Schechter Theory Research Articles

Stochastic gravitational wave background from the collisions of dark matter halos

We investigate the effect of the dark matter (DM) halos collisions, namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, on the stochastic gravitational wave background. We first calculate the gravitational wave signal of a single collision event, assuming point masses and linear perturbation theory. Then we proceed to the calculation of the energy spectrum of the collective effect of all dark matter collisions in the Universe. Concerning the DM halo collision rate, we show that it is given by the product of the number density of DM halos, which is calculated by the extended Press–Schechter (EPS) theory, with the collision rate of a single DM halo, which is given by simulation results, with a function of the linear growth rate of matter density through cosmological evolution. Hence, integrating over all mass and distance ranges, we finally extract the spectrum of the stochastic gravitational wave background created by DM halos collisions. As we show, the resulting contribution to the stochastic gravitational wave background is of the order of hc≈10-29\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{c} \\approx 10^{-29}$$\\end{document} in the band of f≈10-15Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f \\approx 10^{-15} Hz$$\\end{document}. However, in very low frequency band, it is larger. With current observational sensitivity it cannot be detected.

Machine learning and structure formation in modified gravity

ABSTRACT In general relativity, approximations based on the spherical collapse model such as Press–Schechter theory and its extensions are able to predict the number of objects of a certain mass in a given volume. In this paper, we use a machine learning algorithm to test whether such approximations hold in screened modified gravity theories. To this end, we train random forest classifiers on data from N-body simulations to study the formation of structures in lambda cold dark matter (ΛCDM) as well as screened modified gravity theories, in particular f(R) and nDGP gravity. The models are taught to distinguish structure membership in the final conditions from spherical aggregations of density field behaviour in the initial conditions. We examine the differences between machine learning models that have learned structure formation from each gravity, as well as the model that has learned from ΛCDM. We also test the generalizability of the ΛCDM model on data from f(R) and nDGP gravities of varying strengths, and therefore the generalizability of extended Press–Schechter spherical collapse to these types of modified gravity.

Growing the first galaxies’ merger trees

ABSTRACT Modelling the growth histories of specific galaxies often involves generating the entire population of objects that arise in a given cosmology and selecting systems with appropriate properties. This approach is highly inefficient when targeting rare systems such as the extremely luminous high-redshift galaxy candidates detected by JWST. Here, we present a novel framework for generating merger trees with branches that are guaranteed to achieve a desired halo mass at a chosen redshift. This method augments extended Press Schechter theory solutions with constrained random processes known as Brownian bridges and is implemented in the open-source semi-analytic model galacticus. We generate ensembles of constrained merger trees to predict the growth histories of seven high-redshift JWST galaxy candidates, finding that these systems most likely merge ≈2 Gyr after the observation epoch and occupy haloes of mass ≳1014 M⊙ today. These calculations are thousands of times more efficient than existing methods, are analytically controlled, and provide physical insights into the evolution of haloes with rapid early growth. Our constrained merger tree implementation is publicly available at https://github.com/galacticusorg/galacticus.

The brightest galaxies at cosmic dawn

ABSTRACT Recent JWST observations suggest an excess of z ≳ 10 galaxy candidates above most theoretical models. Here, we explore how the interplay between halo formation time-scales, star formation efficiency, and dust attenuation affects the properties and number densities of galaxies observed in the early Universe. To guide intuition, we calculate the theoretical upper limit on the UV luminosity function (LF), assuming star formation is 100 per cent efficient and all gas in haloes is converted into stars, and that galaxies are at the peak age for UV emission (∼10 Myr). This upper limit is ∼4 orders of magnitude greater than current observations, implying no formal tension with star formation in Lambda cold dark matter cosmology. In a more realistic model, we use the distribution of halo formation time-scales derived from extended Press–Schechter theory as a proxy for star formation rate (SFR). We predict that the galaxies observed so far at z ≳ 10 are dominated by those with the fastest formation time-scales, and thus most extreme SFRs and young ages. These galaxies can be upscattered by ∼1.5 mag compared to the median UV magnitude versus halo mass relation. This likely introduces a selection effect at high redshift whereby only the youngest (≲10 Myr), most highly star-forming galaxies (specific SFR$\gtrsim 30\, \mathrm{Gyr}^{-1}$) have been detected so far. Furthermore, our modelling suggests that redshift evolution at the bright end of the UV LF is substantially affected by the build-up of dust attenuation. We predict that deeper JWST observations (reaching m ∼ 30) will reveal more typical galaxies with relatively older ages (∼100 Myr) and less extreme specific SFRs ($\sim 10\, \mathrm{Gyr}^{-1}$ for a MUV ∼ −20 galaxy at z ∼ 10).

