# Posts Tagged mass function

## Recent Postings from mass function

### Direct $N$-body simulations of globular clusters - II. Palomar 4

We use direct $N$-body calculations to study the evolution of the unusually extended outer halo globular cluster Palomar 4 (Pal~4) over its entire lifetime in order to reproduce its observed mass, half-light radius, velocity dispersion and mass function slope at different radii. We find that models evolving on circular orbits, and starting from a non-mass segregated, canonical initial mass function (IMF) can reproduce neither Pal 4′s overall mass function slope nor the observed amount of mass segregation. Including either primordial mass segregation or initially flattened IMFs does not reproduce the observed amount of mass segregation and mass function flattening simultaneously. Unresolved binaries cannot reconcile this discrepancy either. We find that only models with both a flattened IMF and primordial segregation are able to fit the observations. The initial (i.e. after gas expulsion) mass and half-mass radius of Pal~4 in this case are about 57000 M${\odot}$ and 10 pc, respectively. This configuration is more extended than most globular clusters we observe, showing that the conditions under which Pal~4 formed must have been significantly different from that of the majority of globular clusters. We discuss possible scenarios for such an unusual configuration of Pal~4 in its early years.

### Variations in the initial mass function in early-type galaxies: A critical comparison between dynamical and spectroscopic results

I present a comparison between published dynamical (ATLAS3D) and spectroscopic (Conroy & van Dokkum) constraints on the stellar initial mass function (IMF) in early-type galaxies, using the 34 galaxies in common between the two works. Both studies infer an average IMF mass factor $\alpha$ (the stellar mass relative to a Kroupa-IMF population of similar age and metallicity) greater than unity, i.e. both methods favour an IMF which is heavier than that of the Milky Way, on average over the sample. However, on a galaxy-by-galaxy basis, there is no correlation between $\alpha$ inferred from the two approaches. I investigate how the two estimates of $\alpha$ are correlated systematically with the galaxy velocity dispersion, $\sigma$, and with the Mg/Fe abundance ratio. The spectroscopic method, based on the strengths of metal absorption lines, yields a correlation only with metal abundance ratios: at fixed Mg/Fe, there is no residual correlation with $\sigma$. The dynamical method, applied to exactly the same galaxy sample, yields the opposite result: the IMF variation correlates only with dynamics, with no residual correlation with Mg/Fe after controlling for $\sigma$. Hence although both methods indicate a heavy IMF on average in ellipticals, they lead to incompatible results for the systematic trends, when applied to the same set of galaxies. Although other explanations are possible, the sense of the disagreement suggests that one (or both) of the methods has not accounted fully for the main confounding factors, i.e. element abundance ratios or dark matter contributions.

### Gaia photometry for white dwarfs

Context. White dwarfs can be used to study the structure and evolution of the Galaxy by analysing their luminosity function and initial mass function. Among them, the very cool white dwarfs provide the information for the early ages of each population. Because white dwarfs are intrinsically faint only the nearby (about 20 pc) sample is reasonably complete. The Gaia space mission will drastically increase the sample of known white dwarfs through its 5-6 years survey of the whole sky up to magnitude V = 20-25. Aims. We provide a characterisation of Gaia photometry for white dwarfs to better prepare for the analysis of the scientific output of the mission. Transformations between some of the most common photometric systems and Gaia passbands are derived. We also give estimates of the number of white dwarfs of the different galactic populations that will be observed. Methods. Using synthetic spectral energy distributions and the most recent Gaia transmission curves, we computed colours of three different types of white dwarfs (pure hydrogen, pure helium, and mixed composition with H/He= 0.1). With these colours we derived transformations to other common photometric systems (Johnson-Cousins, Sloan Digital Sky Survey, and 2MASS). We also present numbers of white dwarfs predicted to be observed by Gaia. Results. We provide relationships and colour-colour diagrams among different photometric systems to allow the prediction and/or study of the Gaia white dwarf colours. We also include estimates of the number of sources expected in every galactic population and with a maximum parallax error. Gaia will increase the sample of known white dwarfs tenfold to about 200 000. Gaia will be able to observe thousands of very cool white dwarfs for the first time, which will greatly improve our understanding of these stars and early phases of star formation in our Galaxy.

### Cosmology with Galaxy Clusters: Systematic Effects in the Halo Mass Function

We investigate potential systematic effects in constraining the amplitude of primordial fluctuations \sigma_8 arising from the choice of halo mass function in the likelihood analysis of current and upcoming galaxy cluster surveys. We study the widely used N-body simulation fit of Tinker et al. (T08) and, as an alternative, the recently proposed analytical model of Excursion Set Peaks (ESP). We first assess the relative bias between these prescriptions when constraining \sigma_8 by sampling the ESP mass function to generate mock catalogs and using the T08 fit to analyse them, for various choices of survey selection threshold, mass definition and statistical priors. To assess the level of absolute bias in each prescription, we then repeat the analysis on dark matter halo catalogs in N-body simulations designed to mimic the mass distribution in the current data release of Planck SZ clusters. This N-body analysis shows that using the T08 fit without accounting for the scatter introduced when converting between mass definitions (alternatively, the scatter induced by errors on the parameters of the fit) can systematically over-estimate the value of \sigma_8 by as much as 2\sigma\ for current data, while analyses that account for this scatter should be close to unbiased in \sigma_8. With an increased number of objects as expected in upcoming data releases, regardless of accounting for scatter, the T08 fit could over-estimate the value of \sigma_8 by ~1.5\sigma. The ESP mass function leads to systematically more biased but comparable results. A strength of the ESP model is its natural prediction of a weak non-universality in the mass function which closely tracks the one measured in simulations and described by the T08 fit. We suggest that it might now be prudent to build new unbiased ESP-based fitting functions for use with the larger datasets of the near future.

### Setting new Cosmology constraints with ALMA [Replacement]

I make a short revision of Cosmology questions which ALMA was built to address. Without diving into much detail, I point out the ALMA specifications and strategies which are expected to provide a better handle of: the temperature evolution of the Cosmic Microwave Background (CMB) and the properties of its secondary anisotropies (such as the thermal and kinetic Sunyaev-Zel’dovich and the Ostriker-Vishniac effects); variability of dimensionless fundamental constants; Ho and galaxy initial mass function by means of strong gravitational lensing; black hole science with the greatly expected Event Horizon Telescope.

### Setting new Cosmology constraints with ALMA

I make a short revision of Cosmology questions which ALMA was built to address. Without diving into much detail, I point out the ALMA specifications and strategies which are expected to provide a better handle of: the temperature evolution of the Cosmic Microwave Background (CMB) and the properties of its secondary anisotropies (such as the thermal and kinetic Sunyaev-Zel’dovich and the Ostriker-Vishniac effects); variability of dimensionless fundamental constants; Ho and galaxy initial mass function by means of strong gravitational lensing; black hole science with the greatly expected Event Horizon Telescope.

### A Model for Multi-property Galaxy Cluster Statistics

The massive dark matter halos that host groups and clusters of galaxies have observable properties that appear to be log-normally distributed about power-law mean scaling relations in halo mass. Coupling this assumption with either quadratic or cubic approximations to the mass function in log space, we derive closed-form expressions for the space density of halos as a function of multiple observables as well as forms for the low-order moments of properties of observable-selected samples. Using a Tinker mass function in a {\Lambda}CDM cosmology, we show that the cubic analytic model reproduces results obtained from direct, numerical convolution at the 10 percent level or better over nearly the full range of observables covered by current observations and for redshifts extending to z = 1.5. The model provides an efficient framework for estimating effects arising from selection and covariance among observable properties in survey samples.

