Building a predictive model of galaxy formation – I. Phenomenological model constrained to the z = 0 stellar mass function

We conduct Bayesian model inferences from the observed K-band luminosity function of galaxies in the local Universe, using the semi-analytic model (SAM) of galaxy formation introduced in Lu et al (2011). The prior distributions for the 14 free parameters include a large range of possible models. We find that some of the free parameters, e.g. the characteristic scales for quenching star formation in both high-mass and low-mass halos, are already tightly constrained by the single data set. The posterior distribution includes the model parameters adopted in other SAMs. By marginalising over the posterior distribution, we make predictions that include the full inferential uncertainties for the colour-magnitude relation, the Tully-Fisher relation, the conditional stellar mass function of galaxies in halos of different masses, the HI mass function, the redshift evolution of the stellar mass function of galaxies, and the global star formation history. Using posterior predictive checking with the available observational results, we find that the model family (i) predicts a Tully-Fisher relation that is curved; (ii) significantly over predicts the satellite fraction; (iii) vastly over predicts the HI mass function; (iv) predicts high-z stellar mass functions that have too many low mass galaxies and too few high mass ones. and (v) predicts a redshift evolution of the stellar mass density and the star formation history that are in moderate disagreement. These results suggest that some important processes are still missing in the current model family and we discuss a number of possible solutions to solve the discrepancies, such as interactions between galaxies and dark matter halos, tidal stripping, the bimodal accretion of gas, preheating, and a redshift-dependent initial mass function.

We calculate the stellar integrated galactic initial mass function (IGIMF) in the presence of cluster-to-cluster variations of the IMF. Variations of the IMF for a population of coeval clusters that populate the initial cluster mass function (ICLMF) are taken into account in the form of Gaussian distribution functions of the IMF parameters. For the tapered power-law function used in this work, these are the slope at the high-mass end, Γ, the slope at the low-mass end, γ, and the characteristic mass Mch. The level of variations is modeled by varying the width of the Gaussian distributions. The reference values are the standard deviations of the parameters observed for the population of young clusters in the present-day Milky Way, which are σΓ = 0.6, σγ = 0.25, and σMch = 0.27 M⊙. We find that increasing the levels of dispersion for γ and Γ tends to moderately flatten the IGIMF at the low and high-mass end, respectively. The characteristic mass of the IGIMF is, however, strongly impacted by variations in Mch. Increasing the value of σMch shifts the peak of the IGIMF to lower masses, rendering the IGIMF more bottom heavy. This can provide a simple explanation for the bottom-heavy stellar mass function that is inferred for early-type galaxies since these are likely the result of a merger of disk galaxies where the physical conditions of the star-forming gas may vary significantly both in time and space in the merging system. The effect of IMF variations on the IGIMF is compared to the effects of other processes and sources of systematic variations such as those due to variations in the shape of ICLMF, the gas-phase metallicity, and the galactic star formation rate (SFR) which can potentially affect the maximum mass of stellar clusters in a galaxy and set the mean value of the characteristic mass in clusters. For the various dependencies we have explored, we found that the effect of IMF variations is a dominant factor that always affects the characteristic mass of the IGIMF. For the regimes at low metallicity where the IGIMF resembles a single power law, an increased level of IMF variations renders the IGIMF steeper and more bottom heavy, especially at low SFRs. On the other hand, variations in the IMF in the high mass regime can be easily dominated by variations in the slope of the ICLMF. We compare our results of the metallicity and SFR-dependent IGIMF to a sample of Milky Way ultra-faint dwarf (UFD) satellite galaxies that have available metallicity measurements. The present-day stellar mass function of these galaxies is a good analog to the IGIMF at the time their overall population of stars formed. We show that the slope of the stellar mass function of the UFD galaxies measured for stars in the mass range [0.4, 0.8] M⊙ can only be reproduced when IMF variations of the same order as those measured in the present-day Milky Way are included. Our results suggest that the inclusion of IMF variations in models of galaxy formation and evolution is of vital importance in order to improve our understanding of star formation and star formation feedback effects on galactic scales.

