# Posts Tagged mass function

## Recent Postings from mass function

### The Stellar Initial Mass Function in Early-Type Galaxies From Absorption Line Spectroscopy. IV. A Super-Salpeter IMF in the center of NGC 1407 from Non-Parametric Models

It is now well-established that the stellar initial mass function (IMF) can be determined from the absorption line spectra of old stellar systems, and this has been used to measure the IMF and its variation across the early-type galaxy population. Previous work focused on measuring the slope of the IMF over one or more stellar mass intervals, implicitly assuming that this is a good description of the IMF and that the IMF has a universal low-mass cutoff. In this work we consider more flexible IMFs, including two-component power-laws with a variable low-mass cutoff and a general non-parametric model. We demonstrate with mock spectra that the detailed shape of the IMF can be accurately recovered as long as the data quality are high (S/N$\gtrsim300$) and cover a wide wavelength range (0.4um-1.0um). We apply these flexible IMF models to a high S/N spectrum of the center of the massive elliptical galaxy NGC 1407. Fitting the spectrum with non-parametric IMFs, we find that the IMF in the center shows a continuous rise extending toward the hydrogen-burning limit, with a behavior that is well-approximated by a power-law with an index of -2.7. These results provide strong evidence for the existence of extreme (super-Salpeter) IMFs in the cores of massive galaxies.

### The Stellar Initial Mass Function in Early-Type Galaxies from Absorption Line Spectroscopy. III. Radial Gradients [Replacement]

There is good evidence that the centers of massive early-type galaxies have a bottom-heavy stellar initial mass function (IMF) compared to the IMF of the Milky Way. Here we study the radial variation of the IMF within such galaxies, using a combination of high quality Keck spectroscopy and a new suite of stellar population synthesis models that cover a wide range in metallicity. As in the previous studies in this series, the models are fitted directly to the spectra and treat all elemental abundance ratios as free parameters. Using newly obtained spectroscopy for six galaxies, including deep data extending to ~1Re for the galaxies NGC1407, NGC1600, and NGC2695, we find that the IMF varies strongly with galactocentric radius. For all six galaxies the IMF is bottom-heavy in the central regions, with average mass-to-light ratio "mismatch" parameter a~2.5 at r=0. The IMF rapidly becomes more bottom-light with increasing radius, flattening off near the Milky Way value (a~1.1) at R>0.4Re. A consequence is that the luminosity-weighted average IMF depends on the measurement aperture: within R=Re we find <a>=1.3-1.5, consistent with recent lensing and dynamical results from SLACS and ATLAS-3D. Our results are also consistent with several earlier studies that were based on analyses of radial gradients of line indices. The observed IMF gradients support galaxy formation models in which the central regions of massive galaxies had a different formation history than their outer parts. Finally, we make use of the high signal-to-noise central spectra of NGC1407 and NGC2695 to demonstrate that we are indeed measuring IMF effects, not abundance effects.

### The Stellar Initial Mass Function in Early-Type Galaxies from Absorption Line Spectroscopy. III. Radial Gradients

There is good evidence that the centers of massive early-type galaxies have a bottom-heavy stellar initial mass function (IMF) compared to the IMF of the Milky Way. Here we study the radial variation of the IMF within such galaxies, using a combination of high quality Keck spectroscopy and a new suite of stellar population synthesis models that cover a wide range in [Z/H]. As in the previous studies in this series, the models are fit directly to the spectra and treat all elemental abundance ratios as free parameters. Using newly obtained spectroscopy for six galaxies, including deep data extending to ~1Re for the galaxies NGC1407, NGC1600, and NGC2695, we find that the IMF strongly varies with galactocentric radius. For all galaxies the IMF is bottom-heavy in the central regions, with average "mismatch" parameter a~2.5 at r=0. The IMF rapidly becomes more bottom-light with increasing radius, flattening off near the Milky Way value (a~1.1) at R>0.4Re. A consequence is that the luminosity-weighted average IMF depends on the measurement aperture: within R=Re we find <a>=1.3-1.5, that is, the IMF of even the most massive galaxies is only mildly bottom-heavy within the half-light radius. Our results are consistent with several earlier studies that were based on analyses of radial gradients of line indices, and support galaxy formation models in which the central regions of massive galaxies had a different formation history than their outer parts. Finally, we make use of the high signal-to-noise central spectra of NGC1407 and NGC2695 to demonstrate that we are indeed measuring IMF effects, not abundance effects.

### Lensing is Low: Cosmology, Galaxy Formation, or New Physics?

We present high signal-to-noise galaxy-galaxy lensing measurements of the BOSS CMASS sample using 250 square degrees of weak lensing data from CFHTLenS and CS82. We compare this signal with predictions from mock catalogs trained to match observables including the stellar mass function and the projected and two dimensional clustering of CMASS. We show that the clustering of CMASS, together with standard models of the galaxy-halo connection, robustly predicts a lensing signal that is 20-40% larger than observed. Detailed tests show that our results are robust to a variety of systematic effects. Lowering the value of $S_{\rm 8}=\sigma_{\rm 8} \sqrt{\Omega_{\rm m}/0.3}$ compared to Planck2015 reconciles the lensing with clustering. However, given the scale of our measurement ($r<10$ $h^{-1}$ Mpc), other effects may also be at play and need to be taken into consideration. We explore the impact of baryon physics, assembly bias, massive neutrinos, and modifications to general relativity on $\Delta\Sigma$ and show that several of these effects may be non-negligible given the precision of our measurement. Disentangling cosmological effects from the details of the galaxy-halo connection, the effects of baryons, and massive neutrinos, is the next challenge facing joint lensing and clustering analyses. This is especially true in the context of large galaxy samples from Baryon Acoustic Oscillation surveys with precise measurements but complex selection functions.

### On the origin of the Schechter-like mass function of young star clusters in disk galaxies

The mass function of freshly formed star clusters is empirically often described as a power law. However the cluster mass function of populations of young clusters over the scale of a galaxy has been found to be described by a Schechter-function. Here we address this apparent discrepancy. We assume that in an annulus of an isolated self- regulated radially-exponential axially-symmetric disk galaxy, the local mass function of very young (embedded) clusters is a power law with an upper mass limit which depends on the local star formation rate density. Radial integration of this mass function yields a galaxy-wide embedded cluster mass function. This integrated embedded cluster mass function has a Schechter-type form, which results from the addition of many low mass clusters forming at all galactocentric distances and rarer massive clusters only forming close to the center of the galaxy.

### A halo model for cosmological neutral hydrogen : abundances and clustering

We extend the results of previous analyses towards constraining the abundance and clustering of post-reionization ($z \sim 0-5$) neutral hydrogen (HI) systems using a halo model framework. We work with a comprehensive HI dataset including the small-scale clustering, column density and mass function of HI galaxies at low redshifts, intensity mapping measurements at intermediate redshifts and the UV/optical observations of Damped Lyman Alpha (DLA) systems at higher redshifts. We use a Markov Chain Monte Carlo (MCMC) approach to constrain the parameters of the best-fitting models, both for the HI-halo mass relation and the HI radial density profile. We find that a radial exponential profile results in a good fit to the low-redshift HI observations, including the clustering and the column density distribution. The form of the profile is also found to match the high-redshift DLA observations, when used in combination with a three-parameter HI-halo mass relation and a redshift evolution in the HI concentration. The halo model predictions are in good agreement with the observed HI surface density profiles of low-redshift galaxies, and the general trends in the the impact parameter and covering fraction observations of high-redshift DLAs. We provide convenient tables summarizing the best-fit halo model predictions.