### 350 $\mu$m map of the Ophiuchus molecular cloud: core mass function

Stars are born in dense cores of molecular clouds. The core mass function (CMF), which is the mass distribution of dense cores, is important for understanding the stellar initial mass function (IMF). We obtained 350 $\mu$m dust continuum data using the SHARC-II camera at the Caltech Submillimeter Observatory (CSO) telescope. A 350 $\mu$m map covering 0.25 ${deg}^{2}$ of the Ophiuchus molecular cloud was created by mosaicing 56 separate scans. The CSO telescope had an angular resolution of 9 $^{\prime\prime}$, corresponding to $1.2\times {10}^{3}\$AU at the distance of the Ophiuchus molecular cloud (131 pc). The data was reduced using the Comprehensive Reduction Utility for SHARC-II (CRUSH). The flux density map was analyzed using the GaussClumps algorithm, within which 75 cores has been identified. We used the Spitzer c2d catalogs to separate the cores into 63 starless cores and 12 protostellar cores. By locating Jeans instabilities, 55 prestellar cores (a subcategory of starless cores) were also identified. The excitation temperatures, which were derived from FCRAO ${}^{12}$CO data, help to improve the accuracy of the masses of the cores. We adopted a Monte Carlo approach to analyze the CMF with two types of functional forms; power law and log-normal. The whole and prestellar CMF are both well fitted by a log-normal distribution, with $\mu =-1.18\pm0.10,\ \sigma =0.58\pm0.05$ and $\mu =1.40\pm0.10,\ \sigma =0.50\pm0.05$ respectively. This finding suggests that turbulence influences the evolution of the Ophiuchus molecular cloud.

### 350 $\mu$m map of the Ophiuchus molecular cloud: core mass function [Replacement]

Stars are born in dense cores of molecular clouds. The core mass function (CMF), which is the mass distribution of dense cores, is important for understanding the stellar initial mass function (IMF). We obtained 350 $\mu$m dust continuum data using the SHARC-II camera at the Caltech Submillimeter Observatory (CSO) telescope. A 350 $\mu$m map covering 0.25 ${deg}^{2}$ of the Ophiuchus molecular cloud was created by mosaicing 56 separate scans. The CSO telescope had an angular resolution of 9 $^{\prime\prime}$, corresponding to $1.2\times {10}^{3}\$AU at the distance of the Ophiuchus molecular cloud (131 pc). The data was reduced using the Comprehensive Reduction Utility for SHARC-II (CRUSH). The flux density map was analyzed using the GaussClumps algorithm, within which 75 cores has been identified. We used the Spitzer c2d catalogs to separate the cores into 63 starless cores and 12 protostellar cores. By locating Jeans instabilities, 55 prestellar cores (a subcategory of starless cores) were also identified. The excitation temperatures, which were derived from FCRAO ${}^{12}$CO data, help to improve the accuracy of the masses of the cores. We adopted a Monte Carlo approach to analyze the CMF with two types of functional forms; power law and log-normal. The whole and prestellar CMF are both well fitted by a log-normal distribution, with $\mu =-1.18\pm0.10,\ \sigma =0.58\pm0.05$ and $\mu =1.40\pm0.10,\ \sigma =0.50\pm0.05$ respectively. This finding suggests that turbulence influences the evolution of the Ophiuchus molecular cloud.

### 350 $\mu$m map of the Ophiuchus molecular cloud: core mass function [Replacement]

Stars are born in dense cores of molecular clouds. The core mass function (CMF), which is the mass distribution of dense cores, is important for understanding the stellar initial mass function (IMF). We obtained 350 $\mu$m dust continuum data using the SHARC-II camera at the Caltech Submillimeter Observatory (CSO) telescope. A 350 $\mu$m map covering 0.25 ${deg}^{2}$ of the Ophiuchus molecular cloud was created by mosaicing 56 separate scans. The CSO telescope had an angular resolution of 9 $^{\prime\prime}$, corresponding to $1.2\times {10}^{3}\$AU at the distance of the Ophiuchus molecular cloud (131 pc). The data was reduced using the Comprehensive Reduction Utility for SHARC-II (CRUSH). The flux density map was analyzed using the GaussClumps algorithm, within which 75 cores has been identified. We used the Spitzer c2d catalogs to separate the cores into 63 starless cores and 12 protostellar cores. By locating Jeans instabilities, 55 prestellar cores (a subcategory of starless cores) were also identified. The excitation temperatures, which were derived from FCRAO ${}^{12}$CO data, help to improve the accuracy of the masses of the cores. We adopted a Monte Carlo approach to analyze the CMF with two types of functional forms; power law and log-normal. The whole and prestellar CMF are both well fitted by a log-normal distribution, with $\mu =-1.18\pm0.10,\ \sigma =0.58\pm0.05$ and $\mu =1.40\pm0.10,\ \sigma =0.50\pm0.05$ respectively. This finding suggests that turbulence influences the evolution of the Ophiuchus molecular cloud.

### 350 $\mu$m map of the Ophiuchus molecular cloud: core mass function [Replacement]

Stars are born in dense cores of molecular clouds. The core mass function (CMF), which is the mass distribution of dense cores, is important for understanding the stellar initial mass function (IMF). We obtained 350 $\mu$m dust continuum data using the SHARC-II camera at the Caltech Submillimeter Observatory (CSO) telescope. A 350 $\mu$m map covering 0.25 ${deg}^{2}$ of the Ophiuchus molecular cloud was created by mosaicing 56 separate scans. The CSO telescope had an angular resolution of 9 $^{\prime\prime}$, corresponding to $1.2\times {10}^{3}\$AU at the distance of the Ophiuchus molecular cloud (131 pc). The data was reduced using the Comprehensive Reduction Utility for SHARC-II (CRUSH). The flux density map was analyzed using the GaussClumps algorithm, within which 75 cores has been identified. We used the Spitzer c2d catalogs to separate the cores into 63 starless cores and 12 protostellar cores. By locating Jeans instabilities, 55 prestellar cores (a subcategory of starless cores) were also identified. The excitation temperatures, which were derived from FCRAO ${}^{12}$CO data, help to improve the accuracy of the masses of the cores. We adopted a Monte Carlo approach to analyze the CMF with two types of functional forms; power law and log-normal. The whole and prestellar CMF are both well fitted by a log-normal distribution, with $\mu =-1.18\pm0.10,\ \sigma =0.58\pm0.05$ and $\mu =1.40\pm0.10,\ \sigma =0.50\pm0.05$ respectively. This finding suggests that turbulence influences the evolution of the Ophiuchus molecular cloud.

### Salpeter normalization of the Stellar Initial Mass Function for massive galaxies at z~1

The stellar initial mass function (IMF) is a key parameter to study galaxy evolution. Here we measure the IMF mass normalization for a sample of 68 field galaxies in the redshift range 0.7 to 0.9 within the Extended Groth Strip. To do this we derive total (stellar + dark matter) mass-to-light [$(M/L)_{\rm dyn}$] using axisymmetric dynamical models. Within the region where we have kinematics (about one half-light radius), the models assume: (i) that mass-follows-light, implying negligible differences between the stellar and total density profiles; (ii) constant velocity anisotropy ($\beta_{\rm z}\equiv1-\sigma_z^2/\sigma_R^2=0.2$); (iii) that galaxies are seen at the average inclination for random orientations (i.e. $i=60^\circ$, where $i=90^\circ$ represents edge-on). The dynamical models are based on anisotropic Jeans equations, constrained by HST/ACS imaging and the central velocity dispersion of the galaxies, extracted from good-quality spectra taken by the DEEP2 survey. The population $(M/L)_{\rm pop}$ are derived from full-spectrum fitting of the same spectra with a grid of simple stellar population models. Recent dynamical modelling results by the $ATLAS^{3D}$ project and numerical simulations of galaxy evolution indicate that the dark matter fraction within the central regions of our galaxies should be small. This suggest that our derived total $(M/L)_{\rm dyn}$ should closely approximate the stellar $M/L$. Our comparison of $(M/L)_{\rm dyn}$ and $(M/L)_{\rm pop}$ then imply that for galaxies with stellar mass $M_\odot \geq 10^{11}$ $M_{\odot}$, the $average$ normalization of the IMF is consistent with a Salpeter slope, with a substantial scatter. This is similar to what is found within a similar mass range for nearby galaxies.