We believe that a wide range of physical processes conspire to shape the observed galaxy population, but we remain unsure of their detailed interactions. The semi-analytic model (SAM) of galaxy formation uses multidimensional parametrizations of the physical processes of galaxy formation and provides a tool to constrain these underlying physical interactions. Because of the high dimensionality, the parametric problem of galaxy formation may be profitably tackled with a Bayesian-inference-based approach, which allows one to constrain theory with data in a statistically rigorous way. In this paper, we develop a SAM in the framework of Bayesian inference. We show that, with a parallel implementation of an advanced Markov chain Monte Carlo algorithm, it is now possible to rigorously sample the posterior distribution of the high-dimensional parameter space of typical SAMs. As an example, we characterize galaxy formation in the current Λ cold dark matter cosmology using the stellar mass function of galaxies as an observational constraint. We find that the posterior probability distribution is both topologically complex and degenerate in some important model parameters, suggesting that thorough explorations of the parameter space are needed to understand the models. We also demonstrate that because of the model degeneracy, adopting a narrow prior strongly restricts the model. Therefore, the inferences based on SAMs are conditional to the model adopted. Using synthetic data to mimic systematic errors in the stellar mass function, we demonstrate that an accurate observational error model is essential to meaningful inference.

Context. How galaxies form, assemble, and cease their star formation is a central question within the modern landscape of galaxy evolution studies. These processes are indelibly imprinted on the galaxy stellar mass function (SMF), and its measurement and understanding is key to uncovering a unified theory of galaxy evolution. Aims. We present constraints on the shape and evolution of the galaxy SMF, the quiescent galaxy fraction, and the cosmic stellar mass density across 90% of the history of the Universe from z = 7.5 → 0.2 as a means to study the physical processes that underpin galaxy evolution. Methods. The COSMOS survey is an ideal laboratory for studying representative galaxy samples. Now equipped with deeper and more homogeneous near-infrared coverage exploited by the COSMOS2020 catalog, we leverage the large 1.27 deg2 effective area to improve sample statistics and understand spatial variations (cosmic variance) – particularly for rare, massive galaxies – and push to higher redshifts with greater confidence and mass completeness than previous studies. We divide the total stellar mass function into star-forming and quiescent subsamples through NUVrJ color-color selection. The measurements are then fit with single- and double-component Schechter functions to infer the intrinsic galaxy stellar mass function, the evolution of its key parameters, and the cosmic stellar mass density out to z = 7.5. Finally, we compare our measurements to predictions from state-of-the-art cosmological simulations and theoretical dark matter halo mass functions. Results. We find a smooth, monotonic evolution in the galaxy stellar mass function since z = 7.5, in general agreement with previous studies. The number density of star-forming systems have undergone remarkably consistent growth spanning four decades in stellar mass from z = 7.5 → 2 whereupon high-mass systems become predominantly quiescent (“downsizing”). Meanwhile, the assembly and growth of low-mass quiescent systems only occurred recently, and rapidly. An excess of massive systems at z ≈ 2.5 − 5.5 with strikingly red colors, with some being newly identified, increase the observed number densities to the point where the SMF cannot be reconciled with a Schechter function. Conclusions. Systematics including cosmic variance and/or active galactic nuclei contamination are unlikely to fully explain this excess, and so we speculate that they may be dust-obscured populations similar to those found in far infrared surveys. Furthermore, we find a sustained agreement from z ≈ 3 − 6 between the stellar and dark matter halo mass functions for the most massive systems, suggesting that star formation in massive halos may be more efficient at early times.