### Small-scale galaxy clustering in the EAGLE simulation

We study present-day galaxy clustering in the EAGLE cosmological hydrodynamical simulation. EAGLE's galaxy formation parameters were calibrated to reproduce the redshift $z = 0.1$ galaxy stellar mass function, and the simulation also reproduces galaxy colours well. The simulation volume is too small to correctly sample large-scale fluctuations and we therefore concentrate on scales smaller than a few megaparsecs. We find very good agreement with observed clustering measurements from the Galaxy And Mass Assembly (GAMA) survey, when galaxies are binned by stellar mass, colour, or luminosity. However, low-mass red-galaxies are clustered too strongly, which is at least partly due to limited numerical resolution. Apart from this limitation, we conclude that EAGLE galaxies inhabit similar dark matter haloes as observed GAMA galaxies, and that the radial distribution of satellite galaxies as function of stellar mass and colour is similar to that observed as well.

### Verifying the consistency relation for the scale-dependent bias from local primordial non-Gaussianity

We measure the large-scale bias of dark matter halos in simulations with non-Gaussian initial conditions of the local type, and compare this bias to the response of the mass function to a change in the primordial amplitude of fluctuations. The two are found to be consistent, as expected from physical arguments, for three halo-finder algorithms which use different Spherical Overdensity (SO) and Friends-of-Friends (FoF) methods. On the other hand, we find that the commonly used prediction for universal mass functions, that the scale-dependent bias is proportional to the first-order Gaussian Lagrangian bias, does not yield a good agreement with the measurements. For all halo finders, high-mass halos show a non-Gaussian bias suppressed by 10-15% relative to the universal mass function prediction. For SO halos, this deviation changes sign at low masses, where the non-Gaussian bias becomes larger than the universal prediction.

### Discovery of a new accreting millisecond X-ray pulsar in the globular cluster NGC 2808

We report on the discovery of coherent pulsations at a period of 2.9 ms from the X-ray transient MAXI J0911-655 in the globular cluster NGC 2808. We observed X-ray pulsations at a frequency of $\sim339.97$ Hz in three different observations of the source performed with XMM-Newton and NuSTAR during the source outburst. This newly discovered accreting millisecond pulsar is part of an ultra-compact binary system characterised by an orbital period of $44.3$ minutes and a projected semi-major axis of $\sim17.6$ lt-ms. Based on the mass function we estimate a minimum companion mass of 0.024 M$_{\odot}$, which assumes a neutron star mass of 1.4 M$_{\odot}$ and a maximum inclination angle of $75^{\circ}$ (derived from the lack of eclipses and dips in the light-curve of the source). We find that the companion star's Roche-Lobe could either be filled by a hot ($5\times 10^{6}$ K) pure helium white dwarf with a 0.028 M$_{\odot}$ mass (implying $i\simeq58^{\circ}$) or an old (>5 Gyr) brown dwarf with metallicity abundances between solar/sub-solar and mass ranging in the interval 0.065$-$0.085 M$_{\odot}$ (16 < $i$ < 21). During the outburst the broad-band energy spectra are well described by a superposition of a weak black-body component (kT$\sim$ 0.5 keV) and a hard cutoff power-law with photon index $\Gamma \sim$ 1.7 and cut-off at a temperature kT$_e\sim$ 130 keV. Up to the latest Swift-XRT observation performed on 2016 July 19 the source has been observed in outburst for almost 150 days, which makes MAXI J0911-655 the second accreting millisecond X-ray pulsar with outburst duration longer than 100 days.

### Initial mass function of planetesimals formed by the streaming instability [Replacement]

The streaming instability is a mechanism to concentrate solid particles into overdense filaments that undergo gravitational collapse and form planetesimals. However, it remains unclear how the initial mass function of these planetesimals depends on the box dimensions of numerical simulations. To resolve this, we perform simulations of planetesimal formation with the largest box dimensions to date, allowing planetesimals to form simultaneously in multiple filaments that can only emerge within such large simulation boxes. In our simulations, planetesimals with sizes between 80 km and several hundred kilometers form. We find that a power law with a rather shallow exponential cutoff at the high-mass end represents the cumulative birth mass function better than an integrated power law. The steepness of the exponential cutoff is largely independent of box dimensions and resolution, while the exponent of the power law is not constrained at the resolutions we employ. Moreover, we find that the characteristic mass scale of the exponential cutoff correlates with the mass budget in each filament. Together with previous studies of high-resolution simulations with small box domains, our results therefore imply that the cumulative birth mass function of planetesimals is consistent with an exponentially tapered power law with a power-law exponent of approximately -1.6 and a steepness of the exponential cutoff in the range of 0.3-0.4.

### Initial mass function of planetesimals formed by the streaming instability

The streaming instability is a mechanism to concentrate solid particles into overdense filaments that undergo gravitational collapse and form planetesimals. However, it remains unclear how the initial mass function of these planetesimals depends on the box dimensions of numerical simulations. To resolve this, we perform simulations of planetesimal formation with the largest box dimensions to date, allowing planetesimals to form simultaneously in multiple filaments that can only emerge within such large simulation boxes. In our simulations planetesimals with sizes between 80 km and several hundred kilometers form. We find that a power law with a rather shallow exponential cutoff at the high-mass end represents the cumulative birth mass function better than an integrated power law. The steepness of the exponential cutoff is largely independent of box dimensions and resolution, while the exponent of the power law is not constrained at the resolutions we employ. Moreover, we find that the characteristic mass scale of the exponential cutoff correlates with the mass budget in each filament. Together with previous high-resolution simulations of small box domains presented in Johansen et al. (2015) and Simon et al. (2016), our results therefore imply that the cumulative birth mass function of planetesimals is consistent with an exponentially tapered power law with a power-law exponent of approximately -1.6 and a steepness of the exponential cutoff in the range of 0.3-0.4.

### The Black Hole Mass Function from Gravitational Wave Measurements

We examine how future gravitational-wave measurements from merging black holes (BHs) can be used to infer the shape of the black-hole mass function, with important implications for the study of star formation and evolution and the properties of binary BHs. We model the mass function as a power law, inherited from the stellar initial mass function, and introduce lower and upper mass cutoff parameterizations in order to probe the minimum and maximum BH masses allowed by stellar evolution, respectively. We initially focus on the heavier BH in each binary, to minimize model dependence. Taking into account the experimental noise, the mass measurement errors and the uncertainty in the redshift-dependence of the merger rate, we show that the mass function parameters, as well as the total rate of merger events, can be measured to <10% accuracy within a few years of advanced LIGO observations at its design sensitivity. This can be used to address important open questions such as the upper limit on the stellar mass which allows for BH formation and to confirm or refute the currently observed mass gap between neutron stars and BHs. In order to glean information on the progenitors of the merging BH binaries, we then advocate the study of the two-dimensional mass distribution to constrain parameters that describe the two-body system, such as the mass ratio between the two BHs, in addition to the merger rate and mass function parameters. We argue that several years of data collection can efficiently probe models of binary formation, and show, as an example, that the hypothesis that some gravitational-wave events may involve primordial black holes can be tested. Finally, we point out that in order to maximize the constraining power of the data, it may be worthwhile to lower the signal-to-noise threshold imposed on each candidate event and amass a larger statistical ensemble of BH mergers.