### The Panchromatic Hubble Andromeda Treasury V: Ages and Masses of the Year 1 Stellar Clusters

We present ages and masses for 601 star clusters in M31 from the analysis of the six filter integrated light measurements from near ultraviolet to near infrared wavelengths, made as part of the Panchromatic Hubble Andromeda Treasury (PHAT). We derive the ages and masses using a probabilistic technique, which accounts for the effects of stochastic sampling of the stellar initial mass function. Tests on synthetic data show that this method, in conjunction with the exquisite sensitivity of the PHAT observations and their broad wavelength baseline, provides robust age and mass recovery for clusters ranging from $\sim 10^2 – 2 \times 10^6 M_\odot$. We find that the cluster age distribution is consistent with being uniform over the past $100$ Myr, which suggests a weak effect of cluster disruption within M31. The age distribution of older ($>100$ Myr) clusters fall towards old ages, consistent with a power-law decline of index $-1$, likely from a combination of fading and disruption of the clusters. We find that the mass distribution of the whole sample can be well-described by a single power-law with a spectral index of $-1.9 \pm 0.1$ over the range of $10^3-3 \times 10^5 M_\odot$. However, if we subdivide the sample by galactocentric radius, we find that the age distributions remain unchanged. However, the mass spectral index varies significantly, showing best fit values between $-2.2$ and $-1.8$, with the shallower slope in the highest star formation intensity regions. We explore the robustness of our study to potential systematics and conclude that the cluster mass function may vary with respect to environment.

### Chemo-Archeological Downsizing in a Hierarchical Universe: Impact of a Top Heavy IGIMF

We make use of a semi-analytical model of galaxy formation to investigate the origin of the observed correlation between [a/Fe] abundance ratios and stellar mass in elliptical galaxies.We implement a new galaxy-wide stellar initial mass function (Top Heavy Integrated Galaxy Initial Mass Function, TH-IGIMF) in the semi-analytic model SAG and evaluate its impact on the chemical evolution of galaxies. The SFR-dependence of the slope of the TH-IGIMF is found to be key to reproducing the correct [a/Fe]-stellar mass relation. Massive galaxies reach higher [a/Fe] abundance ratios because they are characterized by more top heavy IMFs as a result of their higher SFR. As a consequence of our analysis, the value of the minimum embedded star cluster mass, which is a free parameter involved in the TH-IGIMF theory, is found to be as low as 5 solar masses. A mild downsizing trend is present for galaxies generated assuming either a universal IMF or a variable TH-IGIMF.We find that, regardless of galaxy mass, older galaxies (with formation redshifts > 2) are formed in shorter time-scales (< 2Gyr), thus achieving larger [a/Fe] values. Hence, the time-scale of galaxy formation alone cannot explain the slope of the [a/Fe]-galaxy mass relation, but is responsible for the big dispersion of [a/Fe] abundance ratios at fixed stellar mass.

### Regular rotating black holes and the weak energy condition

We revisit here a recent work on regular rotating black holes. We introduce a new mass function generalizing the commonly used Bardeen and Hayward mass functions and extend the recently proposed solutions in order to accommodate a cosmological constant $\Lambda$. We discuss some aspects of the causal structure (horizons) and the ergospheres of the new proposed solutions. We also show that, in contrast with the spherically symmetrical case, the black hole rotation will unavoidably lead to the violation of the weak energy condition for any physically reasonable choice of the mass function, reinforcing the idea the description of the interior region of a Kerr black hole is much more challenging than in the Schwarzschild case.

### Constraining thawing and freezing models with cluster number counts

Measurements of the cluster abundance as a function of mass and redshift provide an important cosmological test that probe not only the expansion rate but also the growth of perturbations. In this paper we adopt a scalar field scenario which admits both thawing and freezing solutions from an appropriate choice of the model parameters and derived all relevant expressions to calculate the mass function and the cluster number density assuming the well-known Sheth-Torman formalism. We discuss the ability of cluster observations to distinguish between these scalar field behaviours and the standard $\Lambda$CDM scenario by considering the eROSITA and SPT cluster surveys.

### Constraining thawing and freezing models with cluster number counts [Cross-Listing]

Measurements of the cluster abundance as a function of mass and redshift provide an important cosmological test that probe not only the expansion rate but also the growth of perturbations. In this paper we adopt a scalar field scenario which admits both thawing and freezing solutions from an appropriate choice of the model parameters and derived all relevant expressions to calculate the mass function and the cluster number density assuming the well-known Sheth-Torman formalism. We discuss the ability of cluster observations to distinguish between these scalar field behaviours and the standard $\Lambda$CDM scenario by considering the eROSITA and SPT cluster surveys.

### The influence of primordial magnetic fields on the spherical collapse model in cosmology

Despite the ever growing observational evidence for the existence of the large scale magnetic fields, their origin and the evolution are not fully understood. If the magnetic fields are of primordial origin, they result in the generation of the secondary matter density perturbations and the previous studies show that such density perturbations enhance the number of dark matter halos. We extend the conventional spherical collapse model by including the Lorentz force which has not been implemented in the previous analysis to study the evolution of density perturbations produced by primordial magnetic fields. The critical over-density $\delta_{\rm c}$ characterizing the halo mass function turns out to be a bigger value, $\delta_{\rm c}\simeq 1.78$, than the conventional one $\delta_{\rm c}\simeq 1.69$ for the perturbations evolved only by the gravitational force. The difference in $\delta_{\rm c}$ between our model and the fully matter dominated cosmological model is small at a low redshift and, hence, only the high mass tail of the mass function is affected by the magnetic fields. At a high redshift, on the other hand, the difference in $\delta_{\rm c}$ becomes large enough to suppress the halo abundance over a wide range of mass scales. The halo abundance is reduced for instance by as large a factor as $\sim10^5$ at $z=9$.

### Modeling Increased Metal Production in Galaxy Clusters with Pair-Instability Supernovae

Galaxy clusters contain much more metal per star, typically 3 times as much, than is produced in normal galaxies. We set out to determine what changes are needed to the stellar mass function and supernovae rates to account for this excess metal. In particular, we vary the Type Ia supernovae rate, IMF slope, upper and lower mass cutoffs, and the merger rate of massive stars. We then use existing simulation results for metal production from AGB stars, Type Ia SNe and core-collapse SNe to calculate the total amount of each element produced per solar mass of star formation. For models with very massive stars, we also include metal production from pair-instability supernovae (PISNe).We find that including PISNe makes it much easier to increase the amount of metal produced per stellar mass. Therefore a separate population of high-mass stars is not needed to produce the high amounts of metal found in galaxy clusters. We also find that including at least some PISNe increases the abundance of intermediate-mass elements relative to both oxygen and iron, consistent with observations of ICM abundances.

