# Posts Tagged mass function

## Recent Postings from mass function

### 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

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.

### Far-infrared and dust properties of present-day galaxies in the EAGLE simulations

The EAGLE cosmological simulations reproduce the observed galaxy stellar mass function and many galaxy properties. In this work, we study the dust-related properties of present-day EAGLE galaxies through mock observations in the far-infrared and submm wavelength ranges obtained with the 3D dust radiative transfer code SKIRT. To prepare an EAGLE galaxy for radiative transfer processing, we derive a diffuse dust distribution from the gas particles and we re-sample the star-forming gas particles and the youngest star particles into star-forming regions that are assigned dedicated emission templates. We select a set of redshift-zero EAGLE galaxies that matches the K-band luminosity distribution of the galaxies in the Herschel Reference Survey (HRS), a volume-limited sample of about 300 normal galaxies in the Local Universe. We find overall agreement of the EAGLE dust scaling relations with those observed in the HRS, such as the dust-to-stellar mass ratio versus stellar mass and versus NUV-r colour relations. A discrepancy in the f_250/f_350 versus f_350/f_500 submm colour-colour relation implies that part of the simulated dust is insufficiently heated, likely because of limitations in our sub-grid model for star-forming regions. We also investigate the effect of adjusting the metal-to-dust ratio and the covering factor of the photodissociation regions surrounding the star-forming cores. We are able to constrain the important dust-related parameters in our method, informing the calculation of dust attenuation for EAGLE galaxies in the UV and optical domain.

### The stellar mass-halo mass relation of isolated field dwarfs: a critical test of $\Lambda$CDM at the edge of galaxy formation

We fit the rotation curves of isolated dwarf galaxies to directly measure the stellar mass-halo mass relation ($M_*-M_{200}$) over the mass range $5 \times 10^5 < M_{*}/{\rm M}_\odot < 10^{8}$. By accounting for cusp-core transformations due to stellar feedback, we find a monotonic relation with remarkably little scatter. Such monotonicity implies that abundance matching should yield a similar $M_*-M_{200}$ if the cosmological model is correct. Using the 'field galaxy' stellar mass function from the Sloan Digital Sky Survey (SDSS) and the halo mass function from the $\Lambda$ Cold Dark Matter Bolshoi simulation, we find remarkable agreement between the two. This holds down to $M_{200} \sim 5 \times 10^9$ M$_\odot$, and to $M_{200} \sim 5 \times 10^8$ M$_\odot$ if we assume a power law extrapolation of the SDSS stellar mass function below $M_* \sim 10^7$ M$_\odot$. However, if instead of SDSS we use the stellar mass function of nearby galaxy groups, then the agreement is poor. This occurs because the group stellar mass function is shallower than that of the field below $M_* \sim 10^9$ M$_\odot$, recovering the familiar 'missing satellites' and 'too big to fail' problems. Our result demonstrates that both problems are confined to group environments and must, therefore, owe to 'galaxy formation physics' rather than exotic cosmology. Finally, we repeat our analysis for a $\Lambda$ Warm Dark Matter cosmology, finding that it fails at 68% confidence for a thermal relic mass of $m_{\rm WDM} < 1.25$ keV, and $m_{\rm WDM} < 2$ keV if we use the power law extrapolation of SDSS. We conclude by making a number of predictions for future surveys based on these results.

### Merger-driven evolution of the effective stellar initial mass function of massive early-type galaxies

The stellar initial mass function (IMF) of early-type galaxies is the combination of the IMF of the stellar population formed in-situ and that of accreted stellar populations. Using as an observable the effective IMF $\alpha_{IMF}$, defined as the ratio between the true stellar mass of a galaxy and the stellar mass inferred assuming a Salpeter IMF, we present a theoretical model for its evolution as a result of dry mergers. We use a simple dry merger evolution model, based on cosmological $N$-body simulations, together with empirically motivated prescriptions for the IMF to make predictions for how the effective IMF of massive early-type galaxies changes from $z=2$ to $z=0$. We find that the IMF normalization of individual galaxies becomes lighter with time. At fixed velocity dispersion, $\alpha_{IMF}$ is predicted to be constant with redshift. Current constraints on the evolution of the IMF are in slight tension with this prediction, even though systematic uncertainties prevent a conclusive statement. The correlation of $\alpha_{IMF}$ with stellar mass becomes shallower with time, while the correlation between $\alpha_{IMF}$ and velocity dispersion is mostly preserved by dry mergers. We also find that dry mergers can mix the dependence of the IMF on stellar mass and velocity dispersion, making it challenging to infer, from $z=0$ observations of global galactic properties, what is the quantity that is originally coupled with the IMF.

### Constraining $f(R)$ Gravity Theory Using CFHTLenS Weak Lensing Peak Statistics

In this Letter, we report the observational constraints on the Hu-Sawicki $f(R)$ theory derived from weak lensing peak abundances, which are closely related to the mass function of massive halos. In comparison with studies using optical or X-ray clusters of galaxies, weak lensing peak analyses have the advantages of not relying on mass-baryonic observable calibrations. With observations from the Canada-France-Hawaii-Telescope Lensing Survey, our peak analyses give rise to a tight constraint on the model parameter $|f_{R0}|$ for $n=1$. The $95\%$ CL limit is $\log_{10}|f_{R0}| < -4.82$ given WMAP9 priors on $(\Omega_{\rm m}, A_{\rm s})$. With Planck15 priors, the corresponding result is $\log_{10}|f_{R0}| < -5.16$.

### OGLE-2015-BLG-0051/KMT-2015-BLG-0048Lb: a Giant Planet Orbiting a Low-mass Bulge Star Discovered by High-cadence Microlensing Surveys

We report the discovery of an extrasolar planet detected from the combined data of a microlensing event OGLE-2015-BLG-0051/KMT-2015-BLG-0048 acquired by two microlensing surveys. Despite that the short planetary signal occurred in the very early Bulge season during which the lensing event could be seen for just about an hour, the signal was continuously and densely covered. From the Bayesian analysis using models of the mass function, matter and velocity distributions combined with the information of the angular Einstein radius, it is found that the host of the planet is located in the Galactic bulge. The planet has a mass $0.72_{-0.07}^{+0.65}\ M_{\rm J}$ and it is orbiting a low-mass M-dwarf host with a projected separation $d_\perp=0.73 \pm 0.08$ AU. The discovery of the planet demonstrates the capability of the current high-cadence microlensing lensing surveys in detecting and characterizing planets.