### Radial gradients in initial mass function sensitive absorption features in the Coma brightest cluster galaxies

Using the Oxford Short Wavelength Integral Field specTrograph (SWIFT), we trace radial variations of initial mass function (IMF) sensitive absorption features of three galaxies in the Coma cluster. We obtain resolved spectroscopy of the central 5kpc for the two central brightest-cluster galaxies (BCGs) NGC4889, NGC4874, and the BCG in the south-west group NGC4839, as well as unresolved data for NGC4873 as a low-$\sigma_*$ control. We present radial measurements of the IMF-sensitive features sodium NaI$_{\rm{SDSS}}$, calcium triplet CaT and iron-hydride FeH0.99, along with the magnesium MgI0.88 and titanium oxide TiO0.89 features. We employ two separate methods for both telluric correction and sky-subtraction around the faint FeH feature to verify our analysis. Within NGC4889 we find strong gradients of NaI$_{\rm{SDSS}}$ and CaT but a flat FeH profile, which from comparing to stellar population synthesis models, suggests an old, $\alpha$-enhanced population with a Chabrier, or even bottom-light IMF. The age and abundance is in line with previous studies but the normal IMF is in contrast to recent results suggesting an increased IMF slope with increased velocity dispersion. We measure flat NaI$_{\rm{SDSS}}$ and FeH profiles within NGC4874 and determine an old, possibly slightly $\alpha$-enhanced and Chabrier IMF population. We find an $\alpha$-enhanced, Chabrier IMF population in NGC4873. Within NGC4839 we measure both strong NaI$_{\rm{SDSS}}$ and strong FeH, although with a large systematic uncertainty, suggesting a possible heavier IMF. The IMFs we infer for these galaxies are supported by published dynamical modelling. We stress that IMF constraints should be corroborated by further spectral coverage and independent methods on a galaxy-by-galaxy basis.

### Radial gradients in initial mass function sensitive absorption features in the Coma brightest cluster galaxies [Replacement]

Using the Oxford Short Wavelength Integral Field specTrograph (SWIFT), we trace radial variations of initial mass function (IMF) sensitive absorption features of three galaxies in the Coma cluster. We obtain resolved spectroscopy of the central 5kpc for the two central brightest-cluster galaxies (BCGs) NGC4889, NGC4874, and the BCG in the south-west group NGC4839, as well as unresolved data for NGC4873 as a low-$\sigma_*$ control. We present radial measurements of the IMF-sensitive features sodium NaI$_{\rm{SDSS}}$, calcium triplet CaT and iron-hydride FeH0.99, along with the magnesium MgI0.88 and titanium oxide TiO0.89 features. We employ two separate methods for both telluric correction and sky-subtraction around the faint FeH feature to verify our analysis. Within NGC4889 we find strong gradients of NaI$_{\rm{SDSS}}$ and CaT but a flat FeH profile, which from comparing to stellar population synthesis models, suggests an old, $\alpha$-enhanced population with a Chabrier, or even bottom-light IMF. The age and abundance is in line with previous studies but the normal IMF is in contrast to recent results suggesting an increased IMF slope with increased velocity dispersion. We measure flat NaI$_{\rm{SDSS}}$ and FeH profiles within NGC4874 and determine an old, possibly slightly $\alpha$-enhanced and Chabrier IMF population. We find an $\alpha$-enhanced, Chabrier IMF population in NGC4873. Within NGC4839 we measure both strong NaI$_{\rm{SDSS}}$ and strong FeH, although with a large systematic uncertainty, suggesting a possible heavier IMF. The IMFs we infer for these galaxies are supported by published dynamical modelling. We stress that IMF constraints should be corroborated by further spectral coverage and independent methods on a galaxy-by-galaxy basis.

### Constraining the Thin Disc Initial Mass Function using Galactic Classical Cepheids

Context: The Initial Mass Function (IMF) plays a crucial role on galaxy evolution and its implications on star formation theory make it a milestone for the next decade. It is in the intermediate and high mass ranges where the uncertainties of the IMF are larger. This is a major subject of debate and analysis both for Galactic and extragalactic science. Aims: Our goal is to constrain the IMF of the Galactic thin disc population using both Galactic Classical Cepheids and Tycho-2 data. Methods: For the first time the Besan\c{c}on Galaxy Model (BGM) has been used to characterise the Galactic population of the Classical Cepheids. We have modified the age configuration in the youngest populations of the BGM thin disc model to avoid artificial discontinuities in the age distribution of the simulated Cepheids. Three statistical methods, optimized for different mass ranges, have been developed and applied to search for the best IMF that fits the observations. This strategy allows us to quantify variations in the Star Formation History (SFH), the stellar density at Sun position and the thin disc radial scale length. A rigorous treatment of unresolved multiple stellar systems has been undertaken adopting a spatial resolution according to the catalogues used. Results: For intermediate masses, our study favours a composite field-star IMF slope of $\alpha=3.2$ for the local thin disc, excluding flatter values such as the Salpeter IMF ($\alpha=2.35$). Moreover, a constant Star Formation History is definitively excluded, the three statistical methods considered here show that it is inconsistent with the observational data. Conclusions: Using field stars and Galactic Classical Cepheids, we have found, above $1M_\odot$, an IMF steeper than the canonical stellar IMF of associations and young clusters. This result is consistent with the predictions of the Integrated Galactic IMF.

### Does turbulence determine the initial mass function?

We test the hypothesis that the initial mass function (IMF) is determined by the density probability distribution function (PDF) produced by supersonic turbulence. We compare 14 simulations of star cluster formation in 50 solar mass molecular cloud cores where the initial turbulence contains either purely solenoidal or purely compressive modes, in each case resolving fragmentation to the opacity limit to determine the resultant IMF. We find statistically indistinguishable IMFs between the two sets of calculations, despite a factor of two difference in the star formation rate and in the standard deviation of $\log(\rho)$. This suggests that the density PDF, while determining the star formation rate, is not the primary driver of the IMF.

### Does turbulence determine the initial mass function? [Replacement]

We test the hypothesis that the initial mass function (IMF) is determined by the density probability distribution function (PDF) produced by supersonic turbulence. We compare 14 simulations of star cluster formation in 50 solar mass molecular cloud cores where the initial turbulence contains either purely solenoidal or purely compressive modes, in each case resolving fragmentation to the opacity limit to determine the resultant IMF. We find statistically indistinguishable IMFs between the two sets of calculations, despite a factor of two difference in the star formation rate and in the standard deviation of $\log(\rho)$. This suggests that the density PDF, while determining the star formation rate, is not the primary driver of the IMF.

### A simple method to convert sink particles into stars

Hydrodynamical simulations of star formation often do not possess the dynamic range needed to fully resolve the build-up of individual stars and star clusters, and thus have to resort to subgrid models. A popular way to do this is by introducing Lagrangian sink particles, which replace contracting high density regions at the point where the resolution limit is reached. A common problem then is how to assign fundamental stellar properties to sink particles, such as the distribution of stellar masses. We present a new and simple statistical method to assign stellar contents to sink particles. Once the stellar content is specified, it can be used to determine a sink particle's radiative output, supernovae rate or other feedback parameters that may be required in the calculations. Advantages of our method are (i) it is simple to implement, (ii) it guarantees that the obtained stellar populations are good samples of the initial mass function, (iii) it can easily deal with infalling mass accreted at later times, and (iv) it does not put restrictions on the sink particles' masses in order to be used. The method works very well for sink particles that represent large star clusters and for which the stellar mass function is well sampled, but can also handle the transition to sink particles that represent a small number of stars.

### The Mass Function of Unprocessed Dark Matter Halos and Merger Tree Branching Rates

A common approach in semi-analytic modeling of galaxy formation is to construct Monte Carlo realizations of merger histories of dark matter halos whose masses are sampled from a halo mass function. Both the mass function itself, and the merger rates used to construct merging histories are calibrated to N-body simulations. Typically, "backsplash" halos (those which were once subhalos within a larger halo, but which have since moved outside of the halo) are counted in both the halo mass function, and in the merger rates (or, equivalently, progenitor mass functions). This leads to a double-counting of mass in Monte Carlo merger histories which will bias results relative to N-body results. We measure halo mass functions and merger rates with this double-counting removed in a large, cosmological N-body simulation with cosmological parameters consistent with current constraints. Furthermore, we account for the inherently noisy nature of N-body halo mass estimates when fitting functions to N-body data, and show that ignoring these errors leads to a significant systematic bias given the precision statistics available from state-of-the-art N-body cosmological simulations.