### The effect of AGN feedback on the halo mass function

Based on a set of large–scale cosmological simulations, we investigate baryon effects on the halo mass function, with emphasis on the role played by AGN feedback. Halo mass functions are computed after identifying halos with both Friends-of-Friends and Spherical Overdensity halo finding algorithms. We embed the standard SO algorithm into a memory-controlled frame program and present the $\rm P$python spher$\rm I$c$\rm A$l $\rm O$verdensity code — PIAO. The SO halos are identified at three overdensities $\Delta_c = 2500, 500, 200$, and masses computed within the three corresponding radii. We confirm that hydrodynamical simulations based on radiative cooling, star formation and supernova feedback (CSF) produce mass functions higher than from collisionless simulations. In contrast, the effect of AGN feedback is that to suppressing the HMFs to a level even below that of Dark Matter simulations, for both FoF and SO halos. We find that the ratio between the halo mass functions in the AGN and in the DM simulations is ~ 0.8, almost independent of the mass, when estimated at overdensity $\Delta_c=500$, a difference that increases at higher overdensity $\Delta_c=2500$, with no significant redshift dependence for these ratios. We verify that the decrease of the HMF in the AGN simulation is induced by a corresponding decrease of halo masses with respect to the DM case. The shallower inner density profiles of halos in the AGN simulation witnesses that mass reduction is induced by the sudden expulsion of displacement of gas induced by AGN energy feedback. We provide fitting functions to describe halo mass variations at different overdensities. We demonstrate that, using these fitting functions, we recover the DM halo mass function starting from that of hydrodynamical simulations, with a residual random scatter $\leqslant 5$ per cent for halo mass larger than $10^{13} h^{-1} M_{\odot}$.

### The effect of AGN feedback on the halo mass function [Replacement]

We investigate baryon effects on the halo mass function (HMF), with emphasis on the role played by AGN feedback. Halos are identified with both Friends-of-Friends (FoF) and Spherical Overdensity (SO) algorithms. We embed the standard SO algorithm into a memory-controlled frame program and present the {\bf P}ython spher{\bf I}c{\bf A}l {\bf O}verdensity code — {\small PIAO}. For both FoF and SO halos, the effect of AGN feedback is that of suppressing the HMFs to a level even below that of Dark Matter simulations. The ratio between the HMFs in the AGN and in the DM simulations is $\sim 0.8$ at overdensity $\Delta_c=500$, a difference that increases at higher overdensity $\Delta_c=2500$, with no significant redshift and mass dependence. A decrease of the halo masses ratio with respect to the DM case induces the decrease of the HMF in the AGN simulation. The shallower inner density profiles of halos in the AGN simulation witnesses that mass reduction is induced by the sudden displacement of gas induced by thermal AGN feedback. We provide fitting functions to describe halo mass variations at different overdensities, which can recover the HMFs with a residual random scatter $\lt 5$ per cent for halo masses larger than $10^{13} ~h^{-1}\>{\rm M_\odot}$.

### The effect of AGN feedback on the halo mass function [Replacement]

Based on a set of large–scale cosmological simulations, we investigate baryon effects on the halo mass function, with emphasis on the role played by AGN feedback. Halo mass functions are computed after identifying halos with both Friends-of-Friends and Spherical Overdensity halo finding algorithms. We embed the standard SO algorithm into a memory-controlled frame program and present the $\rm P$python spher$\rm I$c$\rm A$l $\rm O$verdensity code — PIAO. The SO halos are identified at three overdensities $\Delta_c = 2500, 500, 200$, and masses computed within the three corresponding radii. We confirm that hydrodynamical simulations based on radiative cooling, star formation and supernova feedback (CSF) produce mass functions higher than from collisionless simulations. In contrast, the effect of AGN feedback is that to suppressing the HMFs to a level even below that of Dark Matter simulations, for both FoF and SO halos. We find that the ratio between the halo mass functions in the AGN and in the DM simulations is ~ 0.8, almost independent of the mass, when estimated at overdensity $\Delta_c=500$, a difference that increases at higher overdensity $\Delta_c=2500$, with no significant redshift dependence for these ratios. We verify that the decrease of the HMF in the AGN simulation is induced by a corresponding decrease of halo masses with respect to the DM case. The shallower inner density profiles of halos in the AGN simulation witnesses that mass reduction is induced by the sudden expulsion of displacement of gas induced by AGN energy feedback. We provide fitting functions to describe halo mass variations at different overdensities. We demonstrate that, using these fitting functions, we recover the DM halo mass function starting from that of hydrodynamical simulations, with a residual random scatter $\leqslant 5$ per cent for halo mass larger than $10^{13} h^{-1} M_{\odot}$.

### Halo Mass Definition and Multiplicity Function

Comparing the excursion set and CUSP formalisms for the derivation of the halo mass function, we investigate the role of the mass definition in the properties of the multiplicity function of cold dark matter (CDM) haloes. We show that the density profile for haloes formed from triaxial peaks that undergo ellipsoidal collapse and virialisation is such that the ratio between the mean inner density and the outer local density is essentially independent of mass. This causes that, for suited values of the spherical overdensity $\Delta$ and the linking length $b$, SO and FoF masses are essentially equivalent to each other and the respective multiplicity functions are essentially the same. The overdensity for haloes having undergone ellipsoidal collapse is the same as if they had formed according to the spherical top-hat model, which leads to a value of $b$ corresponding to the usual virial overdensity, $\Delta_{vir}$, equal to $\sim$ 0.2. The multiplicity function resulting from such mass definitions, expressed as a function of the top-hat height for spherical collapse, is very approximately universal in all CDM cosmologies. The reason for this is that, for such mass definitions, the top-hat density contrast for ellipsoidal collapse and virialisation is close to a universal value, equal to $\sim$ 0.9 times the usual top-hat density contrast for spherical collapse.

### Signatures of very massive stars: supercollapsars and their cosmological rate

We compute the rate of supercollapsars by using cosmological, N-body, hydro, chemistry simulations of structure formation, following detailed stellar evolution according to proper yields (for He, C, N, O, Si, S, Fe, Mg, Ca, Ne, etc.) and lifetimes for stars having different masses and metallicities, and for different stellar populations (population III and population II-I). We find that supercollapsars are usually associated to dense, collapsing gas with little metal pollution and with abundances dominated by oxygen. The resulting supercollapsar rate is about $10^{-2}\,\rm yr^{-1} sr^{-1}$ at redshift $z=0$, and their contribution to the total rate is $< 0.1$ per cent, which explains why they have never been detected so far. Expected rates at redshift $z\simeq 6$ are of the order of $\sim 10^{-3}\,\rm yr^{-1} sr^{-1}$ and decrease further at higher $z$. Because of the strong metal enrichment by massive, short-lived stars, only $\sim 1$ supercollapsar generation is possible in the same star forming region. Given their sensitivity to the high-mass end of the primordial stellar mass function, they are suitable candidates to probe pristine population III star formation and stellar evolution at low metallicities.

### Effects of the initial conditions on cosmological $N$-body simulations

Cosmology is entering an era of percent level precision due to current large observational surveys. This precision in observation is now demanding more accuracy from numerical methods and cosmological simulations. In this paper, we study the accuracy of $N$-body numerical simulations and their dependence on changes in the initial conditions and in the simulation algorithms. For this purpose, we use a series of cosmological $N$-body simulations with varying initial conditions. We test the influence of the initial conditions, namely the pre-initial configuration (preIC), the order of the Lagrangian perturbation theory (LPT), and the initial redshift, on the statistics associated with the large scale structures of the universe such as the halo mass function, the density power spectrum, and the maximal extent of the large scale structures. We find that glass or grid pre-initial conditions give similar results at $z\lesssim 2$. However, the initial excess of power in the glass initial conditions yields a subtle difference in the power spectra and the mass function at high redshifts. The LPT order used to generate the ICs of the simulations is found to play a crucial role. First-order LPT (1LPT) simulations underestimate the number of massive haloes with respect to second-order (2LPT) ones, typically by 2% at $10^{14} h^{-1} M_\odot$ for an initial redshift of 23, and the small-scale power with an underestimation of 6% near the Nyquist frequency for $z_\mathrm{ini} = 23$. Moreover, at higher redshifts, the high-mass end of the mass function is significantly underestimated in 1LPT simulations. On the other hand, when the LPT order is fixed, the starting redshift has a systematic impact on the low-mass end of the halo mass function.