### Spherical collapse of dark matter haloes in tidal gravitational fields

We study the spherical collapse model in the presence of external gravitational tidal shear fields for different dark energy scenarios and investigate the impact on the mass function and cluster number counts. While previous studies of the influence of shear and rotation on $\delta_\mathrm{c}$ have been performed with heuristically motivated models, we try to avoid this model dependence and sample the external tidal shear values directly from the statistics of the underlying linearly evolved density field based on first order Lagrangian perturbation theory. Within this self-consistent approach, in the sense that we restrict our treatment to scales where linear theory is still applicable, only fluctuations larger than the scale of the considered objects are included into the sampling process which naturally introduces a mass dependence of $\delta_\mathrm{c}$. We find that shear effects are predominant for smaller objects and at lower redshifts, i. e. the effect on $\delta_\mathrm{c}$ is at or below the percent level for the $\Lambda$CDM model. For dark energy models we also find small but noticeable differences, similar to $\Lambda$CDM. The virial overdensity $\Delta_\mathrm{V}$ is nearly unaffected by the external shear. The now mass dependent ?c is used to evaluate the mass function for different dark energy scenarios and afterwards to predict cluster number counts, which indicate that ignoring the shear contribution can lead to biases of the order of $1\sigma$ in the estimation of cosmological parameters like $\Omega_\mathrm{m}$, $\sigma_8$ or $w$.

### A simple and general method for solving detailed chemical evolution with delayed production of iron and other chemical elements

In this Letter, we present a new theoretical method for solving the chemical evolution of galaxies, by assuming the instantaneous recycling approximation for chemical elements restored by massive stars and the Delay Time Distribution formalism for the delayed chemical enrichment by Type Ia Supernovae. The galaxy gas mass assembly history, together with the assumed stellar yields and initial mass function, represent the starting point of this method. We derive a very simple and general equation which closely relates the Laplace transforms of the galaxy gas accretion and star formation history, which can be used to simplify the problem of retrieving these quantities in most of current galaxy evolution models. We find that - once the galaxy star formation history has been reconstructed from our assumptions - the differential equation for the evolution of the chemical element $X$ can be suitably solved with classical methods. We apply our model to reproduce the $[\text{O/Fe}]$ and $[\text{Si/Fe}]$ vs. $[\text{Fe/H}]$ chemical abundance patterns as observed in the Milky Way halo and disc stars, by assuming a decaying exponential infall rate of gas and different delay time distributions for Type Ia Supernovae. Although approximate, we conclude that our model with the single degenerate scenario for Type Ia Supernovae provides the best agreement with the observed set of data. Our method will be very useful in cosmological simulations and other complementary stellar population synthesis models to predict the chemical evolution of galaxies.

### A consistent model for both the HI and stellar mass functions of galaxies

Using the L-Galaxies semi-analytic model we simultaneously fit the HI mass function, stellar mass function and galaxy colours. We find good fits to all three observations at z = 0 and to the stellar mass function and galaxy colours at z = 2. Using Markov Chain Monte Carlo (MCMC) techniques we adjust the L-Galaxies parameters to best fit the constraining data. In order to fit the HI mass function we must greatly reduce the gas surface density threshold for star formation, thus lowering the number of low HI mass galaxies. A simultaneous reduction in the star formation efficiency prevents the over production of stellar content. A simplified model in which the surface density threshold is eliminated altogether also provides a good fit to the data. Unfortunately, these changes weaken the fit to the Kennicutt-Schmidt relation and raise the star-formation rate density at recent times, suggesting that a change to the model is required to prevent accumulation of gas onto dwarf galaxies in the local universe.

### Linear perturbation theory for tidal streams and the small-scale CDM power spectrum

Tidal streams in the Milky Way are sensitive probes of the population of dark-matter subhalos predicted in cold-dark-matter (CDM) simulations. We present a new calculus for computing the effect of subhalo fly-bys on cold tidal streams based on the action-angle representation of streams. The heart of this calculus is a line-of-parallel-angle approach that calculates the perturbed distribution function of a given stream segment by undoing the effect of all impacts. This approach allows one to compute the perturbed stream density and track in any coordinate system in minutes for realizations of the subhalo distribution down to 10^5 Msun, accounting for the stream's internal dispersion and overlapping impacts. We study the properties of density and track fluctuations with suites of simulations. The one-dimensional density and track power spectra along the stream trace the subhalo mass function, with higher-mass subhalos producing power only on large scales, while lower mass subhalos cause structure on smaller scales. The time-dependence of impacts and of the evolution of the stream after an impact gives rise to bispectra. We further find that tidal streams are essentially corrugated sheets in the presence of subhalo perturbations: different projections of the track all reflect the same pattern of perturbations, facilitating their observational measurement. We apply this formalism to density data for the Pal 5 stream and make a first rigorous determination of 10^{+11}_{-6} dark-matter subhalos with masses between 3x10^6 and 10^9 Msun within 20 kpc from the Galactic center (corresponding to 1.4^{+1.6}_{-0.9} times the number predicted by CDM-only simulations or to f_{sub}(r<20 kpc) ~ 0.2%). Improved data will allow measurements of the subhalo mass function down to 10^5 Msun, thus definitively testing whether dark matter clumps on the smallest scales relevant for galaxy formation.

### Weighing the Giants V: Galaxy Cluster Scaling Relations

We present constraints on the scaling relations of galaxy cluster X-ray luminosity, temperature and gas mass (and derived quantities) with mass and redshift, employing masses from robust weak gravitational lensing measurements. These are the first such results obtained from an analysis that simultaneously accounts for selection effects and the underlying mass function, and directly incorporates lensing data to constrain total masses. Our constraints on the scaling relations and their intrinsic scatters are in good agreement with previous studies, and reinforce a picture in which departures from self-similar scaling laws are primarily limited to cluster cores. However, the data are beginning to reveal new features that have implications for cluster astrophysics and provide new tests for hydrodynamical simulations. We find a positive correlation in the intrinsic scatters of luminosity and temperature at fixed mass, which is related to the dynamical state of the clusters. While the evolution of the nominal scaling relations is consistent with self similarity, we find tentative evidence that the luminosity and temperature scatters respectively decrease and increase with redshift. Physically, this likely related to the development of cool cores and the rate of major mergers. We also examine the scaling relations of redMaPPer richness and Compton $Y$ from Planck. While the richness--mass relation is in excellent agreement with recent work, the measured $Y$--mass relation departs strongly from that assumed in the Planck cluster cosmology analysis. The latter result is consistent with earlier comparisons of lensing and Planck scaling-relation-derived masses.