We use a large sample of galaxies from the Two Micron All Sky Survey (2MASS) and the Sloan Digital Sky Survey (SDSS) to calculate galaxy luminosity and stellar mass functions in the local Universe. We estimate k-corrections, evolution corrections, and stellar mass-to-light ratios (M/Ls) by fitting the galaxy colors with simple models. Our optical and near-infrared luminosity functions agree with most recent literature optical and near-infrared determinations within the uncertainties. We argue that 2MASS is biased against low surface brightness galaxies, and use SDSS plus our knowledge of stellar populations to estimate that the true K-band luminosity function has a steeper faint end slope than the direct estimate. Assuming a universally-applicable stellar initial mass function (IMF), we derive the stellar mass function of galaxies. The faint end slope slope for the stellar mass function is steeper than -1.1, reflecting the low stellar M/Ls characteristic of low-mass galaxies. We estimate an upper limit to the stellar mass density in the local Universe Omega* h = 0.0020+/-0.0006 by assuming an IMF as rich in low-mass stars as allowed by observations of galaxy dynamics. Finally, we find that the characteristic luminosity or mass of early-type galaxies is larger than for later types, and the faint end slope is steeper for later types than for earlier types. Accounting for typing uncertainties, we estimate that at least half, and perhaps as much as 3/4, of the stellar mass in the Universe is in early-type galaxies. We present also SDSS/2MASS color-M/L correlations, an updated discussion of near-infrared stellar M/L estimates, and the volume-corrected distribution of g and K-band stellar M/Ls as a function of stellar mass. [Abridged]

The evolution of the galaxy stellar mass function is especially useful to test the current model of galaxy formation. Observational data have revealed a few inconsistencies with predictions from the $\Lambda {\rm CDM}$ model. For example, most massive galaxies have already been observed at very high redshifts, and they have experienced only mild evolution since then. In conflict with this, semi-analytical models of galaxy formation predict an insufficient number of massive galaxies at high redshift and a rapid evolution between redshift 1 and 0 . In addition, there is a strong correlation between star formation rate and stellar mass for star-forming galaxies, which can be roughly reproduced with the model, but with a normalization that is too low at high redshift. Furthermore, the stellar mass density obtained from the integral of the cosmic star formation history is higher than the measured one by a factor of 2. In this paper, we study these issues using a semi-analytical model that includes: 1) cold gas accretion in massive halos at high redshift; 2) tidal stripping of stellar mass from satellite galaxies; and 3) an evolving stellar initial mass function (bottom-light) with a higher gas recycle fraction. Our results show that the combined effects from 1) and 2) can predict sufficiently massive galaxies at high redshifts and reproduce their mild evolution at low redshift, While the combined effects of 1) and 3) can reproduce the correlation between star formation rate and stellar mass for star-forming galaxies across wide range of redshifts. A bottom-light/top-heavy stellar IMF could partly resolve the conflict between the stellar mass density and cosmic star formation history.

In the local Universe, the existence of very young galaxies (VYGs), having formed at least half their stellar mass in the last 1 Gyr, is debated. We predict the present-day fraction of VYGs among central galaxies as a function of galaxy stellar mass. For this, we apply to high mass resolution Monte-Carlo halo merger trees (MCHMTs) three (one) analytical models of galaxy formation, where the ratio of stellar to halo mass (mass growth rate) is a function of halo mass and redshift. Galaxy merging is delayed until orbital decay by dynamical friction. With starbursts associated with halo mergers, our models predict typically one percent of VYGs up to galaxy masses of $10^{10}$ M$_\odot$, falling rapidly at higher masses, and VYGs are usually associated with recent major mergers of their haloes. Without these starbursts, two of the models have VYG fractions reduced by 1 or 2 dex at low or intermediate stellar masses, and VYGs are rarely associated with major halo mergers. In comparison, the state-of-the-art semi-analytical model (SAM) of Henriques et al. produces only 0.01% of VYGs at intermediate masses. Finally, the Menci et al. SAM run on MCMHTs with Warm Dark Matter cosmology generates 10 times more VYGs at masses below $10^8$ M$_\odot$ than when run with Cold Dark Matter. The wide range in these VYG fractions illustrates the usefulness of VYGs to constrain both galaxy formation and cosmological models.

