes to the metal retention efficiency. The scatter in the simulated MZR contains a clear correlation with the gas-mass or star formation rate of the system, in agreement with the observed fundamental metallicity relation. The scatter in the MZR is driven by a competition between periods of enrichment- and accretion-dominated metallicity evolution. We expect that while the normalization of the MZR declines with redshift, the strength of the correlation between metallicity and gas-mass at fixed stellar mass is not a strong function of redshift. Our results indicate that the "regulator" style models are best suited for simultaneously explaining the shape, redshift evolution, and existence of correlated scatter with gas fraction about the MZR.

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arXiv2017

The uniformity and time-invariance of the intra-cluster metal distribution in galaxy clusters from the IllustrisTNG simulations

The distribution of metals in the intra-cluster medium encodes important information about the enrichment history and formation of galaxy clusters. Here we explore the metal content of clusters in IllustrisTNG - a new suite of galaxy formation simulations building on the Illustris project. Our cluster sample contains 20 objects in TNG100 - a ~(100 Mpc)^3 volume simulation with 2x1820^3 resolution elements, and 370 objects in TNG300 - a ~(300 Mpc)^3 volume simulation with 2x2500^3 resolution elements. The z=0 metallicity profiles agree with observations, and the enrichment history is consistent with observational data going beyond z~1, showing nearly no metallicity evolution. The abundance profiles vary only minimally within the cluster samples, especially in the outskirts with a relative scatter of ~15%. The average metallicity profile flattens towards the center, where we find a logarithmic slope of -0.1 compared to -0.5 in the outskirts. Cool core clusters have more centrally peaked metallicity profiles (~0.8 solar) compared to non-cool core systems (~0.5 solar), similar to observational trends. Si/Fe and O/Fe radial profiles follow positive gradients. The outer abundance profiles do not evolve below z~2, whereas the inner profiles flatten towards z=0. More than ~80% of the metals in the intra-cluster medium have been accreted from the proto-cluster environment, which has been enriched to ~0.1 solar already at z~2. We conclude that the intra-cluster metal distribution is uniform among our cluster sample, nearly time-invariant in the outskirts for more than 10 Gyr, and forms through a universal enrichment history.

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arXiv2017

Simulating Galaxy Formation with the IllustrisTNG Model

We introduce an updated physical model to simulate the formation and evolution of galaxies in cosmological, large-scale gravity+magnetohydrodynamical simulations with the moving mesh code AREPO. The overall framework builds upon the successes of the Illustris galaxy formation model, and includes prescriptions for star formation, stellar evolution, chemical enrichment, primordial and metal-line cooling of the gas, stellar feedback with galactic outflows, and black hole formation, growth and multi-mode feedback. In this paper we give a comprehensive description of the physical and numerical advances which form the core of the IllustrisTNG (The Next Generation) framework. We focus on the revised implementation of the galactic winds, of which we modify the directionality, velocity, thermal content, and energy scalings, and explore its effects on the galaxy population. As described in earlier works, the model also includes a new black hole driven kinetic feedback at low accretion rates, magnetohydrodynamics, and improvements to the numerical scheme. Using a suite of (25 Mpc $h^{-1}$)$^3$ cosmological boxes we assess the outcome of the new model at our fiducial resolution. The presence of a self-consistently amplified magnetic field is shown to have an important impact on the stellar content of $10^{12} M_{\rm sun}$ haloes and above. Finally, we demonstrate that the new galactic winds promise to solve key problems identified in Illustris in matching observational constraints and affecting the stellar content and sizes of the low mass end of the galaxy population.