### Simulating the dust content of galaxies: successes and failures

We present full volume cosmological simulations using the moving-mesh code AREPO to study the coevolution of dust and galaxies. We extend the dust model in AREPO to include thermal sputtering of grains and investigate the evolution of the dust mass function, the cosmic distribution of dust beyond the interstellar medium, and the dependence of dust-to-stellar mass ratio on galactic properties. The simulated dust mass function is well-described by a Schechter fit and lies closest to observations at $z = 0$. The radial scaling of projected dust surface density out to distances of $10 \, \text{Mpc}$ around galaxies with magnitudes $17 < i < 21$ is similar to that seen in Sloan Digital Sky Survey data. At $z = 0$, the predicted dust density of $\Omega_\text{dust} \approx 1.9 \times 10^{-6}$ lies in the range of $\Omega_\text{dust}$ values seen in low-redshift observations. We find that dust-to-stellar mass ratio anti-correlates with stellar mass for galaxies living along the star formation main sequence. Moreover, we estimate the $850 \, \mu\text{m}$ and $1.1 \, \text{mm}$ number density functions for simulated galaxies at $z = 1$ and analyse the relation between dust-to-stellar flux and mass ratios at $z = 0$. At high redshift, our model fails to produce enough dust-rich galaxies, and this tension is not alleviated by adopting a top-heavy initial mass function. We do not capture a decline in $\Omega_\text{dust}$ from $z = 2$ to $z = 0$, which suggests that dust production mechanisms more strongly dependent on star formation may help to produce the observed number of dusty galaxies near the peak of cosmic star formation.

### Chiral-symmetry breaking and pion structure in the Covariant Spectator Theory

We introduce a covariant approach in Minkowski space for the description of quarks and mesons that exhibits both chiral-symmetry breaking and confinement. In a simple model for the interquark interaction the quark mass function is obtained and used in the calculation of the pion form factor. We study the effects of the mass function and of the different quark pole contributions on the pion form factor.

### Chiral-symmetry breaking and pion structure in the Covariant Spectator Theory [Cross-Listing]

We introduce a covariant approach in Minkowski space for the description of quarks and mesons that exhibits both chiral-symmetry breaking and confinement. In a simple model for the interquark interaction the quark mass function is obtained and used in the calculation of the pion form factor. We study the effects of the mass function and of the different quark pole contributions on the pion form factor.

### Chiral-symmetry breaking and pion structure in the Covariant Spectator Theory [Replacement]

We introduce a covariant approach in Minkowski space for the description of quarks and mesons that exhibits both chiral-symmetry breaking and confinement. In a simple model for the interquark interaction the quark mass function is obtained and used in the calculation of the pion form factor. We study the effects of the mass function and of the different quark pole contributions on the pion form factor.

### Chiral-symmetry breaking and pion structure in the Covariant Spectator Theory [Replacement]

We introduce a covariant approach in Minkowski space for the description of quarks and mesons that exhibits both chiral-symmetry breaking and confinement. In a simple model for the interquark interaction the quark mass function is obtained and used in the calculation of the pion form factor. We study the effects of the mass function and of the different quark pole contributions on the pion form factor.

### Variations of the initial mass function in semi-analytical models: implications for the chemical enrichment of galaxies in the GAEA model

In this work, we investigate the implications of the Integrated Galaxy-wide stellar Initial Mass Function (IGIMF) approach in the framework of the semi-analytic model GAEA (GAlaxy Evolution and Assembly), which features a detailed treatment of chemical enrichment and stellar feedback. The IGIMF provides an analytic description of the dependence of the stellar IMF shape on the rate of star formation in galaxies. We find that our model with a universal IMF predicts a rather flat [$\alpha$/Fe]-stellar mass relation. The model assuming the IGIMF, instead, is able to reproduce the observed increase of $\alpha$-enhancement with stellar mass. This is mainly due to the fact that massive galaxies are characterized by larger SFRs at high-redshift, leading to stronger $\alpha$-enhancement with respect to low-mass galaxies. At the same time, the IGIMF hypothesis does not affect significantly the trend for shorter star formation timescales for more massive galaxies. We argue that in the IGIMF scenario the [$\alpha$/Fe] ratios are good tracers of the highest SFR events, but they do not provide much information on the overall star formation timescales. The final stellar masses and mass-to-light-ratio of our model galaxies are larger than those estimated from the synthetic photometry assuming a universal IMF. This result is in agreement with recent claims of a bottom-heavier IMF in massive galaxies, based on dynamical analyses of local early type galaxies.

### Massive stars reveal variations of the stellar initial mass function in the Milky Way stellar clusters

We investigate whether the stellar initial mass function (IMF) is universal, or whether there are significant cluster-to-cluster variations of the IMF among young stellar clusters in the Milky Way. We propose a method to uncover the range of variation of the parameters that describe the IMF for the population of young clusters in the Milky Way. The method relies exclusively on the high mass content of the clusters, but is able to yield information on the distributions of parameters of the IMF over the entire stellar mass range. This is achieved by appropriately comparing the fractions of single and lonely massive O stars in a recent catalog of the Milky Way clusters with a large library of simulated clusters built with various distribution functions of the IMF parameters. The masses of synthetic clusters are randomly drawn using a power-law distributions function, while stellar masses in the clusters are randomly drawn using a tapered power-law function. The synthetic clusters are further corrected for the effects of binary population, stellar evolution, sample incompleteness, and estimates are made for the effects of ejected O stars. Our findings indicate that broad distributions of the IMF parameters are required in order to reproduce the fractions of single and lonely O stars in the Milky Way clusters and they do not lend support to the existence of a cluster mass-maximum stellar mass relation. We propose a probabilistic formulation of the IMF based on the distribution functions of its parameters.

### Constraining the halo mass function with observations

The abundances of matter halos in the universe are described by the so-called halo mass function (HMF). It enters most cosmological analyses and parametrizes how the linear growth of primordial perturbations is connected to these abundances. Interestingly, this connection can be made approximately cosmology independent. This made it possible to map in detail its near-universal behavior through large-scale simulations. However, such simulations may suffer from systematic effects, especially if baryonic physics is included. In this paper we ask how well observations can constrain directly the HMF. The observables we consider are galaxy cluster number counts, galaxy cluster power spectrum and lensing of type Ia supernovae. Our results show that DES is capable of putting the first meaningful constraints, while both Euclid and J-PAS can give constraints on the HMF parameters which are comparable to the ones from state-of-the-art simulations. We also find that an independent measurement of cluster masses is even more important for measuring the HMF than for constraining the cosmological parameters, and can vastly improve the determination of the halo mass function. Measuring the HMF could thus be used to cross-check simulations and their implementation of baryon physics. It could even, if deviations cannot be accounted for, hint at new physics.