Semi-analytical frameworks for subhaloes from the smallest to the largest scale

ABSTRACT Substructures of dark matter halo, called subhaloes, provide important clues to understand the nature of dark matter. We construct a useful model to describe the properties of subhalo mass functions based on the well-known analytical prescriptions, the extended Press–Schechter theory. The unevolved subhalo mass functions at arbitrary mass scales become describable without introducing free parameters. The different host halo evolution histories are directly recast to their subhalo mass functions. As applications, we quantify the effects from (i) the Poisson fluctuation, (ii) the host-mass scatter, and the (iii) different tidal evolution models on observables in the current Universe with this scheme. The Poisson fluctuation dominates in the number count of the mass ratio to the host of $\sim {\cal O}(10^{-2})$, where the intrinsic scatter is smaller by a factor of a few. The host-mass scatter around its mean does not affect the subhalo mass function. Different models of the tidal evolution predict a factor of ∼2 difference in numbers of subhaloes with $\lesssim {\cal O}(10^{-5})$, while the dependence of the Poisson fluctuation on the tidal evolution models is subtle. The scheme provides a new tool for investigating the smallest scale structures of our Universe which are to be observed in near future experiments.

Public Release of A-SLOTH: Ancient Stars and Local Observables by Tracing Halos

The semianalytical model a-sloth (Ancient Stars and Local Observables by Tracing Halos) is the first public code that connects the formation of the first stars and galaxies to observables. After several successful projects with this model, we publish the source code (https://gitlab.com/thartwig/asloth) and describe the public version in this paper. The model is based on dark matter merger trees that can either be generated based on Extended Press–Schechter theory or be imported from dark matter simulations. On top of these merger trees, a-sloth applies analytical recipes for baryonic physics to model the formation of both metal-free and metal-poor stars and the transition between them with unprecedented precision and fidelity. a-sloth samples individual stars and includes radiative, chemical, and mechanical feedback. It is calibrated based on six observables, such as the optical depth to Thomson scattering, the stellar mass of the Milky Way and its satellite galaxies, the number of extremely metal-poor stars, and the cosmic star formation rate density at high redshift. a-sloth has versatile applications with moderate computational requirements. It can be used to constrain the properties of the first stars and high-z galaxies based on local observables, predicts properties of the oldest and most metal-poor stars in the Milky Way, can serve as a subgrid model for larger cosmological simulations, and predicts next-generation observables of the early universe, such as supernova rates or gravitational wave events.

A Stochastic Theory of the Hierarchical Clustering. II. Halo Progenitor Mass Function and Large-scale Bias

Abstract We generalize the stochastic theory of hierarchical clustering presented in Paper I by Lapi & Danese to derive the (conditional) halo progenitor mass function and the related large-scale bias. Specifically, we present a stochastic differential equation that describes fluctuations in the mass growth of progenitor halos of given descendant mass and redshift, as driven by a multiplicative Gaussian white noise involving the power spectrum and the spherical collapse threshold of density perturbations. We demonstrate that, as cosmic time passes, the noise yields an average drift of the progenitors toward larger masses, which quantitatively renders the expectation from the standard extended Press and Schechter (EPS) theory. We solve the Fokker–Planck equation associated with the stochastic dynamics, and obtain as an exact, stationary solution, the EPS progenitor mass function. Then we introduce a modification of the stochastic equation in terms of a mass-dependent collapse threshold modulating the noise, and solve analytically the associated Fokker–Planck equation for the progenitor mass function. The latter is found to be in excellent agreement with the outcomes of N-body simulations; even more remarkably, this is achieved with the same shape of the collapse threshold used in Paper I to reproduce the halo mass function. Finally, we exploit the above results to compute the large-scale halo bias, and find it in pleasing agreement with the N-body outcomes. All in all, the present paper illustrates that the stochastic theory of hierarchical clustering introduced in Paper I can describe effectively not only halos’ abundance, but also their progenitor distribution and their correlation with the large-scale environment across cosmic times.