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arXiv2017

Single Sources in the Low-Frequency Gravitational Wave Sky: properties and time to detection by pulsar timing arrays

We calculate the properties, occurrence rates and detection prospects of individually resolvable 'single sources' in the low frequency gravitational wave (GW) spectrum. Our simulations use the population of galaxies and massive black hole binaries from the Illustris cosmological hydrodynamic simulations, coupled to comprehensive semi-analytic models of the binary merger process. Using mock pulsar timing arrays (PTA) with, for the first time, varying red-noise models, we calculate plausible detection prospects for GW single sources and the stochastic GW background (GWB). Contrary to previous results, we find that single sources are at least as detectable as the GW background. Using mock PTA, we find that these 'foreground' sources (also 'deterministic'/'continuous') are likely to be detected with $\sim 20 \, \textrm{yr}$ total observing baselines. Detection prospects, and indeed the overall properties of single sources, are only moderately sensitive to binary evolution parameters---namely eccentricity & environmental coupling, which can lead to differences of $\sim 5 \,\textrm{yr}$ in times to detection. Red noise has a stronger effect, roughly doubling the time to detection of the foreground between a white-noise only model ($\sim 10$ - $15 \, \textrm{yr}$) and severe red noise ($\sim 20$ - $30 \, \textrm{yr}$). The effect of red noise on the GWB is even stronger, suggesting that single source detections may be more robust. We find that typical signal-to-noise ratios for the foreground peak near $f = 0.1 \, \textrm{yr}^{-1}$, and are much less sensitive to the continued addition of new pulsars to PTA.

astro-ph.HE↗

arXiv2017

Supermassive black holes and their feedback effects in the IllustrisTNG simulation

We study the population of supermassive black holes (SMBHs) and their effects on massive central galaxies in the IllustrisTNG cosmological hydrodynamical simulations of galaxy formation. The employed model for SMBH growth and feedback assumes a two-mode scenario in which the feedback from active galactic nuclei occurs through a kinetic, comparatively efficient mode at low accretion rates relative to the Eddington limit, and in the form of a thermal, less efficient mode at high accretion rates. We show that the quenching of massive central galaxies happens coincidently with kinetic-mode feedback, consistent with the notion that active supermassive black cause the low specific star formation rates observed in massive galaxies. However, major galaxy mergers are not responsible for initiating most of the quenching events in our model. Up to black hole masses of about $10^{8.5}\,{\rm M}_\odot$, the dominant growth channel for SMBHs is in the thermal mode. Higher mass black holes stay mainly in the kinetic mode and gas accretion is self-regulated via their feedback, which causes their Eddington ratios to drop, with SMBH mergers becoming the main channel for residual mass growth. As a consequence, the quasar luminosity function is dominated by rapidly accreting, moderately massive black holes in the thermal mode. We show that the associated growth history of SMBHs produces a low-redshift quasar luminosity function and a redshift zero black hole mass-stellar bulge mass relation in good agreement with observations, whereas the simulation tends to over-predict the high-redshift quasar luminosity function.

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arXiv2017

Radiative effects during the assembly of direct collapse black holes

We perform a post-processing radiative feedback analysis on a 3D ab initio cosmological simulation of an atomic cooling halo under the direct collapse black hole (DCBH) scenario. We maintain the spatial resolution of the simulation by incorporating native ray-tracing on unstructured mesh data, including Monte Carlo Lyman-alpha (Lyα) radiative transfer. DCBHs are born in gas-rich, metal-poor environments with the possibility of Compton-thick conditions, $N_H \gtrsim 10^{24} {\rm cm}^{-2}$. Therefore, the surrounding gas is capable of experiencing the full impact of the bottled-up radiation pressure. In particular, we find that multiple scattering of Lyα photons provides an important source of mechanical feedback after the gas in the sub-parsec region becomes partially ionized, avoiding the bottleneck of destruction via the two-photon emission mechanism. We provide detailed discussion of the simulation environment, expansion of the ionization front, emission and escape of Lyα radiation, and Compton scattering. A sink particle prescription allows us to extract approximate limits on the post-formation evolution of the radiative feedback. Fully coupled Lyα radiation hydrodynamics will be crucial to consider in future DCBH simulations.