Deviations from a universal, MW-like, Stellar Initial Mass Function (IMF) have been reported for distant galaxies, although the physical reason behind the observed variations is still matter of ongoing debate. In this paper, we present an exploratory study to assess the impact of the proposed IMF evolution on the statistical galaxy properties, as predicted by the Semi-Analytical model of galaxy formation and evolution MORGANA. In particular, we test different dependencies for the IMF shape, as a function of both model galaxy properties (such as star formation rate, velocity dispersion or stellar mass) and environment, and compare the predicted stellar mass functions and star formation rate functions with reference runs at fixed IMF. In most cases, MORGANA predictions show deviations of the order of a few tenths of dex with respect to a run assuming an Universal Kroupa IMF. Among the proposed IMF variations, an increasing Top-Heavy IMF at increasing star formation rates has the largest impact on predicted galaxy properties, while most of the models assuming an increasing Bottom-Heavy IMF at higher masses/velocity dispersion lead to variations in galaxy properties that are of the same order as the uncertainty on the mass and star formation rate determination. By comparing the predicted galaxy stellar mass functions, we conclude that the study of the high-mass end can provide useful constraints to disentangle models assuming an increasing Top-Heavy IMF in high star forming or Bottom-Heavy IMF in massive systems.

We present a new version of the GALFORM semi-analytical model of galaxy formation. This brings together several previous developments of GALFORM into a single unified model, including a different initial mass function (IMF) in quiescent star formation and in starbursts, feedback from active galactic nuclei supressing gas cooling in massive halos, and a new empirical star formation law in galaxy disks based on their molecular gas content. In addition, we have updated the cosmology, introduced a more accurate treatment of dynamical friction acting on satellite galaxies, and updated the stellar population model. The new model is able to simultaneously explain both the observed evolution of the K-band luminosity function and stellar mass function, and the number counts and redshift distribution of sub-mm galaxies selected at 850μm. This was not previously achieved by a single physical model within the ΛCDM framework, but requires having an IMF in starbursts that is somewhat top-heavy. The new model is tested against a wide variety of observational data covering wavelengths from the far-UV to sub-mm, and redshifts from z = 0 to z = 6, and is found to be generally successful. These observations include the optical and near-IR luminosity functions, HI mass function, fraction of early type galaxies, Tully-Fisher, metallicity-luminosity and size-luminosity relations at z = 0, as well as far-IR number counts, and far-UV luminosity functions at z ∼ 3 − 6. Discrepancies are however found in galaxy sizes and metallicities at low luminosities, and in the abundance of low mass galaxies at high-z, suggesting the need for a more sophisticated model of supernova feedback.

We study the stellar haloes of galaxies out to 70-100 kpc as a function of stellar mass and galaxy type by stacking aligned $r$ and $g$ band images from a sample of 45508 galaxies from SDSS DR9 in the redshift range $0.06\,\le\,z\,\le\,0.1$ and in the mass range $10^{10.0} M_{\odot} < M_{*} < 10^{11.4} M_{\odot}$r. We derive surface brightness profiles to a depth of almost $\mu_r \sim 32 \,\mathrm{mag\,arcsec}^{-2}$. We find that the ellipticity of the stellar halo is a function of galaxy stellar mass and that the haloes of high concentration ($C > 2.6$) galaxies are more elliptical than those of low concentration ($C < 2.6$) galaxies. The $g$-$r$ colour profile of high concentration galaxies reveals that the $g$-$r$ colour of the stellar population in the stellar halo is bluer than in the main galaxy, and the colour of the stellar halo is redder for higher mass galaxies. We further demonstrate that the full two-dimensional surface intensity distribution of our galaxy stacks can only be fit through multi-component S\'{e}rsic models. Using the fraction of light in the outer component of the models as a proxy for the fraction of accreted stellar light, we show that this fraction is a function of stellar mass and galaxy type. For high concentration galaxies, the fraction of accreted stellar light rises from $30\%$ to $70\%$ for galaxies in the stellar mass range from $10^{10.0} M_{\odot}$ to $10^{11.4} M_{\odot}$. The fraction of accreted light is much smaller in low concentration systems, increasing from $2\%$ to $25\%$ over the same mass range. This work provides important constraints for the theoretical understanding of the formation of stellar haloes of galaxies.