Universal at Last? The Splashback Mass Function of Dark Matter Halos

Abstract The mass function of dark matter halos is one of the most fundamental statistics in structure formation. Many theoretical models (such as Press–Schechter theory) are based on the notion that it could be universal, meaning independent of redshift and cosmology, when expressed in the appropriate variables. However, simulations exhibit persistent nonuniversalities in the mass functions of the virial mass and other commonly used spherical overdensity definitions. We systematically study the universality of mass functions over a wide range of mass definitions, for the first time including the recently proposed splashback mass, . We confirm that, in ΛCDM cosmologies, all mass definitions exhibit varying levels of nonuniversality that increase with peak height and reach between 20% and 500% at the highest masses we can test. , , and exhibit similar levels of nonuniversality. There are, however, two regimes where the splashback mass functions are significantly more universal. First, they are universal to 10% at , whereas spherical overdensity definitions experience an evolution due to dark energy. Second, when additionally considering self-similar cosmologies with extreme power spectra, splashback mass functions are remarkably universal (to between 40% and 60%), whereas their spherical overdensity counterparts reach nonuniversalities between 180% and 450%. These results strongly support the notion that the splashback radius is a physically motivated definition of the halo boundary. We present a simple and universal fitting formula for splashback mass functions that accurately reproduces our simulation data.

The accretion history of dark matter haloes – III. A physical model for the concentration–mass relation

We present a semi-analytic, physically motivated model for dark matter halo concentration as a function of halo mass and redshift. The semi-analytic model combines an analytic model for the halo mass accretion history (MAH), based on extended Press Schechter (EPS) theory, with an empirical relation between concentration and formation time obtained through fits to the results of numerical simulations. Because the semi-analytic model is based on EPS theory, it can be applied to wide ranges in mass, redshift and cosmology. The resulting concentration-mass (c-M) relations are found to agree with the simulations, and because the model applies only to relaxed halos, they do not exhibit the upturn at high masses or high redshifts found by some recent works. We predict a change of slope in the z=0 c-M relation at a mass scale of $10^{11}\rm{M}_{\odot}$. We find that this is due to the change in the functional form of the halo MAH, which goes from being dominated by an exponential (for high-mass halos) to a power-law (for low-mass halos). During the latter phase, the core radius remains approximately constant, and the concentration grows due to the drop of the background density. We also analyse how the c-M relation predicted by this work affects the power produced by dark matter annihilation, finding that at z = 0 the power is two orders of magnitude lower than that obtained from extrapolating best-fitting c-M relations. We provide fits to the c-M relations as well as numerical routines to compute concentrations and MAHs.

Natural downsizing in hierarchical galaxy formation

Stellar-population analyses of today's galaxies show ‘downsizing’, where the stars in more massive galaxies tend to have formed earlier and over a shorter time-span. We show that this phenomenon is not necessarily ‘antihierarchical’ but rather has its natural roots in the bottom-up clustering process of dark-matter haloes. While the main progenitor does indeed show an opposite effect, the integrated mass in all the progenitors down to a given minimum mass shows a robust downsizing that is qualitatively similar to what has been observed. These results are derived analytically from the standard extended Press–Schechter (EPS) theory, and are confirmed by merger trees based on EPS or drawn from N-body simulations. The downsizing is valid for any minimum mass, as long as it is the same for all haloes at any given time, but the effect is weaker for smaller minimum mass. If efficient star formation is triggered by atomic cooling, then a minimum halo mass arises naturally from the minimum virial temperature for cooling, T≃ 104 K, though for such a small minimum mass the effect is weaker than observed. Baryonic feedback effects, which are expected to stretch the duration of star formation in small galaxies and shut it down in massive haloes at late epochs, are likely to play a subsequent role in shaping up the final downsizing behaviour. Other appearances of downsizing, such as the decline with time of the typical mass of star-forming galaxies, may not be attributed to the gravitational clustering process but rather arise from the gas processes.