### MOCCA code for star cluster simulations - III. Stellar-mass black holes in the globular cluster M22

Using a Monte Carlo code, we construct a dynamic evolutionary model of the Galactic globular cluster M22 (NGC6656). The initial conditions are chosen so that, after about 12Gyr of stellar and dynamical evolution, the model is an approximate fit to the surface brightness and velocity dispersion profiles of the cluster, to its mass function, and to the current binary fraction. Depending on the distribution of black hole natal kicks, we predict that the present-day population of stellar-mass black holes ranges from about 40 (no kicks) down to essentially zero (kicks distributed like those of neutron stars). Provided that natal kicks do not eject all new black holes, it is suggested that clusters with a present-day half-mass relaxation time above about 1Gyr are the ones that may still retain an appreciable population of black holes.

### Probing the primordial initial mass function with high-$z$ supernovae [Replacement]

The first supernovae will soon be visible at the edge of the observable universe, revealing the birthplaces of Population III stars. With upcoming near-infrared missions, a broad analysis of the detectability of Population III supernovae is paramount. We combine cosmological and radiation transport simulations, instrument specifications, and survey strategies to create synthetic observations of primeval core-collapse, Type IIn and pair-instability supernovae with the James Webb Space Telescope (JWST). We show that a dedicated observational campaign with the JWST can detect up to $\sim 15$ pair-instability explosions, $\sim 300$ core-collapse supernovae, but less than one Type IIn explosion per year, depending on the Population III star formation history. Our synthetic survey also shows that $\approx 1-2 \times10^2$ supernovae detections, depending on the accuracy of the classification, are sufficient to discriminate between a Salpeter and flat mass distribution for primordial stars with a confidence level greater than 99.5 per cent.

### Probing the Pop III initial mass function with primordial supernovae

The first supernovae will soon be visible at the edge of the observable universe, revealing the birthplaces of Population III stars. With upcoming near-infrared missions, a broad analysis of the detectability of Population III supernovae is paramount. We combine cosmological and radiation transport simulations, instrument specifications, and survey strategies to create synthetic observations of primeval core-collapse, Type IIn and pair-instability supernovae with the James Webb Space Telescope (JWST). We show that a dedicated observational campaign with the JWST can detect up to $\sim 15$ pair-instability explosions, $\sim 300$ core-collapse supernovae, but less than one Type IIn explosion per year, depending on the Population III star formation history. Our synthetic survey also shows that $\sim 10^2$ supernova detections, properly classified, are sufficient to discriminate between a Salpeter and flat mass distribution for primordial stars with a confidence level greater than 99.5 per cent.

### Jumping the Gap: The Formation Conditions and Mass Function of Pebble-Pile Planetesimals

In a turbulent proto-planetary disk, dust grains undergo large density fluctuations and under the right circumstances, these grain overdensities can overcome shear, turbulent, and gas pressure support to collapse under self-gravity (forming a ‘pebble pile’ planetesimal). Using insights from simulations and a new analytic model for the fluctuations, we calculate the rate-of-formation and mass function of self-gravitating, collapsing planetesimal-mass bodies formed by this mechanism. The statistics of this process depend sensitively on the size/stopping time of the largest grains, disk surface density, and turbulent Mach numbers. However, when it occurs, we predict that the resulting planetesimal mass function is broad and quasi-universal, with a slope dN/dM~1/M, spanning a size/mass range ~10-1e4 km (~1e-9-5.0 M_Earth). Collapse to planetesimal through super-Earth masses is possible. The key condition is that grain density fluctuations reach large amplitudes on large scales, where gravitational instability proceeds most easily (collapse of small grains is strongly suppressed by turbulent vorticity). We show this leads to a new criterion for ‘pebble-pile’ formation in terms of the dimensionless particle stopping time (tau_stop > f(Q,Z,alpha)). In a MMSN, this requires grains larger than a=(50,1,0.1)cm at r=(1,30,100)au. So at large radii, this can easily occur and seed core accretion. At small radii, it would depend on the existence of large boulders. However, because density fluctuations depend super-exponentially on tau_stop (inversely proportional to disk surface density), lower-density disks are more unstable! In fact, we predict that cm-sized grains at ~1au will form pebble piles in a disk with ~10% the MMSN density, so planet formation at ~au may generically occur late, as disks are evaporating.

### A comparison of the ingredients of stellar population models on the inference of the slope of the initial mass function

We perform a direct comparison between two new single stellar population (SSP) models constructed specifically for the purpose of studying old, metal-rich stellar populations. We examine the effect of the underlying model assumptions and ingredients, such as stellar libraries or isochrones, on the inference of the Initial Mass Function (IMF) slope down to 0.1 solar masses in massive early-type galaxies (ETGs). For both models, we study equivalent widths of stellar features with varying ages, metallicity and elemental abundances, as a function of the IMF slope. We show that the use of optical indices, mainly from TiO and CaH molecular absorption lines, permits us to eliminate the uncertainty caused by the different stellar libraries used in the synthesis process. Neither of the two SSP models are able to simultaneously reproduce all the optical stellar features in SDSS ETGs with \sigma > 250km/s if we restrict the analysis to solar abundance patterns and to line-index measurements.We also find that predictions of IMF slope variations and abundance patterns arising from different models differ, especially when sodium lines are used. In particular, we investigate the variation of the two sodium indices with IMF slope and [Na/Fe] abundance. We show that the NaD feature is very sensitive to [Na/Fe] variations whereas the NaI index depends mainly on the IMF slope and only weakly on elemental abundance.

### The bright end of the galaxy luminosity function at z ~ 7: before the onset of mass quenching?

We present the results of a new search for bright star-forming galaxies at z ~ 7 within the UltraVISTA DR2 and UKIDSS UDS DR10 data, which together provide 1.65 sq deg of near-infrared imaging with overlapping optical and Spitzer data. Using a full photo-z analysis to identify high-z galaxies and reject contaminants, we have selected a sample of 34 luminous (-22.7 < M_UV < -21.2) galaxies with the 6.5 < z < 7.5. Crucially, the deeper imaging provided by UltraVISTA DR2 confirms all of the robust objects previously uncovered by Bowler et al. (2012), validating our selection technique. Our sample includes the most massive galaxies known at z ~ 7, with M_* ~ 10^{10} M_sun, and the majority are resolved, consistent with larger sizes (r_{1/2} ~ 1 – 1.5 kpc) than displayed by less massive galaxies. From our final sample, we determine the form of the bright end of the rest-frame UV galaxy luminosity function (LF) at z ~ 7, providing strong evidence that the bright end of the z = 7 LF does not decline as steeply as predicted by the Schechter function fitted to fainter data. We consider carefully, and exclude the possibility that this is due to either gravitational lensing, or significant contamination of our galaxy sample by AGN. Rather, our results favour a double power-law form for the galaxy LF at high z, or, more interestingly, a LF which simply follows the form of the dark-matter halo mass function at bright magnitudes. This suggests that the physical mechanism which inhibits star-formation activity in massive galaxies (i.e. AGN feedback or some other form of mass quenching’) has yet to impact on the observable galaxy LF at z ~ 7, a conclusion supported by the estimated masses of our brightest galaxies which have only just reached a mass comparable to the critical quenching mass’ of M_* = 10 ^{10.2} M_sun derived from studies of the mass function of star-forming galaxies at lower z.