### A theoretical perspective on the formation and fragmentation of protostellar discs

We discuss the factors influencing the formation and gravitational fragmentation of protostellar discs. We start with a review of how observations of prestellar cores can be analysed statistically to yield plausible initial conditions for simulations of their subsequent collapse. Simulations based on these initial conditions show that, despite the low levels of turbulence in prestellar cores, they deliver primary protostars and associated discs which are routinely subject to stochastic impulsive perturbations; consequently misalignment of the spins and orbits of protostars are common. Also, the simulations produce protostars that collectively have a mass function and binary statistics matching those observed in nearby star formation regions, but only if a significant fraction of the turbulent energy in the core is solenoidal, and accretion onto the primary protostar is episodic with a duty cycle > 3000 yr. Under this circumstance a core typically spawns between 4 and 5 protostars, with high efficiency, and the lower-mass protostars are mainly formed by disc fragmentation. The requirement that a proto-fragment in a disc lose thermal energy on a dynamical timescale dictates that {\bf there is a sweet spot for} disc fragmentation at radii 70 AU < R < 100 AU and temperatures 10 K < T < 20 K, and this might explain the Brown Dwarf Desert.

### The bimodal initial mass function in the Orion Nebula Cloud

Due to its youth, proximity and richness the Orion Nebula Cloud (ONC) is an ideal testbed to obtain a comprehensive view on the Initial Mass Function (IMF) down to the planetary mass regime. Using the HAWK-I camera at the VLT, we have obtained an unprecedented deep and wide near-infrared JHK mosaic of the ONC (90% completeness at K~19.0mag, 22'x28). Applying the most recent isochrones and accounting for the contamination of background stars and galaxies, we find that ONC's IMF is bimodal with distinct peaks at about 0.25 and 0.025 M_sun separated by a pronounced dip at the hydrogen burning limit (0.08 M_sun), with a depth of about a factor 2-3 below the log-normal distribution. Apart from ~920 low-mass stars (M < 1.4 M_sun) the IMF contains ~760 brown dwarf (BD) candidates and ~160 isolated planetary mass object (IPMO) candidates with M > 0.005 M_sun, hence about ten times more substellar candidates than known before. The substellar IMF peak at 0.025 M_sun could be caused by BDs and IPMOs which have been ejected from multiple systems during the early star-formation process or from circumstellar disks.

### The millisecond pulsar mass distribution: Evidence for bimodality and constraints on the maximum neutron star mass

The mass function of neutron stars (NSs) contains information about the late evolution of massive stars, the supernova explosion mechanism, and the equation-of-state of cold, nuclear matter beyond the nuclear saturation density. A number of recent NS mass measurements in binary millisecond pulsar (MSP) systems increase the fraction of massive NSs (with $M > 1.8$ M$_{\odot}$) to $\sim 20\%$ of the observed population. In light of these results, we employ a Bayesian framework to revisit the MSP mass distribution. We find that a single Gaussian model does not sufficiently describe the observed population. We test alternative empirical models and infer that the MSP mass distribution is strongly asymmetric. The diversity in spin and orbital properties of high-mass NSs suggests that this is most likely not a result of the recycling process, but rather reflects differences in the NS birth masses. The asymmetry is best accounted for by a bimodal distribution with a low mass component centred at $1.393_{-0.029}^{+0.031}$ M$_{\odot}$ and dispersed by $0.064_{-0.025}^{+0.064}$ M$_{\odot}$, and a high-mass component with a mean of $1.807_{-0.132}^{+0.081}$ and a dispersion of $0.177_{-0.072}^{+0.115}$ M$_{\odot}$. We also establish a lower limit of $M_{max} \ge 2.018$ M$_{\odot}$ at 98% C.L. for the maximum NS mass, from the absence of a high-mass truncation in the observed masses. Using our inferred model, we find that the measurement of 350 MSP masses, expected after the conclusion of pulsar surveys with the Square-Kilometre Array, can result in a precise localization of a maximum mass up to 2.15 M$_{\odot}$, with a 5% accuracy. Finally, we identify possible massive NSs within the known pulsar population and discuss birth masses of MSPs.

### The millisecond pulsar mass distribution: Evidence for bimodality and constraints on the maximum neutron star mass [Cross-Listing]

The mass function of neutron stars (NSs) contains information about the late evolution of massive stars, the supernova explosion mechanism, and the equation-of-state of cold, nuclear matter beyond the nuclear saturation density. A number of recent NS mass measurements in binary millisecond pulsar (MSP) systems increase the fraction of massive NSs (with $M > 1.8$ M$_{\odot}$) to $\sim 20\%$ of the observed population. In light of these results, we employ a Bayesian framework to revisit the MSP mass distribution. We find that a single Gaussian model does not sufficiently describe the observed population. We test alternative empirical models and infer that the MSP mass distribution is strongly asymmetric. The diversity in spin and orbital properties of high-mass NSs suggests that this is most likely not a result of the recycling process, but rather reflects differences in the NS birth masses. The asymmetry is best accounted for by a bimodal distribution with a low mass component centred at $1.393_{-0.029}^{+0.031}$ M$_{\odot}$ and dispersed by $0.064_{-0.025}^{+0.064}$ M$_{\odot}$, and a high-mass component with a mean of $1.807_{-0.132}^{+0.081}$ and a dispersion of $0.177_{-0.072}^{+0.115}$ M$_{\odot}$. We also establish a lower limit of $M_{max} \ge 2.018$ M$_{\odot}$ at 98% C.L. for the maximum NS mass, from the absence of a high-mass truncation in the observed masses. Using our inferred model, we find that the measurement of 350 MSP masses, expected after the conclusion of pulsar surveys with the Square-Kilometre Array, can result in a precise localization of a maximum mass up to 2.15 M$_{\odot}$, with a 5% accuracy. Finally, we identify possible massive NSs within the known pulsar population and discuss birth masses of MSPs.

### A large difference in the progenitor masses of active and passive galaxies in the EAGLE simulation [Replacement]

Cumulative number density matching of galaxies is a method to observationally connect descendent galaxies to their typical main progenitors at higher redshifts and thereby to assess the evolution of galaxy properties. The accuracy of this method is limited due to galaxy merging and scatter in the stellar mass growth history of individual galaxies. Behroozi et al. (2013) have introduced a refinement of the method, based on abundance matching of observed galaxies to the Bolshoi dark-matter-only simulation. The EAGLE cosmological hydro-simulation is well suited to test this method, because it reproduces the observed evolution of the galaxy stellar mass function and the passive fraction. We find agreement with the Behroozi et al. (2013) method for the complete sample of main progenitors of z = 0 galaxies, but we also find a strong dependence on the current star formation rate. Passive galaxies with a stellar mass up to 10^10.75 Msun have a completely different median mass history than active galaxies of the same mass. This difference persists if we only select central galaxies. This means that the cumulative number density method should be applied separately to active and passive galaxies. Even then, the typical main progenitor of a z = 0 galaxy already spans two orders of magnitude in stellar mass at z = 2.