Early structure in CDM

We use a novel technique to simulate the growth of one of the most massive progenitors of a supercluster region from redshift z 80, when its mass was about 10 M, until the present day. Our nested sequence of N-body resimulations allows us to study in detail the structure both of the dark matter object itself and of its environment. Our effective resolution is optimal at redshifts of 49, 29, 12, 5 and 0 when the dominant object has mass 1.2 × 105, 5 × 107, 2 × 1010, 3 × 1012 and 8 × 1014 h1 M, respectively, and contains 106 simulation particles within its virial radius. Extended Press–Schechter (EPS) theory correctly predicts both this rapid growth and the substantial overabundance of massive haloes we find at early times in regions surrounding the dominant object. Although the large-scale structure in these regions differs dramatically from a scaled version of its present-day counterpart, the internal structure of the dominant object is remarkably similar. Molecular hydrogen cooling could start as early as z 49 in this object, while cooling by atomic hydrogen becomes effective at z 39. If the first stars formed in haloes with virial temperature 2000 K, their comoving abundance at z= 49 should be similar to that of dwarf galaxies today, while their comoving correlation length should be 2.5 h1 Mpc.

Ultraviolet background radiation from cosmic structure formation

We calculate the contribution to the ultraviolet background (UVB) from thermal emission from gas shock heated during cosmic structure formation. Our main calculation is based on an updated version of Press–Schechter theory. It is consistent with a more empirical estimate based on the observed properties of galaxies and the observed cosmic star formation history. Thermal UVB emission is characterized by a hard spectrum extending well beyond 4 Ryd. The bulk of the radiation is produced by objects in the mass range 1011–1013 M⊙, i.e. large galaxies and small groups. We compute a composite UVB spectrum due to quasi-stellar object (QSO), stellar and thermal components. The ratio of the UVB intensities at the H and He Lyman limits increases from 60 at z = 2 to more than 300 at z = 6. A comparison of the resulting photoionization rates to the observed Gunn–Peterson effect at high redshifts constrains the escape fraction of ionizing photons from galaxies to be less than a few per cent. Near 1 Ryd, thermal and stellar emission are comparable, amounting to about 10, 20 and 35 per cent of the total flux at redshifts of 3, 4.5 and higher, respectively. However, near the ionization threshold for He ii, the thermal contribution is much stronger. It is comparable to the QSO intensity already at redshift ∼3 and dominates at redshifts above 4. Thermal photons alone are enough to produce and sustain He ii reionization already at z≈ 6. We discuss the possible implications of our results for the thermal history of the intergalactic medium, in particular for He ii reionization.

Extended Press-Schechter theory and the density profiles of dark matter haloes

An inside–out model for the formation of haloes in a hierarchical clustering scenario is studied. The method combines the picture of the spherical infall model and a modification of the extended Press–Schechter theory. The mass accretion rate of a halo is defined to be the rate of its mass increase due to minor mergers. The accreted mass is deposited at the outer shells without changing the density profile of the halo inside its current virial radius. We applied the method to a flat Λ-cold dark matter universe. The resulting density profiles are compared with analytical models proposed in the literature, and a very good agreement is found. A trend is found of the inner density profile to become steeper for larger halo mass, which also results from recent N-body simulations. Additionally, present-day concentrations as well as their time evolution are derived and it is shown that they reproduce the results of large cosmological N-body simulations.