It is well known that the mass function for_halos_ in CDM cosmology is a relatively steep power law for low masses, possibly too steep to be consistent with observations. But how steep is the_galaxy_ mass function? We have analyzed the stellar and gas mass functions of the first massive luminous objects formed in a \Lambda CDM universe, as calculated in the numerical simulation described in Gnedin (2000ab). We found that while the dark matter mass function is steep, the stellar and gas mass functions are flatter for low mass objects. The stellar mass function is consistently flat at the low mass end. Moreover, while the gas mass function follows the dark matter mass function until reionization at z~7, between z=7 and z=4, the gas mass function also flattens considerably at the low mass end. At z=4, the gas and stellar mass functions are fit by a Schechter function with \alpha ~ -1.2 +/- 0.1, significantly shallower than the dark matter halo mass function and consistent with some recent observations. The baryonic mass functions are shallower because (a) the dark matter halo mass function is consistent with the Press-Schechter formulation at low masses n(M) M^-2 and (b) heating/cooling and ionization processes appear to cause baryons to collect in halos with the relationship M_b M_d^4 at low masses. Combining (a) and (b) gives n(M_b) M_b^-5/4, comparable to the simulation results. Thus, the well known observational fact that low mass galaxies are underabundant as compared to expectations from numerical dark matter simulations or Press-Schechter modeling of CDM universes emerges naturally from these results, implying that perhaps no ``new physics'' beyond the standard model is needed.

In this paper, we present a new derivation of the shape and evolution of the integrated galaxy-wide initial mass function (IGIMF), incorporating explicitly the effects of cosmic rays (CRs) as regulators of the chemical and thermal state of the gas in the dense cores of molecular clouds. We predict the shape of the IGIMF as a function of star formation rate and CR density and show that it can be significantly different with respect to local estimates. In particular, we focus on the physical conditions corresponding to IGIMF shapes that are simultaneously shallower at high-mass end and steeper at the low-mass end than those of a Kroupa IMF. These solutions can explain both the levels of α-enrichment and the excess of low-mass stars as a function of stellar mass, observed for local spheroidal galaxies. As a preliminary test of our scenario, we use idealized star formation histories to estimate the mean IMF shape for galaxies of different |$z$| = 0 stellar mass. We show that the fraction of low-mass stars as a function of galaxy stellar mass predicted by these mean IMFs agrees with the values derived from high-resolution spectroscopic surveys.

We compare the semi-analytic models of galaxy formation of Fu et al. (2010), which track the evolution of the radial profiles of atomic and molecular gas in galaxies, with gas fraction scaling relations derived from the COLD GASS survey (Saintonge et al 2011). The models provide a good description of how condensed baryons in galaxies with gas are partitioned into stars, atomic and molecular gas as a function of galaxy stellar mass and surface density. The models do not reproduce the tight observed relation between stellar surface density and bulge-to-disk ratio for this population. We then turn to an analysis of the"quenched" population of galaxies without detectable cold gas. The current implementation of radio-mode feedback in the models disagrees strongly with the data. In the models, gas cooling shuts down in nearly all galaxies in dark matter halos above a mass of 10**12 M_sun. As a result, stellar mass is the observable that best predicts whether a galaxy has little or no neutral gas. In contrast, our data show that quenching is largely independent of stellar mass. Instead, there are clear thresholds in bulge-to-disk ratio and in stellar surface density that demarcate the location of quenched galaxies. We propose that processes associated with bulge formation play a key role in depleting the neutral gas in galaxies and that further gas accretion is suppressed following the formation of the bulge, even in dark matter halos of low mass.

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.

We compare predictions of a number of empirical models and numerical simulations of galaxy formation to the conditional stellar mass functions (CSMF)of galaxies in groups of different masses obtained recently by Lan et al. to test how well different models accommodate the data. The observational data clearly prefer a model in which star formation in low-mass halos changes behavior at a characteristic redshift $z_c\sim 2$. There is also tentative evidence that this characteristic redshift depends on environment, becoming $z_c\sim 4$ in regions that eventually evolve into rich clusters of galaxies. The constrained model is used to understand how galaxies form and evolve in dark matter halos, and to make predictions for other statistical properties of the galaxy population, such as the stellar mass functions of galaxies at high $z$, the star formation and stellar mass assembly histories in dark matter halos. A comparison of our model predictions with those of other empirical models shows that different models can make vastly different predictions, even though all of them are tuned to match the observed stellar mass functions of galaxies.