### Are Some Milky Way Globular Clusters Hosted by Undiscovered Galaxies?

The confirmation of a globular cluster (GC) in the recently discovered ultrafaint galaxy Eridanus II (Eri II) motivated us to examine the question posed in the title. After estimating the halo mass of Eri II using a published stellar mass - halo mass relation, the one GC in this galaxy supports extending the relationship between the number of GCs hosted by a galaxy and the galaxy's total mass about two orders of magnitude in stellar mass below the previous limit. For this empirically determined specific frequency of between 0.06 and 0.39 globular clusters per 10$^9$ $M_\odot$ of total mass, the surviving Milky Way (MW) subhalos with masses smaller than $10^{10} M_\odot$ could host as many as 5 to 31 GCs, broadly consistent with the actual population of outer halo MW GCs, although matching the radial distribution in detail remains a challenge. Using a subhalo mass function from published high resolution numerical simulations and a Poissonian model for populating those halos with the aforementioned empirically constrained frequency, we find that about 90$\%$ of these GCs lie in lower-mass subhalos than that of Eri II. From what we know about the stellar mass-halo mass function, the subhalo mass function, and the mass-normalized GC specific frequency, we conclude that some of the MW's outer halo GCs are likely to be hosted by undetected subhalos with extremely modest stellar populations.

### The EAGLE simulations: atomic hydrogen associated with galaxies

We examine the properties of atomic hydrogen (HI) associated with galaxies in the EAGLE simulations of galaxy formation. EAGLE's feedback parameters were calibrated to reproduce the stellar mass function and galaxy sizes at $z=0.1$, and we assess whether this calibration also yields realistic HI properties. We estimate the self-shielding density with a fitting function calibrated using radiation transport simulations, and correct for molecular hydrogen with empirical or theoretical relations. The `standard-resolution' simulations systematically underestimate HI column densities, leading to an HI deficiency in low-mass ($M_\star < 10^{10}M_\odot$) galaxies and poor reproduction of the observed HI mass function. These shortcomings are largely absent from EAGLE simulations featuring a factor of 8 (2) better mass (spatial) resolution, within which the HI mass of galaxies evolves more mildly from $z=1$ to $0$ than in the standard-resolution simulations. The largest-volume simulation reproduces the observed clustering of HI systems, and its dependence on HI-richness. At fixed $M_\star$, galaxies acquire more HI in simulations with stronger feedback, as they become associated with more massive haloes and higher infall rates. They acquire less HI in simulations with a greater star formation efficiency, since the star formation and feedback necessary to balance the infall rate is produced by smaller gas reservoirs. The simulations indicate that the HI of present-day galaxies was acquired primarily by the smooth accretion of ionized, intergalactic gas at $z\simeq1$, which later self-shields, and that only a small fraction is contributed by the reincorporation of gas previously heated strongly by feedback. HI reservoirs are highly dynamic: over $40$ percent of HI associated with $z=0.1$ galaxies is converted to stars or ejected by $z=0$.

### The Next Generation Virgo Cluster Survey (NGVS). XIII. The Luminosity and Mass Function of Galaxies in the Core of the Virgo Cluster and the Contribution from Disrupted Satellites

We present measurements of the galaxy luminosity and stellar mass function in a 3.71 deg$^2$ (0.3 Mpc$^2$) area in the core of the Virgo cluster, based on $ugriz$ data from the Next Generation Virgo Cluster Survey (NGVS). The galaxy sample consists of 352 objects brighter than $M_g=-9.13$ mag, the 50% completeness limit of the survey. Using a Bayesian analysis, we find a best-fit faint end slope of $\alpha=-1.33 \pm 0.02$ for the g-band luminosity function; consistent results are found for the stellar mass function as well as the luminosity function in the other four NGVS bandpasses. We discuss the implications for the faint-end slope of adding 92 ultra compact dwarfs galaxies (UCDs) -- previously compiled by the NGVS in this region -- to the galaxy sample, assuming that UCDs are the stripped remnants of nucleated dwarf galaxies. Under this assumption, the slope of the luminosity function (down to the UCD faint magnitude limit, $M_g = -9.6$ mag) increases dramatically, up to $\alpha = -1.60 \pm 0.06$ when correcting for the expected number of disrupted non-nucleated galaxies. We also calculate the total number of UCDs and globular clusters that may have been deposited in the core of Virgo due to the disruption of satellites, both nucleated and non-nucleated. We estimate that ~150 objects with $M_g\lesssim-9.6$ mag and that are currently classified as globular clusters, might, in fact, be the nuclei of disrupted galaxies. We further estimate that as many as 40% of the (mostly blue) globular clusters in the core of Virgo might once have belonged to such satellites; these same disrupted satellites might have contributed ~40% of the total luminosity in galaxies observed in the core region today. Finally, we use an updated Local Group galaxy catalog to provide a new measurement of the luminosity function of Local Group satellites, $\alpha=-1.21\pm0.05$.

### Constraints on primordial black holes from Galactic gamma-ray background [Cross-Listing]

The fraction of the Universe going into primordial black holes (PBHs) with initial mass M_* \approx 5 \times 10^{14} g, such that they are evaporating at the present epoch, is strongly constrained by observations of both the extragalactic and Galactic gamma-ray backgrounds. However, while the dominant contribution to the extragalactic background comes from the time-integrated emission of PBHs with initial mass M_*, the Galactic background is dominated by the instantaneous emission of those with initial mass slightly larger than M_* and current mass below M_*. Also, the instantaneous emission of PBHs smaller than 0.4 M_* mostly comprises secondary particles produced by the decay of directly emitted quark and gluon jets. These points were missed in the earlier analysis by Lehoucq et al. using EGRET data. For a monochromatic PBH mass function, with initial mass (1+\mu) M_* and \mu << 1, the current mass is (3\mu)^{1/3} M_* and the Galactic background constrains the fraction of the Universe going into PBHs as a function of \mu. However, the initial mass function cannot be precisely monochromatic and even a tiny spread of mass around M_* would generate a current low-mass tail of PBHs below M_*. This tail would be the main contributor to the Galactic background, so we consider its form and the associated constraints for a variety of scenarios with both extended and nearly-monochromatic initial mass functions. In particular, we consider a scenario in which the PBHs form from critical collapse and have a mass function which peaks well above M_*. In this case, the largest PBHs could provide the dark matter without the M_* ones exceeding the gamma-ray background limits.