### The properties of the Malin 1 galaxy giant disk: A panchromatic view from the NGVS and GUViCS surveys

Low surface brightness galaxies (LSBGs) represent a significant percentage of local galaxies but their formation and evolution remain elusive. They may hold crucial information for our understanding of many key issues (i.e., census of baryonic and dark matter, star formation in the low density regime, mass function). The most massive examples - the so called giant LSBGs - can be as massive as the Milky Way, but with this mass being distributed in a much larger disk. Malin 1 is an iconic giant LSBG, perhaps the largest disk galaxy known. We attempt to bring new insights on its structure and evolution on the basis of new images covering a wide range in wavelength. We have computed surface brightness profiles (and average surface brightnesses in 16 regions of interest), in six photometric bands (FUV, NUV, u, g, i, z). We compared these data to various models, testing a variety of assumptions concerning the formation and evolution of Malin 1. We find that the surface brightness and color profiles can be reproduced by a long and quiet star-formation history due to the low surface density; no significant event, such as a collision, is necessary. Such quiet star formation across the giant disk is obtained in a disk model calibrated for the Milky Way, but with an angular momentum approximately 20 times larger. Signs of small variations of the star-formation history are indicated by the diversity of ages found when different regions within the galaxy are intercompared.For the first time, panchromatic images of Malin 1 are used to constrain the stellar populations and the history of this iconic example among giant LSBGs. Based on our model, the extreme disk of Malin 1 is found to have a long history of relatively low star formation (about 2 Msun/yr). Our model allows us to make predictions on its stellar mass and metallicity.

### Constraints on galaxy formation models from the galaxy stellar mass function and its evolution

We explore the parameter space of the semi-analytic galaxy formation model GALFORM, studying the constraints imposed by measurements of the galaxy stellar mass function (GSMF) and its evolution. We use the Bayesian Emulator method to quickly eliminate vast implausible volumes of the parameter space and zoom in on the most interesting regions, allowing us to identify a set of models that match the observational data within the model uncertainties. We find that the GSMF strongly constrains parameters related to the quiescent star formation in discs, stellar and AGN feedback and the threshold for disc instabilities, but more weakly restricts other parameters. Constraining the model using the local data alone does not usually select models that match the evolution of the mass function well. Nevertheless, we show that a small subset of models provides an acceptable match to GSMF data out to redshift 1.5, without introducing an explicit redshift dependence of feedback parameters. We explore the physical significance of the parameters of these models, in particular exploring whether the model can provide a better description if the mass loading of the galactic winds generated by starbursts ($\beta_{0,\text{burst}}$) and quiescent disks ($\beta_{0,\text{disc}}$) is different. Performing a principal component analysis of the plausible volume of the parameter space, we write a set of relations between parameters obeyed by plausible models with respect to the GSMF evolution. We find that while $\beta_{0,\text{disc}}$ is strongly constrained by GSMF evolution data, constraints on $\beta_{0,\text{burst}}$ are weak. We discuss the implications of these results.

### Star Formation Quenching Timescale of Central Galaxies in a Hierarchical Universe

Central galaxies make up the majority of the galaxy population, including the majority of the quiescent population at $\mathcal{M}_* > 10^{10}\mathrm{M}_\odot$. Thus, the mechanism(s) responsible for quenching central galaxies plays a crucial role in galaxy evolution as whole. We combine a high resolution cosmological $N$-body simulation with observed evolutionary trends of the "star formation main sequence," quiescent fraction, and stellar mass function at $z < 1$ to construct a model that statistically tracks the star formation histories and quenching of central galaxies. Comparing this model to the distribution of central galaxy star formation rates in a group catalog of the SDSS Data Release 7, we constrain the timescales over which physical processes cease star formation in central galaxies. Over the stellar mass range $10^{9.5}$ to $10^{11} \mathrm{M}_\odot$ we infer quenching e-folding times that span $1.5$ to $0.5\; \mathrm{Gyr}$ with more massive central galaxies quenching faster. For $\mathcal{M}_* = 10^{10.5}\mathrm{M}_\odot$, this implies a total migration time of $\sim 4~\mathrm{Gyrs}$ from the star formation main sequence to quiescence. Compared to satellites, central galaxies take $\sim 2~\mathrm{Gyrs}$ longer to quench their star formation, suggesting that different mechanisms are responsible for quenching centrals versus satellites. Finally, the central galaxy quenching timescale we infer provides key constraints for proposed star formation quenching mechanisms. Our timescale is generally consistent with gas depletion timescales predicted by quenching through strangulation. However, the exact physical mechanism(s) responsible for this still remain unclear.

### High proper motion objects towards the inner Milky Way: characterisation of newly identified nearby stars from the VISTA Variables in the Via Lactea Survey [Replacement]

The census of the Solar neighbourhood is still incomplete, as demonstrated by recent discoveries of many objects within 5-10 pc from the Sun. The area around the mid-plane and bulge of the Milky Way presents the most difficulties in searches for such nearby objects, and is therefore deficient in the known population. This is largely due to high stellar densities encountered. Spectroscopic, photometric and kinematic characterization of these objects allows better understand the local mass function, the binary fraction, and provides new interesting targets for more detailed studies. We report the spectroscopic follow-up and characterisation of 12 bright high PM objects, identified from the VISTA Variables in Via Lactea survey (VVV). We used the 1.9-m telescope of the South African Astronomical Observatory (SAAO) for low-resolution optical spectroscopy and spectral classification, and the MPG/ESP 2.2m telescope Fiber-fed Extended Range Optical Spectrograph (FEROS) high-resolution optical spectroscopy to obtain the radial and space velocities for three of them. Six of our objects have co-moving companions. We derived optical spectral types and photometric distances, and classified all of them as K and M dwarfs within 27-264 pc of the Sun. Finally, we found that one of the sources, VVV J141421.23-602326.1 (a co-moving companion of VVV J141420.55-602337.1), appears to be a rare massive white dwarf that maybe close to the ZZ Ceti instability strip. Many of the objects in our list are interesting targets for exoplanet searches.

### The impact of mass segregation and star-formation on the rates of gravitational-wave sources from extreme mass ratio inspirals

Compact stellar objects inspiralling into massive black holes (MBHs) in galactic nuclei are some of the most promising gravitational wave (GWs) sources for next generation GW-detectors. The rates of such extreme mass ratio inspirals (EMRIs) depend on the dynamics and distribution of compact objects around the MBH. Here we study the impact of mass-segregation processes on EMRI rates. In particular, we provide the expected mass function of EMRIs, given an initial mass function of stellar BHs (SBHs), and relate it to the mass-dependent detection rate of EMRIs. We then consider the role of star formation on the distribution of compact objects and its implication on EMRI rates. We find that the existence of a wide spectrum of SBH masses lead to the overall increase of EMRI rates, and to high rates of the EMRIs from the most-massive SBHs. However, it also leads to a relative quenching of EMRI rates from lower-mass SBHs, and together produces a steep dependence of the EMRI mass function on the highest-mass SBHs. Star-formation history plays a relatively small role in determining the EMRI rates of SBHs, since most of them migrate close to the MBH through mass-segregation rather than forming in-situ. However, the EMRI rate of neutron stars can be significantly increased when they form in-situ close to the MBH, as they can inspiral before relaxation processes significantly segregates them outwards. A reverse but weaker effect of decreasing the EMRI rates from neutron stars and white dwarfs occurs when star-formation proceeds far from the MBH.