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arXiv2017

Galaxies with Shells in the Illustris Simulation: Metallicity Signatures

Stellar shells are low surface brightness arcs of overdense stellar regions, extending to large galactocentric distances. In a companion study, we identified 39 shell galaxies in a sample of 220 massive ellipticals ($\mathrm{M}_{\mathrm{200crit}}>6\times10^{12}\,\mathrm{M}_\odot$) from the Illustris cosmological simulation. We used stellar history catalogs to trace the history of each individual star particle inside the shell substructures, and we found that shells in high-mass galaxies form through mergers with massive satellites (stellar mass ratios $μ_{\mathrm{stars}}\gtrsim1:10$). Using the same sample of shell galaxies, the current study extends the stellar history catalogs in order to investigate the metallicity of stellar shells around massive galaxies. Our results indicate that outer shells are often times more metal-rich than the surrounding stellar material in a galaxy's halo. For a galaxy with two different satellites forming $z=0$ shells, we find a significant difference in the metallicity of the shells produced by each progenitor. We also find that shell galaxies have higher mass-weighted logarithmic metallicities ([Z/H]) at $2$-$4\,\mathrm{R}_{\mathrm{eff}}$ compared to galaxies without shells. Our results indicate that observations comparing the metallicities of stars in tidal features, such as shells, to the average metallicities in the stellar halo can provide information about the assembly histories of galaxies.

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arXiv2017

Galaxy Formation with BECDM: I. Turbulence and relaxation of idealised haloes

We present a theoretical analysis of some unexplored aspects of relaxed Bose-Einstein condensate dark matter (BECDM) haloes. This type of ultralight bosonic scalar field dark matter is a viable alternative to the standard cold dark matter (CDM) paradigm, as it makes the same large-scale predictions as CDM and potentially overcomes CDM's small-scale problems via a galaxy-scale de Broglie wavelength. We simulate BECDM halo formation through mergers, evolved under the Schrödinger-Poisson equations. The formed haloes consist of a soliton core supported against gravitational collapse by the quantum pressure tensor and an asymptotic $r^{-3}$ NFW-like profile. We find a fundamental relation of the core=to-halo mass with the dimensionless invariant $Ξ\equiv \lvert E \rvert/M^3/(Gm/\hbar)^2$ or $M_{\rm c}/M \simeq 2.6 Ξ^{1/3}$, linking the soliton to global halo properties. For $r \geq 3.5 \,r_{\rm c}$ core radii, we find equipartition between potential, classical kinetic, and quantum gradient energies. The haloes also exhibit a conspicuous turbulent behavior driven by the continuous reconnection of vortex lines due to wave interference. We analyse the turbulence 1D velocity power spectrum and find a $k^{-1.1}$ power-law. This suggests the vorticity in BECDM haloes is homogeneous, similar to thermally-driven counterflow BEC systems from condensed matter physics, in contrast to a $k^{-5/3}$ Kolmogorov power-law seen in mechanically-driven quantum systems. The mode where the power spectrum peaks is approximately the soliton width, implying the soliton-sized granules carry most of the turbulent energy in BECDM haloes.