### Constraints on primordial black holes from Galactic gamma-ray background [Cross-Listing]

The fraction of the Universe going into primordial black holes (PBHs) with initial mass M_* \approx 5 \times 10^{14} g, such that they are evaporating at the present epoch, is strongly constrained by observations of both the extragalactic and Galactic gamma-ray backgrounds. However, while the dominant contribution to the extragalactic background comes from the time-integrated emission of PBHs with initial mass M_*, the Galactic background is dominated by the instantaneous emission of those with initial mass slightly larger than M_* and current mass below M_*. Also, the instantaneous emission of PBHs smaller than 0.4 M_* mostly comprises secondary particles produced by the decay of directly emitted quark and gluon jets. These points were missed in the earlier analysis by Lehoucq et al. using EGRET data. For a monochromatic PBH mass function, with initial mass (1+\mu) M_* and \mu << 1, the current mass is (3\mu)^{1/3} M_* and the Galactic background constrains the fraction of the Universe going into PBHs as a function of \mu. However, the initial mass function cannot be precisely monochromatic and even a tiny spread of mass around M_* would generate a current low-mass tail of PBHs below M_*. This tail would be the main contributor to the Galactic background, so we consider its form and the associated constraints for a variety of scenarios with both extended and nearly-monochromatic initial mass functions. In particular, we consider a scenario in which the PBHs form from critical collapse and have a mass function which peaks well above M_*. In this case, the largest PBHs could provide the dark matter without the M_* ones exceeding the gamma-ray background limits.

### Constraints on primordial black holes from Galactic gamma-ray background

The fraction of the Universe going into primordial black holes (PBHs) with initial mass M_* \approx 5 \times 10^{14} g, such that they are evaporating at the present epoch, is strongly constrained by observations of both the extragalactic and Galactic gamma-ray backgrounds. However, while the dominant contribution to the extragalactic background comes from the time-integrated emission of PBHs with initial mass M_*, the Galactic background is dominated by the instantaneous emission of those with initial mass slightly larger than M_* and current mass below M_*. Also, the instantaneous emission of PBHs smaller than 0.4 M_* mostly comprises secondary particles produced by the decay of directly emitted quark and gluon jets. These points were missed in the earlier analysis by Lehoucq et al. using EGRET data. For a monochromatic PBH mass function, with initial mass (1+\mu) M_* and \mu << 1, the current mass is (3\mu)^{1/3} M_* and the Galactic background constrains the fraction of the Universe going into PBHs as a function of \mu. However, the initial mass function cannot be precisely monochromatic and even a tiny spread of mass around M_* would generate a current low-mass tail of PBHs below M_*. This tail would be the main contributor to the Galactic background, so we consider its form and the associated constraints for a variety of scenarios with both extended and nearly-monochromatic initial mass functions. In particular, we consider a scenario in which the PBHs form from critical collapse and have a mass function which peaks well above M_*. In this case, the largest PBHs could provide the dark matter without the M_* ones exceeding the gamma-ray background limits.

### Constraints on primordial black holes from Galactic gamma-ray background [Cross-Listing]

The fraction of the Universe going into primordial black holes (PBHs) with initial mass M_* \approx 5 \times 10^{14} g, such that they are evaporating at the present epoch, is strongly constrained by observations of both the extragalactic and Galactic gamma-ray backgrounds. However, while the dominant contribution to the extragalactic background comes from the time-integrated emission of PBHs with initial mass M_*, the Galactic background is dominated by the instantaneous emission of those with initial mass slightly larger than M_* and current mass below M_*. Also, the instantaneous emission of PBHs smaller than 0.4 M_* mostly comprises secondary particles produced by the decay of directly emitted quark and gluon jets. These points were missed in the earlier analysis by Lehoucq et al. using EGRET data. For a monochromatic PBH mass function, with initial mass (1+\mu) M_* and \mu << 1, the current mass is (3\mu)^{1/3} M_* and the Galactic background constrains the fraction of the Universe going into PBHs as a function of \mu. However, the initial mass function cannot be precisely monochromatic and even a tiny spread of mass around M_* would generate a current low-mass tail of PBHs below M_*. This tail would be the main contributor to the Galactic background, so we consider its form and the associated constraints for a variety of scenarios with both extended and nearly-monochromatic initial mass functions. In particular, we consider a scenario in which the PBHs form from critical collapse and have a mass function which peaks well above M_*. In this case, the largest PBHs could provide the dark matter without the M_* ones exceeding the gamma-ray background limits.

### Do open star clusters evolve toward energy equipartition?

We investigate whether open clusters (OCs) tend to energy equipartition, by means of direct N-body simulations with a broken power-law mass function. We find that the simulated OCs become strongly mass segregated, but the local velocity dispersion does not depend on the stellar mass for most of the mass range: the curve of the velocity dispersion as a function of mass is nearly flat even after several half-mass relaxation times, regardless of the adopted stellar evolution recipes and Galactic tidal field model. This result holds both if we start from virialized King models and if we use clumpy sub-virial initial conditions. The velocity dispersion of the most massive stars and stellar remnants tends to be higher than the velocity dispersion of the lighter stars. This trend is particularly evident in simulations without stellar evolution. We interpret this result as a consequence of the strong mass segregation, which leads to Spitzer's instability. Stellar winds delay the onset of the instability. Our simulations strongly support the result that OCs do not attain equipartition, for a wide range of initial conditions.

### Internal Structure of Charged AdS Black Holes [Replacement]

When an electrically charged black hole is perturbed its inner horizon becomes a singularity, often referred to as the Poisson-Israel mass inflation singularity. Ori constructed a model of this phenomenon for asymptotically flat black holes, in which the metric can be determined explicitly in the mass inflation region. In this paper we implement the Ori model for charged AdS black holes. We find that the mass function inflates faster than the flat space case as the inner horizon is approached. Nevertheless, the mass inflation singularity is still a weak singularity: although spacetime curvature becomes infinite, tidal distortions remain finite on physical objects attempting to cross it.