### Quintessential Scale Dependence from Separate Universe Simulations [Replacement]

By absorbing fluctuations into a local background, separate universe simulations provide a powerful technique to characterize the response of small-scale observables to the long-wavelength density fluctuations, for example those of the power spectrum and halo mass function which lead to the squeezed-limit $n$-point function and halo bias, respectively. Using quintessence dark energy as the paradigmatic example, we extend these simulation techniques to cases where non-gravitational forces in other sectors establish a Jeans scale across which the growth of density fluctuations becomes scale dependent. By characterizing the separate universes with matching background expansion histories, we show that the power spectrum and mass function responses depend on whether the long-wavelength mode is above or below the Jeans scale. Correspondingly, the squeezed bispectrum and halo bias also become scale dependent. Models of bias that are effectively local in the density field at a single epoch, initial or observed, cannot describe this effect which highlights the importance of temporal nonlocality in structure formation. Validated by these quintessence tests, our techniques are applicable to a wide range of models where the complex dynamics of additional fields affect the clustering of matter in the linear regime and it would otherwise be difficult to simulate their impact in the nonlinear regime.

### Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function [Replacement]

The recent discovery of gravitational waves from mergers of $\sim 10 \, M_{\odot}$ black hole binaries has stimulated interested in Primordial Black Hole dark matter in this mass range. Microlensing and dynamical constraints exclude all of the dark matter being in compact objects with a delta function mass function in the range $10^{-7} \lesssim M/ M_{\odot} \lesssim 10^{5}$. However it has been argued that all of the dark matter could be composed of compact objects in this range with an extended mass function. We explicitly recalculate the microlensing and dynamical constraints for compact objects with an extended mass function which replicates the PBH mass function produced by inflation models. We find that the microlensing and dynamical constraints place conflicting constraints on the width of the mass function, and do not find a mass function which satisfies both constraints.

### Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function

The recent discovery of gravitational waves from mergers of $\sim 10 \, M_{\odot}$ black hole binaries has stimulated interested in Primordial Black Hole dark matter in this mass range. Microlensing and dynamical constraints exclude all of the dark matter being in compact objects with a delta function mass function in the range $10^{-7} \lesssim M/ M_{\odot} \lesssim 10^{5}$. However it has been argued that all of the dark matter could be composed of compact objects in this range with an extended mass function. We explicitly recalculate the microlensing and dynamical constraints for compact objects with an extended mass function which replicates the PBH mass function produced by inflation models. We find that the microlensing and dynamical constraints place conflicting constraints on the width of the mass function, and do not find a mass function which satisfies both constraints.

### Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function [Cross-Listing]

The recent discovery of gravitational waves from mergers of $\sim 10 \, M_{\odot}$ black hole binaries has stimulated interested in Primordial Black Hole dark matter in this mass range. Microlensing and dynamical constraints exclude all of the dark matter being in compact objects with a delta function mass function in the range $10^{-7} \lesssim M/ M_{\odot} \lesssim 10^{5}$. However it has been argued that all of the dark matter could be composed of compact objects in this range with an extended mass function. We explicitly recalculate the microlensing and dynamical constraints for compact objects with an extended mass function which replicates the PBH mass function produced by inflation models. We find that the microlensing and dynamical constraints place conflicting constraints on the width of the mass function, and do not find a mass function which satisfies both constraints.

### The long-term dynamical evolution of disc-fragmented multiple systems in the Solar Neighborhood

The origin of very low-mass hydrogen-burning stars, brown dwarfs, and planetary-mass objects at the low-mass end of the initial mass function is not yet fully understood. Gravitational fragmentation of circumstellar discs provides a possible mechanism for the formation of such low-mass objects. The kinematic and binary properties of very low-mass objects formed through disc fragmentation at early times (< 10 Myr) were discussed in Li et al. (2015). In this paper we extend the analysis by following the long-term evolution of disc-fragmented systems, up to an age of 10 Gyr, covering the ages of the stellar and substellar population in the Galactic field. We find that the systems continue to decay, although the rates at which companions escape or collide with each other are substantially lower than during the first 10 Myr, and that dynamical evolution is limited beyond 1 Gyr. By t = 10 Gyr, about one third of the host stars is single, and more than half have only one companion left. Most of the other systems have two companions left that orbit their host star in widely separated orbits. A small fraction of companions have formed binaries that orbit the host star in a hierarchical triple configuration. The majority of such double companion systems have internal orbits that are retrograde with respect to their orbits around their host stars. Our simulations allow a comparison between the predicted outcomes of disc-fragmentation with the observed low-mass hydrogen-burning stars, brown dwarfs, and planetary-mass objects in the Solar neighborhood. Imaging and radial velocity surveys for faint binary companions among nearby stars are necessary for verification or rejection for the formation mechanism proposed in this paper.

### Constraints on shear and rotation with massive galaxy clusters

A precise determination of the mass function is an important tool to verify cosmological predictions of the $\Lambda$CDM model and to infer more precisely the better model describing the evolution of the Universe. Galaxy clusters have been currently used to infer cosmological parameters, in particular the matter density parameter $\Omega_{\rm m}$, the matter power spectrum normalization $\sigma_8$ and the equation of state parameter $w_{\rm de}$ of the dark energy fluid. In this work, using data on massive galaxy clusters ($M>8\times 10^{14}~h^{-1}~M_{\odot}$) in the redshift range $0.05\lesssim z\lesssim 0.83$ we put constraints on the parameter $\alpha$ introduced within the formalism of the extended spherical collapse model to quantify deviations from sphericity due to shear and rotation. Since at the moment there is no physical model describing its functional shape, we assume it to be a logarithmic function of the cluster mass. By holding $\sigma_8$ fixed and restricting our analysis to a $\Lambda$CDM model, we find, at $1-\sigma$ confidence level, $\Omega_{\rm m}=0.284\pm0.0064$, $h=0.678\pm0.017$ and $\beta=0.0019^{+0.0008}_{-0.0015}$, where $\beta$ represents the slope of the parameter $\alpha$. This results translates into a $9\%$ decrement of the number of massive clusters with respect to a standard $\Lambda$CDM mass function, but better data are required to better constrain this quantity, since at the $2-\sigma$ and $3-\sigma$ confidence level we are only able to infer upper limits.

### Constraints on shear and rotation with massive galaxy clusters [Cross-Listing]

A precise determination of the mass function is an important tool to verify cosmological predictions of the $\Lambda$CDM model and to infer more precisely the better model describing the evolution of the Universe. Galaxy clusters have been currently used to infer cosmological parameters, in particular the matter density parameter $\Omega_{\rm m}$, the matter power spectrum normalization $\sigma_8$ and the equation of state parameter $w_{\rm de}$ of the dark energy fluid. In this work, using data on massive galaxy clusters ($M>8\times 10^{14}~h^{-1}~M_{\odot}$) in the redshift range $0.05\lesssim z\lesssim 0.83$ we put constraints on the parameter $\alpha$ introduced within the formalism of the extended spherical collapse model to quantify deviations from sphericity due to shear and rotation. Since at the moment there is no physical model describing its functional shape, we assume it to be a logarithmic function of the cluster mass. By holding $\sigma_8$ fixed and restricting our analysis to a $\Lambda$CDM model, we find, at $1-\sigma$ confidence level, $\Omega_{\rm m}=0.284\pm0.0064$, $h=0.678\pm0.017$ and $\beta=0.0019^{+0.0008}_{-0.0015}$, where $\beta$ represents the slope of the parameter $\alpha$. This results translates into a $9\%$ decrement of the number of massive clusters with respect to a standard $\Lambda$CDM mass function, but better data are required to better constrain this quantity, since at the $2-\sigma$ and $3-\sigma$ confidence level we are only able to infer upper limits.