astro-ph.CO↗

arXiv2017

The formation and evolution of star clusters in interacting galaxies

Observations of globular clusters show that they have universal lognormal mass functions with a characteristic peak at $\sim 2\times 10^{5}\, {\rm{M_{\odot}}}$, but the origin of this peaked distribution is highly debated. Here we investigate the formation and evolution of star clusters in interacting galaxies using high-resolution hydrodynamical simulations performed with two different codes in order to mitigate numerical artifacts. We find that massive star clusters in the range of $\sim 10^{5.5} - 10^{7.5}\, {\rm{M_{\odot}}}$ form preferentially in the highly-shocked regions produced by galaxy interactions. The nascent cluster-forming clouds have high gas pressures in the range of $P/k \sim 10^8 - 10^{12}\, \rm{K}\,\rm{cm^{-3}}$, which is $\sim 10^4 - 10^8$ times higher than the typical pressure of the interstellar medium but consistent with recent observations of a pre-super star cluster cloud in the Antennae Galaxies. Furthermore, these massive star clusters have quasi-lognormal initial mass functions with a peak around $\sim 10^{6}\, {\rm{M_{\odot}}}$. The number of clusters declines with time due to destructive processes, but the shape and the peak of the mass functions do not change significantly during the course of galaxy collisions. Our results suggest that gas-rich galaxy mergers may provide a favorable environment for the formation of massive star clusters such as globular clusters, and that the lognormal mass functions and the unique peak may originate from the extreme high-pressure conditions of the birth clouds and may survive the dynamical evolution.

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arXiv2017

Unveiling the Role of the Magnetic Field at the Smallest Scales of Star Formation

We report ALMA observations of polarized dust emission from the protostellar source Ser-emb 8 at a linear resolution of 140 au. Assuming models of dust-grain alignment hold, the observed polarization pattern gives a projected view of the magnetic field structure in this source. Contrary to expectations based on models of strongly magnetized star formation, the magnetic field in Ser-emb 8 does not exhibit an hourglass morphology. Combining the new ALMA data with previous observational studies, we can connect magnetic field structure from protostellar core (~80,000 au) to disk (~100 au) scales. We compare our observations with four magnetohydrodynamic gravo-turbulence simulations made with the AREPO code that have initial conditions ranging from super-Alfvénic (weakly magnetized) to sub-Alfvénic (strongly magnetized). These simulations achieve the spatial dynamic range necessary to resolve the collapse of protostars from the parsec scale of star-forming clouds down to the ~100 au scale probed by ALMA. Only in the very strongly magnetized simulation do we see both the preservation of the field direction from cloud to disk scales and an hourglass-shaped field at < 1000 au scales. We conduct an analysis of the relative orientation of the magnetic field and the density structure in both the Ser-emb 8 ALMA observations and the synthetic observations of the four AREPO simulations. We conclude that the Ser-emb 8 data are most similar to the weakly magnetized simulations, which exhibit random alignment, in contrast to the strongly magnetized simulation, where the magnetic field plays a role in shaping the density structure in the source. In the weak-field case, it is turbulence -- not the magnetic field -- that shapes the material that forms the protostar, highlighting the dominant role that turbulence can play across many orders of magnitude in spatial scale.

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arXiv2017

Gas Cooling in Hydrodynamic Simulations with An Exact Time Integration Scheme

We implement and test the exact time integration method proposed by Townsend 2009 for gas cooling in cosmological hydrodynamic simulations. The errors using this time integrator for the internal energy are limited by the resolution of the cooling tables and are insensitive to the size of the timestep, improving accuracy relative to explicit or implicit schemes when the cooling time is short. We compare results with different time integrators for gas cooling in cosmological hydrodynamic simulations. We find that the temperature of the gas in filaments before accreting into dark matter halos to form stars, obtained with the exact cooling integration, lies close to the equilibrium where radiative cooling balances heating from the UV background. For comparison, the gas temperature without the exact integrator shows substantial deviations from the equilibrium relation. Galaxy stellar masses with the exact cooling technique agree reasonably well, but are systematically lower than the results obtained by the other integration schemes, reducing the need for feedback to suppress star formation. Our implementation of the exact cooling technique is provided and can be easily incorporated into any hydrodynamic code.