### Internal Structure of Charged AdS Black Holes [Cross-Listing]

When an electrically charged black hole is perturbed its inner horizon becomes a singularity, often referred to as the Poisson-Israel mass inflation singularity. Ori constructed a model of this phenomenon for asymptotically flat black holes, in which the metric can be determined explicitly in the mass inflation region. In this paper we implement the Ori model for charged AdS black holes. We find that the mass function inflates faster than the flat space case as the inner horizon is approached. Nevertheless, the mass inflation singularity is still a weak singularity: although spacetime curvature becomes infinite, tidal distortions remain finite on physical objects attempting to cross it.

### Internal Structure of Charged AdS Black Holes [Replacement]

When an electrically charged black hole is perturbed its inner horizon becomes a singularity, often referred to as the Poisson-Israel mass inflation singularity. Ori constructed a model of this phenomenon for asymptotically flat black holes, in which the metric can be determined explicitly in the mass inflation region. In this paper we implement the Ori model for charged AdS black holes. We find that the mass function inflates faster than the flat space case as the inner horizon is approached. Nevertheless, the mass inflation singularity is still a weak singularity: although spacetime curvature becomes infinite, tidal distortions remain finite on physical objects attempting to cross it.

### Internal Structure of Charged AdS Black Holes

When an electrically charged black hole is perturbed its inner horizon becomes a singularity, often referred to as the Poisson-Israel mass inflation singularity. Ori constructed a model of this phenomenon for asymptotically flat black holes, in which the metric can be determined explicitly in the mass inflation region. In this paper we implement the Ori model for charged AdS black holes. We find that the mass function inflates faster than the flat space case as the inner horizon is approached. Nevertheless, the mass inflation singularity is still a weak singularity: although spacetime curvature becomes infinite, tidal distortions remain finite on physical objects attempting to cross it.

### The high mass end of the stellar mass function: Dependence on stellar population models and agreement between fits to the light profile

We quantify the systematic effects on the stellar mass function which arise from assumptions about the stellar population, as well as how one fits the light profiles of the most luminous galaxies at z ~ 0.1. When comparing results from the literature, we are careful to separate out these effects. Our analysis shows that while systematics in the estimated comoving number density which arise from different treatments of the stellar population remain of order < 0.5 dex, systematics in photometry are now about 0.1 dex, despite recent claims in the literature. Compared to these more recent analyses, previous work based on Sloan Digital Sky Survey (SDSS) pipeline photometry leads to underestimates of rho_*(> M_*) by factors of 3-10 in the mass range 10^11 - 10^11.6 M_Sun, but up to a factor of 100 at higher stellar masses. This impacts studies which match massive galaxies to dark matter halos. Although systematics which arise from different treatments of the stellar population remain of order < 0.5 dex, our finding that systematics in photometry now amount to only about 0.1 dex in the stellar mass density is a significant improvement with respect to a decade ago. Our results highlight the importance of using the same stellar population and photometric models whenever low and high redshift samples are compared.

### Stellar mass functions: methods, systematics and results for the local Universe

We present a comprehensive method for determining stellar mass functions, and apply it to samples in the local Universe. We combine the classical 1/Vmax approach with STY, a parametric maximum likelihood method and SWML, a non-parametric maximum likelihood technique. In the parametric approach, we are assuming that the stellar mass function can be modelled by either a single or a double Schechter function and we use a likelihood ratio test to determine which model provides a better fit to the data. We discuss how the stellar mass completeness as a function of z biases the three estimators and how it can affect, especially the low mass end of the stellar mass function. We apply our method to SDSS DR7 data in the redshift range from 0.02 to 0.06. We find that the entire galaxy sample is best described by a double Schechter function with the following parameters: $\log (M^{*}/M_\odot) = 10.79 \pm 0.01$, $\log (\Phi^{*}_1/\mathrm{h^3\ Mpc^{-3}}) = -3.31 \pm 0.20$, $\alpha_1 = -1.69 \pm 0.10$, $\log (\Phi^{*}_2/\mathrm{h^3\ Mpc^{-3}}) = -2.01 \pm 0.28$ and $\alpha_2 = -0.79 \pm 0.04$. We also use morphological classifications from Galaxy Zoo and halo mass, overdensity, central/satellite, colour and sSFR measurements to split the galaxy sample into over 130 subsamples. We determine and present the stellar mass functions and the best fit Schechter function parameters for each of these subsamples.

### Dynamical ejections of massive stars from young star clusters under diverse initial conditions [Replacement]

We study the effects of initial conditions of star clusters and their massive star population on dynamical ejections of massive stars from star clusters up to an age of 3 Myr. We use a large set of direct N-body calculations for moderately massive star clusters (Mecl=$10^{3.5}$ Msun). We vary the initial conditions of the calculations such as the initial half-mass radius of the clusters, initial binary populations for massive stars and initial mass segregation. We find that the initial density is the most influential parameter for the ejection fraction of the massive systems. The clusters with an initial half-mass radius of 0.1 (0.3) pc can eject up to 50% (30%) of their O-star systems on average. Most of the models show that the average ejection fraction decreases with decreasing stellar mass. For clusters efficient at ejecting O stars, the mass function of the ejected stars is top-heavy compared to the given initial mass function (IMF), while the mass function of stars that remain in the cluster becomes slightly steeper (top-light) than the IMF. The top-light mass functions of stars in 3 Myr old clusters in our N-body models agree well with the mean mass function of young intermediate-mass clusters in M31, as reported previously. We show that the multiplicity fraction of the ejected massive stars can be as high as 60%, that massive high-order multiple systems can be dynamically ejected, and that high-order multiples become common especially in the cluster. We also discuss binary populations of the ejected massive systems. When a large kinematic survey of massive field stars becomes available, for instance through Gaia, our results may be used to constrain the birth configuration of massive stars in star clusters. (Abridged)

### Dynamical ejections of massive stars from young star clusters under diverse initial conditions

We study the effects of initial conditions of star clusters and their massive star population on dynamical ejections of stars from star clusters up to an age of 3 Myr, particularly focusing on massive systems, using a large set of direct N-body calculations for moderately massive star clusters (Mecl=$10^{3.5}$ Msun). We vary the initial conditions of the calculations such as the initial half-mass radius of the clusters, initial binary populations for massive stars and initial mass segregation. We find that the initial density is the most influential parameter for the ejection fraction of the massive systems. The clusters with an initial half-mass radius of 0.1 (0.3) pc can eject up to 50% (30)% of their O-star systems on average. Most of the models show that the average ejection fraction decreases with decreasing stellar mass. For clusters efficient at ejecting O stars, the mass function of the ejected stars is top-heavy compared to the given initial mass function (IMF), while the mass function of stars remaining in the cluster becomes slightly steeper (top-light) than the IMF. The top-light mass functions of stars in 3 Myr old clusters in our N-body models are in good agreement with the mean mass function of young intermediate mass clusters in M31 as found by Weisz et al.. We show that the multiplicity fraction of the ejected massive stars can be as high as 60%, that massive high-order multiple systems can be dynamically ejected, and that high-order multiples become common especially in the cluster. Furthermore, binary populations of the ejected massive systems are discussed. When a large survey of the kinematics of the field massive stars becomes available, e.g through Gaia, our results may be used to constrain the birth configuration of massive stars in star clusters. (Abridged)