### Constraints on shear and rotation with massive galaxy clusters [Replacement]

A precise determination of the mass function is an important tool to verify cosmological predictions of the $\Lambda$CDM model and to infer more precisely the better model describing the evolution of the Universe. Galaxy clusters have been currently used to infer cosmological parameters, in particular the matter density parameter $\Omega_{\rm m}$, the matter power spectrum normalization $\sigma_8$ and the equation of state parameter $w_{\rm de}$ of the dark energy fluid. In this work, using data on massive galaxy clusters ($M>8\times 10^{14}~h^{-1}~M_{\odot}$) in the redshift range $0.05\lesssim z\lesssim 0.83$ we put constraints on the parameter $\alpha$ introduced within the formalism of the extended spherical collapse model to quantify deviations from sphericity due to shear and rotation. Since at the moment there is no physical model describing its functional shape, we assume it to be a logarithmic function of the cluster mass. By holding $\sigma_8$ fixed and restricting our analysis to a $\Lambda$CDM model, we find, at $1-\sigma$ confidence level, $\Omega_{\rm m}=0.284\pm0.0064$, $h=0.678\pm0.017$ and $\beta=0.0019^{+0.0008}_{-0.0015}$, where $\beta$ represents the slope of the parameter $\alpha$. This results translates into a $9\%$ decrement of the number of massive clusters with respect to a standard $\Lambda$CDM mass function, but better data are required to better constrain this quantity, since at the $2-\sigma$ and $3-\sigma$ confidence level we are only able to infer upper limits.

### Constraints on shear and rotation with massive galaxy clusters [Replacement]

A precise determination of the mass function is an important tool to verify cosmological predictions of the $\Lambda$CDM model and to infer more precisely the better model describing the evolution of the Universe. Galaxy clusters have been currently used to infer cosmological parameters, in particular the matter density parameter $\Omega_{\rm m}$, the matter power spectrum normalization $\sigma_8$ and the equation of state parameter $w_{\rm de}$ of the dark energy fluid. In this work, using data on massive galaxy clusters ($M>8\times 10^{14}~h^{-1}~M_{\odot}$) in the redshift range $0.05\lesssim z\lesssim 0.83$ we put constraints on the parameter $\alpha$ introduced within the formalism of the extended spherical collapse model to quantify deviations from sphericity due to shear and rotation. Since at the moment there is no physical model describing its functional shape, we assume it to be a logarithmic function of the cluster mass. By holding $\sigma_8$ fixed and restricting our analysis to a $\Lambda$CDM model, we find, at $1-\sigma$ confidence level, $\Omega_{\rm m}=0.284\pm0.0064$, $h=0.678\pm0.017$ and $\beta=0.0019^{+0.0008}_{-0.0015}$, where $\beta$ represents the slope of the parameter $\alpha$. This results translates into a $9\%$ decrement of the number of massive clusters with respect to a standard $\Lambda$CDM mass function, but better data are required to better constrain this quantity, since at the $2-\sigma$ and $3-\sigma$ confidence level we are only able to infer upper limits.

### SUSY Method for the Three-Dimensional Schr\"odinger Equation with Effective Mass

The three-dimensional Schr\"odinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharges, and the corresponding wave functions are connected by intertwining relations. All models are partially integrable by construction: each of them has at least one second order symmetry operator.

### Radial Variation in the Stellar Mass Functions of Star Clusters

A number of recent observational studies of Galactic globular clusters have measured the variation in the slope of a cluster's stellar mass function $\alpha$ with clustercentric distance $r$. In order to gather a deeper understanding of the information contained in such observations, we have explored the evolution of $\alpha(r)$ for star clusters with a variety of initial conditions using a large suite of $N$-body simulations. We have specifically studied how the time evolution of $\alpha(r)$ is affected by initial size, mass, binary fraction, primordial mass segregation, black hole retention, an external tidal field, and the initial mass function itself. Previous studies have shown that the evolution of $\alpha_G$ is closely related to the amount of mass loss suffered by a cluster. Hence for each simulation we have also followed the evolution of the slope of the cluster's global stellar mass function, $\alpha_G$, and have shown that clusters follow a well-defined track in the $\alpha_G$-$d\alpha(r)/d(ln(r/r_m))$ plane. The location of a cluster on the $\alpha_G-d\alpha(r)/d(ln(r/r_m))$ plane can therefore constrain its dynamical history and, in particular, constrain possible variations in the stellar initial mass function. The $\alpha_G$-$d\alpha(r)/d(ln(r/r_m))$ plane thus serves as a key tool for fully exploiting the information contained in wide field studies of cluster stellar mass functions.

### Radial Variation in the Stellar Mass Functions of Star Clusters [Replacement]

A number of recent observational studies of Galactic globular clusters have measured the variation in the slope of a cluster's stellar mass function $\alpha$ with clustercentric distance $r$. In order to gather a deeper understanding of the information contained in such observations, we have explored the evolution of $\alpha(r)$ for star clusters with a variety of initial conditions using a large suite of $N$-body simulations. We have specifically studied how the time evolution of $\alpha(r)$ is affected by initial size, mass, binary fraction, primordial mass segregation, black hole retention, an external tidal field, and the initial mass function itself. Previous studies have shown that the evolution of $\alpha_G$ is closely related to the amount of mass loss suffered by a cluster. Hence for each simulation we have also followed the evolution of the slope of the cluster's global stellar mass function, $\alpha_G$, and have shown that clusters follow a well-defined track in the $\alpha_G$-$d\alpha(r)/d(ln(r/r_m))$ plane. The location of a cluster on the $\alpha_G-d\alpha(r)/d(ln(r/r_m))$ plane can therefore constrain its dynamical history and, in particular, constrain possible variations in the stellar initial mass function. The $\alpha_G$-$d\alpha(r)/d(ln(r/r_m))$ plane thus serves as a key tool for fully exploiting the information contained in wide field studies of cluster stellar mass functions.

### Galaxy And Mass Assembly (GAMA): the Stellar Mass Budget of Galaxy Spheroids and Disks

We build on a recent photometric decomposition analysis of 7506 Galaxy and Mass Assembly (GAMA) survey galaxies to derive stellar mass function fits to individual spheroid and disk component populations down to a lower mass limit of log(M_*/M_sun)= 8. We find that the spheroid/disk mass distributions for individual galaxy morphological types are well described by single Schechter function forms. We derive estimates of the total stellar mass densities in spheroids (rho_spheroid = 1.24+/-0.49 * 10^8 M_sun Mpc^-3 h_0.7) and disks (rho_disk = 1.20+/-0.45 * 10^8 M_sun Mpc^-3 h_0.7), which translates to approximately 50% of the local stellar mass density in spheroids and 48% in disks. The remaining stellar mass is found in the dwarf "little blue spheroid" class, which is not obviously similar in structure to either classical spheroid or disk populations. We also examine the variation of component mass ratios across galaxy mass and group halo mass regimes, finding the transition from spheroid to disk mass dominance occurs near galaxy stellar mass ~10^11 M_sun and group halo mass ~10^12.5 M_sun/h. We further quantify the variation in spheroid-to-total mass ratio with group halo mass for central and satellite populations as well as the radial variation of this ratio within groups.

### Thermal Effects in Dense Matter Beyond Mean Field Theory

The formalism of next-to-leading order Fermi Liquid Theory is employed to calculate the thermal properties of symmetric nuclear and pure neutron matter in a relativistic many-body theory beyond the mean field level which includes two-loop effects. For all thermal variables, the semi-analytical next-to-leading order corrections reproduce results of the exact numerical calculations for entropies per baryon up to 2. This corresponds to excellent agreement down to sub-nuclear densities for temperatures up to $20$ MeV. In addition to providing physical insights, a rapid evaluation of the equation of state in the homogeneous phase of hot and dense matter is achieved through the use of the zero-temperature Landau effective mass function and its derivatives.