Galaxy formation using halo merger histories taken from N-body simulations

We develop a hybrid galaxy formation model that uses outputs from an N-body simulation to follow the merger histories (or ‘merger trees’) of dark matter haloes and treats baryonic processes, such as the cooling of gas within haloes and subsequent star formation, using the semi-analytic model of Cole et al. We compare this hybrid model with an otherwise identical model that utilizes merger-tree realizations generated using a Monte Carlo algorithm and find that, apart from the limited mass resolution imposed by the N-body particle mass, the only significant differences between the models are caused by the known discrepancy between the distribution of halo progenitor masses predicted by the extended Press—Schechter theory and that found in N-body simulations. We investigate the effect of limited mass resolution on the hybrid model by comparing with a purely semi-analytic model that has greatly improved mass resolution. We find that the mass resolution of the simulation we use, which has a particle mass of 1.4 × 1010h−1 M⊙, is insufficient to produce a reasonable luminosity function for galaxies with magnitudes in the bJ band fainter than −17.

The build-up of haloes within Press-Schechter theory

Modelling the build-up of haloes is important for linking the formation of galaxies with cosmological models. A simple model of halo growth is provided by Press–Schechter (PS) theory, where the initial field of density fluctuations is smoothed using spherically symmetric filters centred on a given position to obtain information about the likelihood of later collapse on varying scales. In this paper the predicted halo mass growth is compared for three filter shapes: Gaussian, top-hat and sharp k-space. Preliminary work is also presented analysing the build-up of haloes within numerical simulations using a friends-of-friends group finder. The best-fit to the simulation mass function was obtained using PS theory with a top-hat filter. By comparing both the backwards conditional mass function, which gives the distribution of halo progenitors, and the distribution of halo mergers in time, the build-up of haloes in the simulations is shown to be better fitted by PS theory with a sharp k-space filter. This strengthens previous work, which also found the build-up of haloes in simulations to be well matched to PS theory with a sharp k-space filter by providing a direct comparison of different filters and by extending the statistical tools used to analyse halo mass growth. The usefulness of this work is illustrated by showing that the cosmological evolution in the proportion of haloes that have undergone recent merger is predicted to be independent of mass and power spectrum and to only depend upon cosmology. Recent results from observations of field galaxies are shown to match the evolution expected, but are not sufficiently accurate to distinguish usefully between cosmological parameters.

Merging history trees for dark matter haloes: tests of the Merging Cell Model in a CDM cosmology

The merging history of dark matter halos is computed with the Merging Cell Model proposed by Rodrigues & Thomas (1996). While originally discussed in the case of scale–free power spectra, it is developed and tested here in the framework of the cold dark matter cosmology. The halo mass function, the mass distribution of progenitors and child halos, as well as the probability distribution of formation times, have been computed and compared to the available analytic predictions. The halo auto– correlation function has also been obtained (a first for a semi–analytic merging tree), and tested against analytic formulae. An overall good agreement is found between results of the model, and the predictions derived from the Press & Schechter theory and its extensions. More severe discrepancies appear when formulae that better describe N-body simulations are used for comparison. In many instances, the model can be a useful tool for following the hierarchical growth of structures. In particular, it is suitable for addressing the issue of the formation and evolution of galaxy clusters, as well as the population of Lyman–break galaxies at high redshift, and their clustering properties.

Cosmological evolution and hierarchical galaxy formation

We calculate the rate at which dark matter haloes merge to form higher mass systems. Two complementary derivations using Press—Schechter theory are given, both of which result in the same equation for the formation rate. First, a derivation using the properties of the Brownian random walks within the framework of Press—Schechter theory is presented. We then use Bayes' theorem to obtain the same result from the standard Press—Schechter mass function. The rate obtained is shown to be in good agreement with results from Monte Carlo and N-body simulations. We illustrate the usefulness of this formula by calculating the expected cosmological evolution in the rate of star formation that is due to short-lived, merger-induced starbursts. The calculated evolution is well-matched to the observed evolution in ultraviolet luminosity density, in contrast to the lower rates of evolution that are derived from semi-analytic models that do not include a dominant contribution from starbursts. Hence we suggest that the bulk of the observed ultraviolet starlight at z>1 arises from merger-induced starbursts. Finally, we show that a simple merging-halo model can also account for the bulk of the observed evolution in the comoving quasar space density.