### Building up the Population III initial mass function from cosmological initial conditions

We perform simulations of the growth of a Population III stellar system, starting from cosmological initial conditions at z=100. We self-consistently follow the formation of a minihalo and the subsequent collapse of its central gas to high densities, resolving scales as small as ~ 1 AU. Using sink particles to represent the growing protostars, we model the growth of the photodissociating and ionizing region around the first sink, continuing the simulation for ~ 5000 years after initial protostar formation. In addition to the first-forming sink, several tens of secondary sinks form before an ionization front develops around the most massive star. The resulting cluster has high rates of sink formation, ejections from the stellar disk, and sink mergers during the first ~ 2000 yr, before the onset of radiative feedback. By this time a warm ~ 5000 K phase of neutral gas has expanded to roughly the disk radius of 2000 AU, slowing mass flow onto the disk and sinks. By the end of the simulation, the most massive star grows to 20 M_Sun, while the total stellar mass approaches 75 M_Sun. Out of the ~ 40 sinks, approximately 30 are low-mass (M_*< 1 M_Sun). We therefore find that protostellar radiative feedback is insufficient to prevent rapid disk fragmentation and the formation of a high-member Pop III cluster before an ionization front emerges. Throughout the simulation, the majority of stellar mass is contained within the most massive stars, further implying that the Pop III initial mass function is top-heavy.

### On maximal analytical extension of the Vaidya metric [Replacement]

The classical Vaidya metric is transformed to the special diagonal coordinates in the case of the linear mass function allowing rather easy treatment. We find the exact analytical expressions for metric functions in these diagonal coordinates. Using these coordinates, we elaborate the maximum analytic extension of the Vaidya metric with a linear growth of the black hole mass and construct the corresponding Carter-Penrose diagrams for different specific cases. The derived global geometry seemingly is valid also for a more general behavior of the black hole mass in the Vaidya metric.

### On maximal analytical extension of the Vaidya metric [Replacement]

The classical Vaidya metric is transformed to the special diagonal coordinates in the case of the linear mass function allowing rather easy treatment. We find the exact analytical expressions for metric functions in these diagonal coordinates. Using these coordinates, we elaborate the maximum analytic extension of the Vaidya metric with a linear growth of the black hole mass and construct the corresponding Carter-Penrose diagrams for different specific cases. The derived global geometry seemingly is valid also for a more general behavior of the black hole mass in the Vaidya metric.

### On maximal analytical extension of the Vaidya metric [Cross-Listing]

The classical Vaidya metric is transformed to the special diagonal coordinates in the case of the linear mass function allowing rather easy treatment. Using these coordinates, we elaborate the maximum analytic extension of the Vaidya metric with a linear growth of the black hole mass and construct the corresponding Carter-Penrose diagrams for different specific cases. The derived global geometry seemingly is valid also for a more general behavior of the black hole mass in the Vaidya metric.

### On maximal analytical extension of the Vaidya metric

The classical Vaidya metric is transformed to the special diagonal coordinates in the case of the linear mass function allowing rather easy treatment. Using these coordinates, we elaborate the maximum analytic extension of the Vaidya metric with a linear growth of the black hole mass and construct the corresponding Carter-Penrose diagrams for different specific cases. The derived global geometry seemingly is valid also for a more general behavior of the black hole mass in the Vaidya metric.

### The very low-mass stellar content of the young supermassive Galactic star cluster Westerlund 1

We present deep near-infrared HST/WFC3 observations of the young supermassive Galactic star cluster Westerlund 1 and an adjacent control field. The depth of the data is sufficient to derive the mass function for the cluster as a function of radius down to 0.15 M$_\odot$ in the outer parts of the cluster. We identify for the first time a flattening in the mass function (in logarithmic units) at a mass range that is consistent with that of the field and nearby embedded clusters. Through log-normal functional fits to the mass functions we find the nominal peak mass to be comparable to that of the field and nearby embedded star clusters. The width of a log-normal fit appears slightly narrow compared to the width of the field IMF, closer to the values found for globular clusters. The subsolar content within the cluster does not appear to be mass segregated in contrast to the findings for the supersolar content. The total mass of Westerlund 1 is estimated to be 44-57 $\times 10^3$ M$_\odot$ where the main uncertainty is the choice of the isochrone age and the higher mass slope. Comparing the photometric mass with the dynamically determined mass, Westerlund 1 is sufficiently massive to remain bound and could potentially evolve into a low-mass globular cluster.

### A census of dense cores in the Taurus L1495 cloud from the Herschel Gould Belt Survey [Replacement]

We present a catalogue of dense cores in a $\sim 4^\circ\times2^\circ$ field of the Taurus star-forming region, inclusive of the L1495 cloud, derived from Herschel SPIRE and PACS observations in the 70 $\mu$m, 160 $\mu$m, 250 $\mu$m, 350 $\mu$m, and 500 $\mu$m continuum bands. Estimates of mean dust temperature and total mass are derived using modified blackbody fits to the spectral energy distributions. We detect 528 starless cores of which $\sim10$-20% are gravitationally bound and therefore presumably prestellar. Our census of unbound objects is $\sim85$% complete for $M>0.015\,M_\odot$ in low density regions ($A_V\stackrel{<}{_\sim}5$ mag), while the bound (prestellar) subset is $\sim85$% complete for $M>0.1\,M_\odot$ overall. The prestellar core mass function (CMF) is consistent with lognormal form, resembling the stellar system initial mass function, as has been reported previously. All of the inferred prestellar cores lie on filamentary structures whose column densities exceed the expected threshold for filamentary collapse, in agreement with previous reports. Unlike the prestellar CMF, the unbound starless CMF is not lognormal, but instead is consistent with a power-law form below $0.3\,M_\odot$ and shows no evidence for a low-mass turnover. It resembles previously reported mass distributions for CO clumps at low masses ($M\stackrel{<}{_\sim}0.3\,M_\odot$). The volume density PDF, however, is accurately lognormal except at high densities. It is consistent with the effects of self-gravity on magnetized supersonic turbulence. The only significant deviation from lognormality is a high-density tail which can be attributed unambiguously to prestellar cores.