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arXiv2017

The Gravitational Wave Background from Massive Black Hole Binaries in Illustris: spectral features and time to detection with pulsar timing arrays

Pulsar Timing Arrays (PTA) around the world are using the incredible consistency of millisecond pulsars to measure low frequency gravitational waves from (super)Massive Black Hole (MBH) binaries. We use comprehensive MBH merger models based on cosmological hydrodynamic simulations to predict the spectrum of the stochastic Gravitational-Wave Background (GWB). We use real Time-of-Arrival (TOA) specifications from the European, NANOGrav, Parkes, and International PTA (IPTA) to calculate realistic times to detection of the GWB across a wide range of model parameters. In addition to exploring the parameter space of environmental hardening processes (in particular: stellar scattering efficiencies), we have expanded our models to include eccentric binary evolution which can have a strong effect on the GWB spectrum. Our models show that strong stellar scattering and high characteristic eccentricities enhance the GWB strain amplitude near the PTA sensitive "sweet-spot" (near the frequency $f = 1 \, \mathrm{yr}^{-1}$), slightly improving detection prospects in these cases. While the GWB $amplitude$ is degenerate between cosmological and environmental parameters, the location of a spectral turnover at low frequencies ($f \lesssim 0.1 \, \mathrm{yr}^{-1}$) is strongly indicative of environmental coupling. At high frequencies ($f\gtrsim 1 \, \mathrm{yr}^{-1}$), the GWB spectral index can be used to infer the number density of sources and possibly their eccentricity distribution. Even with merger models that use pessimistic environmental and eccentricity parameters, if the current rate of PTA expansion continues, we find that the International PTA is highly likely to make a detection within about 10 years.

astro-ph.HE↗

arXiv2017

The inner structure of early-type galaxies in the Illustris simulation

Early-type galaxies provide unique tests for the predictions of the cold dark matter cosmology and the baryonic physics assumptions entering models for galaxy formation. In this work, we use the Illustris simulation to study correlations of three main properties of early-type galaxies, namely, the stellar orbital anisotropies, the central dark matter fractions and the central radial density slopes, as well as their redshift evolution since $z=1.0$. We find that lower-mass galaxies or galaxies at higher redshift tend to be bluer in rest-frame colour, have higher central gas fractions, and feature more tangentially anisotropic orbits and steeper central density slopes than their higher-mass or lower-redshift counterparts, respectively. The projected central dark matter fraction within the effective radius shows a very mild mass dependence but positively correlates with galaxy effective radii due to the aperture effect. The central density slopes obtained by combining strong lensing measurements with single aperture kinematics are found to differ from the true density slopes. We identify systematic biases in this measurement to be due to two common modelling assumptions, isotropic stellar orbital distributions and power-law density profiles. We also compare the properties of early-type galaxies in Illustris to those from existing galaxy and strong lensing surveys, we find in general broad agreement but also some tension, which poses a potential challenge to the stellar formation and feedback models adopted by the simulation.

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arXiv2017

Lens galaxies in the Illustris simulation: power-law models and the bias of the Hubble constant from time-delays

A power-law density model, i.e., $ρ(r) \propto r^{-γ'}$ has been commonly employed in strong gravitational lensing studies, including the so-called time-delay technique used to infer the Hubble constant $H_0$. However, since the radial scale at which strong lensing features are formed corresponds to the transition from the dominance of baryonic matter to dark matter, there is no known reason why galaxies should follow a power law in density. The assumption of a power law artificially breaks the mass-sheet degeneracy, a well-known invariance transformation in gravitational lensing which affects the product of Hubble constant and time delay and can therefore cause a bias in the determination of $H_0$ from the time-delay technique. In this paper, we use the Illustris hydrodynamical simulations to estimate the amplitude of this bias, and to understand how it is related to observational properties of galaxies. Investigating a large sample of Illustris galaxies that have velocity dispersion $σ_{SIE}$>160 km/s at redshifts below $z=1$, we find that the bias on $H_0$ introduced by the power-law assumption can reach 20%-50%, with a scatter of $10\%-30\%$ (rms). However, we find that by selecting galaxies with an inferred power-law model slope close to isothermal, it is possible to reduce the bias on $H_0$ to <5%, and the scatter to <10%. This could potentially be used to form less biased statistical samples for $H_0$ measurements in the upcoming large survey era.

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