### The bright end of the galaxy luminosity function at z ~ 7: before the onset of mass quenching? [Replacement]

We present the results of a new search for bright star-forming galaxies at z ~ 7 within the UltraVISTA DR2 and UKIDSS UDS DR10 data, which together provide 1.65 sq deg of near-infrared imaging with overlapping optical and Spitzer data. Using a full photo-z analysis to identify high-z galaxies and reject contaminants, we have selected a sample of 34 luminous (-22.7 < M_UV < -21.2) galaxies with 6.5 < z < 7.5. Crucially, the deeper imaging provided by UltraVISTA DR2 confirms all of the robust objects previously uncovered by Bowler et al. (2012), validating our selection technique. Our sample includes the most massive galaxies known at z ~ 7, with M_* ~ 10^{10} M_sun, and the majority are resolved, consistent with larger sizes (r_{1/2} ~ 1 – 1.5 kpc) than displayed by less massive galaxies. From our final sample, we determine the form of the bright end of the rest-frame UV galaxy luminosity function (LF) at z ~ 7, providing strong evidence that the bright end of the z = 7 LF does not decline as steeply as predicted by the Schechter function fitted to fainter data. We consider carefully, and exclude the possibility that this is due to either gravitational lensing, or significant contamination of our galaxy sample by AGN. Rather, our results favour a double power-law form for the galaxy LF at high z, or, more interestingly, a LF which simply follows the form of the dark-matter halo mass function at bright magnitudes. This suggests that the physical mechanism which inhibits star-formation activity in massive galaxies (i.e. AGN feedback or some other form of mass quenching’) has yet to impact on the observable galaxy LF at z ~ 7, a conclusion supported by the estimated masses of our brightest galaxies which have only just reached a mass comparable to the critical quenching mass’ of M_* = 10 ^{10.2} M_sun derived from studies of the mass function of star-forming galaxies at lower z.

### The Origin and Universality of the Stellar Initial Mass Function

We review current theories for the origin of the Stellar Initial Mass Function (IMF) with particular focus on the extent to which the IMF can be considered universal across various environments. To place the issue in an observational context, we summarize the techniques used to determine the IMF for different stellar populations, the uncertainties affecting the results, and the evidence for systematic departures from universality under extreme circumstances. We next consider theories for the formation of prestellar cores by turbulent fragmentation and the possible impact of various thermal, hydrodynamic and magneto-hydrodynamic instabilities. We address the conversion of prestellar cores into stars and evaluate the roles played by different processes: competitive accretion, dynamical fragmentation, ejection and starvation, filament fragmentation and filamentary accretion flows, disk formation and fragmentation, critical scales imposed by thermodynamics, and magnetic braking. We present explanations for the characteristic shapes of the Present-Day Prestellar Core Mass Function and the IMF and consider what significance can be attached to their apparent similarity. Substantial computational advances have occurred in recent years, and we review the numerical simulations that have been performed to predict the IMF directly and discuss the influence of dynamics, time-dependent phenomena, and initial conditions.

### The Origin and Universality of the Stellar Initial Mass Function [Replacement]

We review current theories for the origin of the Stellar Initial Mass Function (IMF) with particular focus on the extent to which the IMF can be considered universal across various environments. To place the issue in an observational context, we summarize the techniques used to determine the IMF for different stellar populations, the uncertainties affecting the results, and the evidence for systematic departures from universality under extreme circumstances. We next consider theories for the formation of prestellar cores by turbulent fragmentation and the possible impact of various thermal, hydrodynamic and magneto-hydrodynamic instabilities. We address the conversion of prestellar cores into stars and evaluate the roles played by different processes: competitive accretion, dynamical fragmentation, ejection and starvation, filament fragmentation and filamentary accretion flows, disk formation and fragmentation, critical scales imposed by thermodynamics, and magnetic braking. We present explanations for the characteristic shapes of the Present-Day Prestellar Core Mass Function and the IMF and consider what significance can be attached to their apparent similarity. Substantial computational advances have occurred in recent years, and we review the numerical simulations that have been performed to predict the IMF directly and discuss the influence of dynamics, time-dependent phenomena, and initial conditions.

### The Origin and Universality of the Stellar Initial Mass Function [Replacement]

We review current theories for the origin of the Stellar Initial Mass Function (IMF) with particular focus on the extent to which the IMF can be considered universal across various environments. To place the issue in an observational context, we summarize the techniques used to determine the IMF for different stellar populations, the uncertainties affecting the results, and the evidence for systematic departures from universality under extreme circumstances. We next consider theories for the formation of prestellar cores by turbulent fragmentation and the possible impact of various thermal, hydrodynamic and magneto-hydrodynamic instabilities. We address the conversion of prestellar cores into stars and evaluate the roles played by different processes: competitive accretion, dynamical fragmentation, ejection and starvation, filament fragmentation and filamentary accretion flows, disk formation and fragmentation, critical scales imposed by thermodynamics, and magnetic braking. We present explanations for the characteristic shapes of the Present-Day Prestellar Core Mass Function and the IMF and consider what significance can be attached to their apparent similarity. Substantial computational advances have occurred in recent years, and we review the numerical simulations that have been performed to predict the IMF directly and discuss the influence of dynamics, time-dependent phenomena, and initial conditions.

### The stellar mass function and star formation rate${\bf{-}}$stellar mass relation of galaxies at ${\bf{z\sim4-7}}$

We investigate the evolution of the star formation rate$-$stellar mass relation (SFR$-{\rm M}_{\star}$) and Galaxy Stellar Mass Function (GSMF) of $z\sim 4-7$ galaxies, using cosmological simulations run with the smoothed particle hydrodynamics code P-GADGET3(XXL). We explore the effects of different feedback prescriptions (supernova driven galactic winds and AGN feedback), initial stellar mass functions and metal cooling. We show that our fiducial model, with strong energy-driven winds and early AGN feedback, is able to reproduce the observed stellar mass function obtained from UV-selected samples of galaxies at redshift $6\le z\le7$. At $z\sim4$ our simulations are more consistent with recent results from IR-selected samples, which provide a better proxy of stellar masses. Despite this success, there is a tension between simulated and observed (UV) SFR$-{\rm M}_{\star}$ relations that leads to a disagreement between the GSMF recovered from simulations and UV observations. By combining the simulated SFR(M$_{\star}$) relationship with the observed star formation rate function at a given redshift, we argue that this disagreement may be the result of the uncertainty in the observed $L_{\rm UV}-{\rm M}_{\star}$ conversion and in the normalization of the observed SFR$-{\rm M}_{\star}$ relation. Our simulations predict a population of faint galaxies not seen by current observations. We find that there is an inconsistency between the observed SFR$-{\rm M}_{\star}$ relations based on galaxies selected in the IR and UV at redshift $z \sim 4-5$. The main reason for this tension is that high redshift surveys assume different dust corrections to recover SFRs. Our simulated SFR(M$_{\star}$) is more consistent with IR-selected samples of galaxies at all the redshifts considered.

### Almost-Killing conserved currents: a general mass function

A new class of conserved currents, describing non-gravitational energy-momentum density, is presented. The proposed currents do not require the existence of a (timelike) Killing vector, and are not restricted to spherically symmetric spacetimes (or similar ones, in which the Kodama vector can be defined). They are based instead on almost-Killing vectors, which could in principle be defined on generic spacetimes. We provide local arguments, based on energy density profiles in highly simplified (stationary, rigidly-rotating) star models, which confirm the physical interest of these ‘almost-Killing currents’. A mass function is defined in this way for the spherical case, qualitatively different from the Hern\’andez-Misner mass function. An elliptic equation determining the new mass function is derived for the Tolman-Bondi spherically symmetric dust metrics, including a simple solution for the Oppenheimer-Schneider collapse. The equations for the non-symmetric case are shown to be of a mixed elliptic-hyperbolic nature.

### The relation between accretion rates and the initial mass function in hydrodynamical simulations of star formation

We analyse a hydrodynamical simulation of star formation. Sink particles in the simulations which represent stars show episodic growth, which is presumably accretion from a core that can be regularly replenished in response to the fluctuating conditions in the local environment. The accretion rates follow $\dot{m} \propto m^{2/3}$, as expected from accretion in a gas-dominated potential, but with substantial variations over-laid on this. The growth times follow an exponential distribution which is tapered at long times due to the finite length of the simulation. The initial collapse masses have an approximately lognormal distribution with already an onset of a power-law at large masses. The sink particle mass function can be reproduced with a non-linear stochastic process, with fluctuating accretion rates $\propto m^{2/3}$, a distribution of seed masses and a distribution of growth times. All three factors contribute equally to the form of the final sink mass function. We find that the upper power law tail of the IMF is unrelated to Bondi-Hoyle accretion.