### Star cluster formation in cosmological simulations. I. properties of young clusters

We present a new implementation of star formation in cosmological simulations, by considering star clusters as a unit of star formation. Cluster particles grow in mass over several million years at the rate determined by local gas properties, with high time resolution. The particle growth is terminated by its own energy and momentum feedback on the interstellar medium. We test this implementation for Milky Way-sized galaxies at high redshift, by comparing the properties of model clusters with observations of young star clusters. We find that the cluster initial mass function is best described by a Schechter function rather than a single power law. In agreement with observations, at low masses the logarithmic slope is $\alpha\approx 1.8-2$, while the cutoff at high mass scales with the star formation rate. A related trend is a positive correlation between the surface density of star formation rate and fraction of stars contained in massive clusters. Both trends indicate that the formation of massive star clusters is preferred during bursts of star formation. These bursts are often associated with major merger events. We also find that the median timescale for cluster formation ranges from 0.5 to 4 Myr and decreases systematically with increasing star formation efficiency. Local variations in the gas density and cluster accretion rate naturally lead to the scatter of the overall formation efficiency by an order of magnitude, even when the instantaneous efficiency is kept constant. Comparison of the formation timescale with the observed age spread of young star clusters provides an additional important constraint on the modeling of star formation and feedback schemes.

### Star cluster formation in cosmological simulations. I. Properties of young clusters [Replacement]

We present a new implementation of star formation in cosmological simulations, by considering star clusters as a unit of star formation. Cluster particles grow in mass over several million years at the rate determined by local gas properties, with high time resolution. The particle growth is terminated by its own energy and momentum feedback on the interstellar medium. We test this implementation for Milky Way-sized galaxies at high redshift, by comparing the properties of model clusters with observations of young star clusters. We find that the cluster initial mass function is best described by a Schechter function rather than a single power law. In agreement with observations, at low masses the logarithmic slope is $\alpha\approx 1.8-2$, while the cutoff at high mass scales with the star formation rate. A related trend is a positive correlation between the surface density of star formation rate and fraction of stars contained in massive clusters. Both trends indicate that the formation of massive star clusters is preferred during bursts of star formation. These bursts are often associated with major merger events. We also find that the median timescale for cluster formation ranges from 0.5 to 4 Myr and decreases systematically with increasing star formation efficiency. Local variations in the gas density and cluster accretion rate naturally lead to the scatter of the overall formation efficiency by an order of magnitude, even when the instantaneous efficiency is kept constant. Comparison of the formation timescale with the observed age spread of young star clusters provides an additional important constraint on the modeling of star formation and feedback schemes.

### Evolution of star cluster systems in isolated galaxies: first results from direct $N$-body simulations

The evolution of star clusters is largely affected by the tidal field generated by the host galaxy. It is thus in principle expected that under the assumption of an "universal" initial cluster mass function the properties of the evolved present-day mass function of star cluster systems should show a dependency on the properties of the galactic environment in which they evolve. To explore this expectation a sophisticated model of the tidal field is required in order to study the evolution of star cluster systems in realistic galaxies. Along these lines, in the present work we first describe a method developed for coupling $N$-body simulations of galaxies and star clusters. We then generate a database of galaxy models along the Hubble sequence and calibrate evolutionary equations to the results of direct $N$-body simulations of star clusters in order to predict the clusters' mass evolution as function of the galactic environment. We finally apply our methods to explore the properties of evolved "universal" initial cluster mass functions and any dependence on the host galaxy morphology and mass distribution. The preliminary results show that an initial power-law distribution of the masses "universally" evolves into a log-normal distribution, with the properties correlated with the stellar mass and stellar mass density density of the host galaxy.

### Dark-ages reionization and galaxy formation simulation--VII. The sizes of high-redshift galaxies

We investigate high-redshift galaxy sizes using a semi-analytic model constructed for the Dark-ages Reionization And Galaxy-formation Observables from Numerical Simulation project. Our fiducial model, including strong feedback from supernovae and photoionization background, accurately reproduces the evolution of the stellar mass function and luminosity function. Using this model, we study the size--luminosity relation of galaxies and find that the effective radius scales with UV luminosity as $R_\mathrm{e}\propto L^{0.25}$ at $z{\sim}5$--$9$. We show that recently discovered very luminous galaxies at $z{\sim}7$ (Bowler et al. 2016) and $z{\sim}11$ (Oesch et al. 2016) lie on our predicted size--luminosity relations. We find that a significant fraction of galaxies at $z>6$ will not be resolved by JWST, but GMT will have the ability to resolve all galaxies in haloes above the atomic cooling limit. We show that our fiducial model successfully reproduces the redshift evolution of average galaxy sizes at $z>5$. We also explore galaxy sizes in models without supernova feedback. The no-supernova feedback models produce galaxy sizes that are smaller than observations. We therefore conclude that supernova feedback plays an important role in determining the size--luminosity relation of galaxies and its redshift evolution during reionization.

### A Chandra X-ray study of the young star cluster NGC 6231: low-mass population and initial mass function

NGC6231 is a massive young star cluster, near the center of the Sco OB1 association. While its OB members are well studied, its low-mass population has received little attention. We present high-spatial resolution Chandra ACIS-I X-ray data, where we detect 1613 point X-ray sources. Our main aim is to clarify global properties of NGC6231 down to low masses through a detailed membership assessment, and to study the cluster stars' spatial distribution, the origin of their X-ray emission, the cluster age and formation history, and initial mass function. We use X-ray data, complemented by optical/IR data, to establish cluster membership. The spatial distribution of different stellar subgroups also provides highly significant constraints on cluster membership, as does the distribution of X-ray hardness. We perform spectral modeling of group-stacked X-ray source spectra. We find a large cluster population down to ~0.3 Msun (complete to ~1 Msun), with minimal non-member contamination, with a definite age spread (1-8 Myrs) for the low-mass PMS stars. We argue that low-mass cluster stars also constitute the majority of the few hundreds unidentified X-ray sources. We find mass segregation for the most massive stars. The fraction of circumstellar-disk bearing members is found to be ~5%. Photoevaporation of disks under the action of massive stars is suggested by the spatial distribution of the IR-excess stars. We also find strong Halpha emission in 9% of cluster PMS stars. The dependence of X-ray properties on mass, stellar structure, and age agrees with extrapolations based on other young clusters. The cluster initial mass function, computed over ~2 dex in mass, has a slope Gamma~-1.14. The total mass of cluster members above 1 Msun is 2280 Msun, and the inferred total mass is 4380 Msun. We also study the peculiar, hard X-ray spectrum of the Wolf-Rayet star WR79.

### Cluster abundance in chameleon $f(R)$ gravity I: toward an accurate halo mass function prediction

We refine the mass and environment dependent spherical collapse model of chameleon $f(R)$ gravity by calibrating a phenomenological correction inspired by the parameterized post-Friedmann framework against high-resolution $N$-body simulations. We employ our method to predict the corresponding modified halo mass function, and provide fitting formulas to calculate the fractional enhancement of the $f(R)$ halo abundance with respect to that of General Relativity (GR) within a precision of $\lesssim 5\%$ from the results obtained in the simulations. Similar accuracy can be achieved for the full $f(R)$ mass function on the condition that the modeling of the reference GR abundance of halos is accurate at the percent level. We use our fits to forecast constraints on the additional scalar degree of freedom of the theory, finding that upper bounds competitive with current Solar System tests are within reach of cluster number count analyses from ongoing and upcoming surveys at much larger scales. Importantly, the flexibility of our method allows also for this to be applied to other scalar-tensor theories characterized by a mass and environment dependent spherical collapse.