Cosmological mass function with 1D gravity

The cosmological mass function problem is analyzed in full detail in the case of 1D gravity, with analytical, semianalytical, and numerical techniques. The extended Press and Schechter theory is improved by detailing the relation between smoothing radius and mass of the objects. This is done by introducing in the formalism the concept of a growth curve for the objects. The predictions of the extended Press and Schechter theory are compared to large N-body simulations of flat expanding 1D universes with scale-free power spectra of primordial perturbations. The collapsed objects in the simulations are located with a clump-finding algorithm designed to find regions that have undergone orbit crossing or that are in the multistream regime (these are different as an effect of the finite size of the multistream regions). It is found that the semianalytical mass function theory, which has no free parameters, is able to recover the properties of collapsed objects both statistically and object by object. In particular, the predictions of regions in orbit crossing are optimized by the use of Gaussian filtering, while the use of sharp k-space filtering apparently allows to reproduce the larger multistream regions. The mass function theory does not reproduce well the clumps found with the standard friends-of-friends algorithm; however, the performance of this algorithm has not been thoroughly tested in the 1D cosmology. Our preliminary analyses of the 3D case confirms that the techniques developed in this paper are precious in understanding the cosmological mass function problem in 3D.

Number Counts of Clusters of Galaxies in X-ray and Submm Bands

We compute the number counts of clusters of galaxies, the logN-logS relation, in several X-ray and submm bands on the basis of the Press—Schechter theory (Kitayama et al. 1998). We pay particular attention to a set of theoretical models which well reproduce the ROSAT 0.5–2 keV band logN-logS (Ebeling et al. 1997; Rosati et al. 1997), and explore possibilities to further constrain the models from future observations with ASCA and/or at submm bands. The latter is closely related to the European PLANCK mission and the Japanese LMSA project. We exhibit that one can break the degeneracy in an acceptable parameter region on the Ω0–σ8 plane by combining the ROSAT logN-logS and the submm number counts. Models which reproduce the ROSAT band logN-logS will have N(> S) ∼ (150–300)(S/10−12 erg cm−2 s−) −1.3 str−1 at S ≳ 10−12 erg cm−2 s−1 in the ASCA 2–10 keV band, and N(> Sv) ∼ (102–104)(Sv/100 mJy)−1.5 str−1 at Sv ≳ 100m J y in the submm (0.85mm) band. The amplitude of the logN-logS is very sensitive to the model parameters in the submm band. We also compute the redshift evolution of the cluster number counts and compare with that of the X-ray brightest Abell-type clusters (Ebeling et al. 1996). The results, although still preliminary, point to low density (Ω0 ∼ 0.3) universes. The contribution of clusters to the X-ray and submm background radiations is shown to be insignificant in any model compatible with the ROSAT logN-logS.

The cluster distribution as a test of dark matter models - I. Clustering properties

We present simulations of the cluster distribution in several dark matter models, using an optimized version of the truncated Zel'dovich approximation (TZA). We compare them with N-body cluster simulations and find that the TZA provides a very accurate description of the cluster distribution as long as fluctuations on the cluster mass scale are in the mildly non-linear regime. The simulated dark matter models are: Standard CDM (SCDM), Tilted CDM (TCDM) with n=0.7, Cold+Hot DM (CHDM) with 30% of hot component, low Hubble constant (h=0.3) CDM (LOWH) and a spatially flat low-density CDM model with \Omega_0=0.2. We compare the simulations with a redshift sample of Abell/ACO clusters, using the integral of the 2-point correlation function and the probability density function. We find that the best models at reproducing the data are CHDM and LCDM. All the other models are ruled out. The reduced skewness S_3 is fairly constant with S_3=1.9, independent of the DM model and consistent with observational data. The abundances of clusters predicted using the Press--Schechter theory provide strong constraints: only the CHDM, LOWH and LCDM models appear to produce the correct number-density of clusters.