### Computation of the Halo Mass Function Using Physical Collapse Parameters: Application to Non-Standard Cosmologies

In this article we compare the halo mass function predicted by the excursion set theory with a drifting diffusive barrier against the results of N-body simulations for several cosmological models. This includes the standard LCDM case for a large range of halo masses, models with different types of primordial non-Gaussianity, and a dark energy model. We show that in all those cosmological scenarios, the abundance of dark matter halos can be described by a drifting diffusive barrier, where the two parameters describing the barrier have physical content. In the case of the Gaussian LCDM, the statistics is precise enough to actually predict those parameters from the initial conditions. Furthermore, we found that the stochasticity in the barrier is non-negligible making the simple deterministic spherical collapse model a bad approximation even at very high halo masses. We also show that using the standard excursion set approach with a barrier inspired by peak patches leads to inconsistent predictions of the halo mass function.

### Computation of the Halo Mass Function Using Physical Collapse Parameters: Application to Non-Standard Cosmologies [Cross-Listing]

In this article we compare the halo mass function predicted by the excursion set theory with a drifting diffusive barrier against the results of N-body simulations for several cosmological models. This includes the standard LCDM case for a large range of halo masses, models with different types of primordial non-Gaussianity, and a dark energy model. We show that in all those cosmological scenarios, the abundance of dark matter halos can be described by a drifting diffusive barrier, where the two parameters describing the barrier have physical content. In the case of the Gaussian LCDM, the statistics is precise enough to actually predict those parameters from the initial conditions. Furthermore, we found that the stochasticity in the barrier is non-negligible making the simple deterministic spherical collapse model a bad approximation even at very high halo masses. We also show that using the standard excursion set approach with a barrier inspired by peak patches leads to inconsistent predictions of the halo mass function.

### Halo modelling in chameleon theories

We analyse modelling techniques for the large-scale structure formed in scalar-tensor theories of constant Brans-Dicke parameter which match the concordance model background expansion history and produce a chameleon suppression of the gravitational modification in high-density regions. Thereby, we use a mass and environment dependent chameleon spherical collapse model, the Sheth-Tormen halo mass function and linear halo bias, the Navarro-Frenk-White halo density profile, and the halo model. Furthermore, using the spherical collapse model, we extrapolate a chameleon mass-concentration scaling relation from a LCDM prescription calibrated to N-body simulations. We also provide constraints on the model parameters to ensure viability on local scales. We test our description of the halo mass function and nonlinear matter power spectrum against the respective observables extracted from large-volume and high-resolution N-body simulations in the limiting case of f(R) gravity, corresponding to a vanishing Brans-Dicke parameter. We find good agreement between the two; the halo model provides a good qualitative description of the shape of the relative enhancement of the f(R) matter power spectrum with respect to LCDM caused by the extra attractive gravitational force but fails to recover the correct amplitude. Introducing an effective linear power spectrum in the computation of the two-halo term to account for an underestimation of the chameleon suppression at intermediate scales in our approach, we accurately reproduce the measurements from the N-body simulations.

### Halo modelling in chameleon theories [Cross-Listing]

We analyse modelling techniques for the large-scale structure formed in scalar-tensor theories of constant Brans-Dicke parameter which match the concordance model background expansion history and produce a chameleon suppression of the gravitational modification in high-density regions. Thereby, we use a mass and environment dependent chameleon spherical collapse model, the Sheth-Tormen halo mass function and linear halo bias, the Navarro-Frenk-White halo density profile, and the halo model. Furthermore, using the spherical collapse model, we extrapolate a chameleon mass-concentration scaling relation from a LCDM prescription calibrated to N-body simulations. We also provide constraints on the model parameters to ensure viability on local scales. We test our description of the halo mass function and nonlinear matter power spectrum against the respective observables extracted from large-volume and high-resolution N-body simulations in the limiting case of f(R) gravity, corresponding to a vanishing Brans-Dicke parameter. We find good agreement between the two; the halo model provides a good qualitative description of the shape of the relative enhancement of the f(R) matter power spectrum with respect to LCDM caused by the extra attractive gravitational force but fails to recover the correct amplitude. Introducing an effective linear power spectrum in the computation of the two-halo term to account for an underestimation of the chameleon suppression at intermediate scales in our approach, we accurately reproduce the measurements from the N-body simulations.

### The ages of young star clusters, massive blue stragglers and the upper mass limit of stars: analysing age dependent stellar mass functions

Massive stars rapidly change their masses through strong stellar winds and mass transfer in binary systems. We show that such mass changes leave characteristic signatures in stellar mass functions of young star clusters which can be used to infer their ages and to identify products of binary evolution. We model the observed present day mass functions of the young Galactic Arches and Quintuplet star clusters using our rapid binary evolution code. We find that shaping of the mass function by stellar wind mass loss allows us to determine the cluster ages to 3.5$\pm$0.7 Myr and 4.8$\pm$1.1 Myr, respectively. Exploiting the effects of binary mass exchange on the cluster mass function, we find that the most massive stars in both clusters are rejuvenated products of binary mass transfer, i.e. the massive counterpart of classical blue straggler stars. This resolves the problem of an apparent age spread among the most luminous stars exceeding the expected duration of star formation in these clusters. We perform Monte Carlo simulations to probe stochastic sampling, which support the idea of the most massive stars being rejuvenated binary products. We find that the most massive star is expected to be a binary product after 1.0$\pm$0.7 Myr in Arches and after 1.7$\pm$1.0 Myr in Quintuplet. Today, the most massive 9$\pm$3 stars in Arches and 8$\pm$3 in Quintuplet are expected to be such objects. Our findings have strong implications for the stellar upper mass limit and solve the discrepancy between the claimed 150 $\mathrm{M}_\odot$ limit and observations of fours stars with initial masses of 165-320 $\mathrm{M}_\odot$ in R136 and of SN 2007bi, which is thought to be a pair-instability supernova from an initial 250 $\mathrm{M}_\odot$ star. Using the stellar population of R136, we revise the upper mass limit to values in the range 200-500 $\mathrm{M}_\odot$.

### On the mass function of stars growing in a flocculent medium

Stars form in regions of very inhomogeneous densities and may have chaotic orbital motions. This leads to a time variation of the accretion rate, which will spread the masses over some mass range. We investigate the mass distribution functions that arise from fluctuating accretion rates in non-linear accretion, $\dot{m} \propto m^{\alpha}$. The distribution functions evolve in time and develop a power law tail attached to a lognormal body, like in numerical simulations of star formation. Small fluctuations may be modelled by a Gaussian and develop a power-law tail $\propto m^{-\alpha}$ at the high-mass side for $\alpha > 1$ and at the low-mass side for $\alpha < 1$. Large fluctuations require that their distribution is strictly positive, for example, lognormal. For positive fluctuations the mass distribution function develops the power-law tail always at the high-mass hand side, independent of $\alpha$ larger or smaller than unity. Furthermore, we discuss Bondi-Hoyle accretion in a supersonically turbulent medium, the range of parameters for which non-linear stochastic growth could shape the stellar initial mass function, as well as the effects of a distribution of initial masses and growth times.