### On Shearing Fluids with Homogeneous Densities

In this paper, we study shearing spherically symmetric homogeneous density fluids in comoving coordinates. It is found that the expansion of the four-velocity of a perfect fluid is homogeneous, whereas its shear is generated by an arbitrary function of time M(t), related to the mass function of the distribution. This function is found to bear a functional relationship with density. The field equations are reduced to two coupled first order ordinary differential equations for the metric coefficients, g 11 and g 22. We have explored a class of solutions assuming that M is a linear function of the density. This class embodies, as a subcase, the complete class of shear-free solutions. We have discussed the off quoted work of Kustaanheimo (1947) and have noted that it deals with shear-free fluids having anisotropic pressure. It is shown that the anisotropy of the fluid is characterized by an arbitrary function of time. We have discussed some issues of historical priorities and credentials related to shear-free solutions. Recent controversial claims by Mitra (2011, 2012) have also been addressed. We found that the singularity and the shearing motion of the fluid are closely related. Hence, there is a need for fresh look to the solutions obtained earlier in comoving coordinates. Keywords (separated by '-') Shearing solution - Perfect fluids - Homogeneous density

### Distribution of star formation rates during the rapid assembly of NGC 1399 as deduced from its globular cluster system

Ultra-compact dwarf galaxies (UCDs) share many properties with globular clusters (GCs) and are found in similar environments. A large sample of UCDs and GCs around NGC 1399, the central giant elliptical of the Fornax galaxy cluster, is used to infer their formation history and also that of NGC 1399. We assumed that all GCs and UCDs in our sample are star clusters (SCs) and used them as tracers of past star formation activities. After correcting our GC/UCD sample for mass loss, we interpreted their overall mass function to be a superposition of SC populations that formed coevally during different times. The SC masses of each population were distributed according to the embedded cluster mass function (ECMF), a pure power law with the slope $-\beta$ and a stellar upper mass limit, $M_{\mathrm{max}}$, which depended on the star formation rate (SFR). We decomposed the observed GC/UCD mass function into individual SC populations and converted $M_{\mathrm{max}}$ of each SC population to an SFR. The overall distribution of SFRs reveals how the GC/UCD sample formed. Considering the age of the GCs/UCDs and the present stellar mass of NGC 1399, we found that the formation of the GCs/UCDs can be well explained for $\beta<2.3$. This agrees very well with the observation in young SCs where $\beta\approx2.0$ is usually found. Even if taking into account that some of the most massive objects might not be genuine SCs and applying different corrections for the mass loss, the outcome is not influenced much. We found peak SFRs between approximately 300 and 3000 $M_{\odot}\mathrm{yr}^{-1}$, which matches the SFRs observed in massive high-$z$ sub-mm galaxies and an SFR estimate inferred from NGC 1399 based on "downsizing", i.e. more massive galaxies must have formed over shorter times. Our results suggest that NGC 1399 and its GC/UCD system formed in an early, short, and intense starburst.

### Primordial Black Holes as Dark Matter [Replacement]

The possibility that the dark matter comprises primordial black holes (PBHs) is considered, with particular emphasis on the currently allowed mass windows at $10^{16}$ -- $10^{17}\,$g, $10^{20}$ -- $10^{24}\,$g and $1$ -- $10^{3}\,M_{\odot}$. The Planck mass relics of smaller evaporating PBHs are also considered. All relevant constraints (lensing, dynamical, large-scale structure and accretion) are reviewed and various effects necessary for a precise calculation of the PBH abundance (non-Gaussianity, non-sphericity, critical collapse and merging) are accounted for. It is difficult to put all the dark matter in PBHs if their mass function is monochromatic but this is still possible if the mass function is extended, as expected in many scenarios. A novel procedure for confronting observational constraints with an extended PBH mass spectrum is therefore introduced. This applies for arbitrary constraints and a wide range of PBH formation models, and allows us to identify which model-independent conclusions can be drawn from constraints over all mass ranges. We focus particularly on PBHs generated by inflation, pointing out which effects in the formation process influence the mapping from the inflationary power spectrum to the PBH mass function. We then apply our scheme to two specific inflationary models in which PBHs provide the dark matter. The possibility that the dark matter is in intermediate-mass PBHs of $1$ -- $10^{3}\,M_{\odot}$ is of special interest in view of the recent detection of black-hole mergers by LIGO. The possibility of Planck relics is also intriguing but virtually untestable.

### Primordial Black Holes as Dark Matter

The possibility that the dark matter comprises primordial black holes (PBHs) is considered, with particular emphasis on the currently allowed mass windows at $10^{16}$ - $10^{17}\,$g, $10^{20}$ - $10^{24}\,$g and $1$ - $10^{3}\,M_{\odot}$. The Planck mass relics of smaller evaporating PBHs are also considered. All relevant constraints (lensing, dynamical, large-scale structure and accretion) are reviewed and various effects necessary for a precise calculation of the PBH abundance (non-Gaussianity, non-sphericity, critical collapse and merging) are accounted for. It is difficult to put all the dark matter in PBHs if their mass function is monochromatic but this is still possible if the mass function is extended, as expected in many scenarios. A novel procedure for confronting observational constraints with an extended PBH mass spectrum is therefore introduced. This applies for arbitrary constraints and a wide range of PBH formation models, and allows us to identify which model-independent conclusions can be drawn from constraints over all mass ranges. We focus particularly on PBHs generated by inflation, pointing out which effects in the formation process influence the mapping from the inflationary power spectrum to the PBH mass function. We then apply our scheme to two specific inflationary models in which PBHs provide the dark matter. The possibility that the dark matter is in intermediate mass PBHs of $1$ - $10^{3}\,M_{\odot}$ is of special interest in view of the recent detection of black hole mergers by LIGO. The possibility of Planck relics is also intriguing but virtually untestable.

### Primordial Black Holes as Dark Matter [Cross-Listing]

The possibility that the dark matter comprises primordial black holes (PBHs) is considered, with particular emphasis on the currently allowed mass windows at $10^{16}$ - $10^{17}\,$g, $10^{20}$ - $10^{24}\,$g and $1$ - $10^{3}\,M_{\odot}$. The Planck mass relics of smaller evaporating PBHs are also considered. All relevant constraints (lensing, dynamical, large-scale structure and accretion) are reviewed and various effects necessary for a precise calculation of the PBH abundance (non-Gaussianity, non-sphericity, critical collapse and merging) are accounted for. It is difficult to put all the dark matter in PBHs if their mass function is monochromatic but this is still possible if the mass function is extended, as expected in many scenarios. A novel procedure for confronting observational constraints with an extended PBH mass spectrum is therefore introduced. This applies for arbitrary constraints and a wide range of PBH formation models, and allows us to identify which model-independent conclusions can be drawn from constraints over all mass ranges. We focus particularly on PBHs generated by inflation, pointing out which effects in the formation process influence the mapping from the inflationary power spectrum to the PBH mass function. We then apply our scheme to two specific inflationary models in which PBHs provide the dark matter. The possibility that the dark matter is in intermediate mass PBHs of $1$ - $10^{3}\,M_{\odot}$ is of special interest in view of the recent detection of black hole mergers by LIGO. The possibility of Planck relics is also intriguing but virtually untestable.

### Primordial Black Holes as Dark Matter [Replacement]

The possibility that the dark matter comprises primordial black holes (PBHs) is considered, with particular emphasis on the currently allowed mass windows at $10^{16}$ -- $10^{17}\,$g, $10^{20}$ -- $10^{24}\,$g and $1$ -- $10^{3}\,M_{\odot}$. The Planck mass relics of smaller evaporating PBHs are also considered. All relevant constraints (lensing, dynamical, large-scale structure and accretion) are reviewed and various effects necessary for a precise calculation of the PBH abundance (non-Gaussianity, non-sphericity, critical collapse and merging) are accounted for. It is difficult to put all the dark matter in PBHs if their mass function is monochromatic but this is still possible if the mass function is extended, as expected in many scenarios. A novel procedure for confronting observational constraints with an extended PBH mass spectrum is therefore introduced. This applies for arbitrary constraints and a wide range of PBH formation models, and allows us to identify which model-independent conclusions can be drawn from constraints over all mass ranges. We focus particularly on PBHs generated by inflation, pointing out which effects in the formation process influence the mapping from the inflationary power spectrum to the PBH mass function. We then apply our scheme to two specific inflationary models in which PBHs provide the dark matter. The possibility that the dark matter is in intermediate-mass PBHs of $1$ -- $10^{3}\,M_{\odot}$ is of special interest in view of the recent detection of black-hole mergers by LIGO. The possibility of Planck relics is also intriguing but virtually untestable.