### The busy function: a new analytic function for describing the integrated 21-cm spectral profile of galaxies

Accurate parametrization of galaxies detected in the 21-cm HI emission is of fundamental importance to the measurement of commonly used indicators of galaxy evolution, including the Tully-Fisher relation and the HI mass function. Here, we propose a new analytic function, named the ‘busy function’, that can be used to accurately describe the characteristic double-horn HI profile of many galaxies. The busy function is a continuous, differentiable function that consists of only two basic functions, the error function, erf(x), and a polynomial, |x|^n, of degree n >= 2. We present the basic properties of the busy function and illustrate its great flexibility in fitting a wide range of HI profiles from the Gaussian profiles of dwarf galaxies to the broad, asymmetric double-horn profiles of spiral galaxies. Applications of the busy function include the accurate and efficient parametrization of observed HI spectra of galaxies and the construction of spectral templates for simulations and matched filtering algorithms. We demonstrate the busy function’s power by automatically fitting it to the HI spectra of 1000 galaxies from the HIPASS Bright Galaxy Catalog, using our own C/C++ implementation, and comparing the resulting parameters with the catalogued ones. We also present two methods for determining the uncertainties of observational parameters derived from the fit.

### The stellar mass function and efficiency of galaxy formation with a varying initial mass function

Several recent observational studies have concluded that the initial mass function (IMF) of stars varies systematically with galaxy properties such as velocity dispersion. In this paper, we investigate the effect of linking the circular velocity of galaxies, as determined from the Fundamental Plane and Tully-Fisher relations, to the slope of the IMF with parameterizations guided by several of these studies. For each empirical relation, we generate stellar masses of ~600,000 SDSS galaxies at z ~ 0.1, by fitting the optical photometry to large suites of synthetic stellar populations that sample the full range of galaxy parameters. We generate stellar mass functions and examine the stellar-to-halo mass relations using sub-halo abundance matching. At the massive end, the stellar mass functions become a power law, instead of the familiar exponential decline. As a result, it is a generic feature of these models that the central galaxy stellar-to-halo mass relation is significantly flatter at high masses (slope ~ -0.3 to -0.4) than in the case of a universal IMF (slope ~ -0.6). We find that regardless of whether the IMF varies systematically in all galaxies or just early types, there is still a well-defined peak in the central stellar-to-halo mass ratio at halo masses of ~ 10E12 solar masses. In general, the IMF variations explored here lead to significantly higher integrated stellar densities if the assumed dependence on circular velocity applies to all galaxies, including late-types; in fact the more extreme cases can be ruled out, as they imply an unphysical situation in which the stellar fraction exceeds the universal baryon fraction.

### The stellar mass function and efficiency of galaxy formation with a varying initial mass function [Replacement]

Several recent observational studies have concluded that the initial mass function (IMF) of stars varies systematically with galaxy properties such as velocity dispersion. In this paper, we investigate the effect of linking the circular velocity of galaxies, as determined from the Fundamental Plane and Tully-Fisher relations, to the slope of the IMF with parameterizations guided by several of these studies. For each empirical relation, we generate stellar masses of ~600,000 SDSS galaxies at z ~ 0.1, by fitting the optical photometry to large suites of synthetic stellar populations that sample the full range of galaxy parameters. We generate stellar mass functions and examine the stellar-to-halo mass relations using sub-halo abundance matching. At the massive end, the stellar mass functions become a power law, instead of the familiar exponential decline. As a result, it is a generic feature of these models that the central galaxy stellar-to-halo mass relation is significantly flatter at high masses (slope ~ -0.3 to -0.4) than in the case of a universal IMF (slope ~ -0.6). We find that regardless of whether the IMF varies systematically in all galaxies or just early types, there is still a well-defined peak in the central stellar-to-halo mass ratio at halo masses of ~ 10E12 solar masses. In general, the IMF variations explored here lead to significantly higher integrated stellar densities if the assumed dependence on circular velocity applies to all galaxies, including late-types; in fact the more extreme cases can be ruled out, as they imply an unphysical situation in which the stellar fraction exceeds the universal baryon fraction.

### Generating Merger Trees for Dark Matter Haloes: A Comparison of Methods

Halo merger trees describe the hierarchical mass assembly of dark matter haloes, and are the backbone for modeling galaxy formation and evolution. Merger trees constructed using Monte Carlo algorithms based on the extended Press-Schechter (EPS) formalism are complementary to those extracted from N-body simulations, and have the advantage that they are not trammeled by limited numerical resolution and uncertainties in identifying (sub)haloes and linking them between snapshots. This paper compares multiple EPS-based merger tree algorithms to simulation results using four diagnostics: progenitor mass function (PMF), mass assembly history (MAH), merger rate per descendant halo, and the unevolved subhalo mass function (USMF). In general, algorithms based on spherical collapse yield major-merger rates that are too high by a factor of two, resulting in MAHs that are systematically offset. Assuming ellipsoidal collapse solves most of these issues, but the particular algorithm investigated here that incorporates ellipsoidal collapse dramatically overpredicts the minor-merger rate for massive haloes. The only algorithm in our comparison that yields MAHs, merger rates, and USMFs in good agreement with simulations, is that by Parkinson et al. (2008). However, this is not a true EPS-based algorithm as it draws its progenitor masses from a PMF calibrated against simulations, rather than `predicted’ by EPS. Finally we emphasize that the benchmarks used to test the EPS algorithms are obtained from simulations and are hampered by significant uncertainties themselves. In particular, MAHs and halo merger rates obtained from simulations by different authors reveal discrepancies that easily exceed 50 percent, even when based on the same simulation. Given this status quo, merger trees constructed using the Parkinson et al. algorithm are as accurate as those extracted from N-body simulations.

### Lyman-{\alpha} Forest and Cosmic Weak Lensing in a Warm Dark Matter Universe

We review the current state of the theory of large scale structure in a warm dark matter (WDM) cosmological model. In particular, we focus on the non-linear modelling of the matter power spectrum and on the mass function of dark matter haloes. We describe the results of N-body simulations with WDM and mention the effects that could be induced by baryonic physics. We also examine the halo model of large scale structure and its recently suggested modifications for a WDM cosmology, which account for the small scale smoothness of the initial matter density field and better fit the results of N-body simulations. Having described the theoretical models, we discuss the current lower limits on the WDM particle mass, m_w, which correspond to upper limits on the WDM temperature under the assumption that the particles are thermal relics. The best such constraints come from the Ly{\alpha} forest and exclude all masses below 3.3 keV at the 2{\sigma} confidence level. We finally review the forecasts for future lensing surveys, which will be of the same order of magnitude as the already existing constraints from the Ly{\alpha} forest data but explore a different redshift regime.

### On the shape of the mass-function of dense clumps in the Hi-GAL fields. II. Using Bayesian inference to study the clump mass function

Context. Stars form in dense, dusty clumps of molecular clouds, but little is known about their origin, their evolution and their detailed physical properties. In particular, the relationship between the mass distribution of these clumps (also known as the "clump mass function", or CMF) and the stellar initial mass function (IMF), is still poorly understood. Aims. In order to better understand how the CMF evolve toward the IMF, and to discern the "true" shape of the CMF, large samples of bona-fide pre- and proto-stellar clumps are required. Two such datasets obtained from the Herschel infrared GALactic Plane Survey (Hi-GAL) have been described in paper I. Robust statistical methods are needed in order to infer the parameters describing the models used to fit the CMF, and to compare the competing models themselves. Methods. In this paper we apply Bayesian inference to the analysis of the CMF of the two regions discussed in Paper I. First, we determine the Bayesian posterior probability distribution for each of the fitted parameters. Then, we carry out a quantitative comparison of the models used to fit the CMF. Results. We have compared the results from several methods implementing Bayesian inference, and we have also analyzed the impact of the choice of priors and the influence of various constraints on the statistical conclusions for the preferred values of the parameters. We find that both parameter estimation and model comparison depend on the choice of parameter priors. Conclusions. Our results confirm our earlier conclusion that the CMFs of the two Hi-GAL regions studied here have very similar shapes but different mass scales. Furthermore, the lognormal model appears to better describe the CMF measured in the two Hi-GAL regions studied here. However, this preliminary conclusion is dependent on the choice of parameters priors.