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«Theoretical Modeling of the High Redshift Galaxy Population David H. Weinberg Department of Astronomy, Ohio State University, Columbus, OH 43210 ...»

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Theoretical Modeling of the High Redshift Galaxy


David H. Weinberg

Department of Astronomy, Ohio State University, Columbus, OH 43210

Romeel Dav´


Astrophysical Sciences, Princeton University, Princeton, NJ 08544

Jeffrey P. Gardner

Department of Astronomy, University of Washington, Seattle, WA


Lars Hernquist

Department of Astronomy, Harvard University, Cambridge, MA 02138

Neal Katz

Department of Physics and Astronomy, University of Massachusetts,

Amherst, MA 98195 Abstract. We review theoretical approaches to the study of galaxy formation, with emphasis on the role of hydrodynamic cosmological sim- ulations in modeling the high redshift galaxy population. We present new predictions for the abundance of star-forming galaxies in the LCDM model (inflation + cold dark matter, with Ωm = 0.4, ΩΛ = 0.6), combin- ing results from several simulations to probe a wide range of redshift. At a threshold density of one object per arcmin2 per unit redshift, these sim- ulations predict galaxies with star formation rates of 2M /yr (z = 10), 5M /yr (z = 8), 20M /yr (z = 6), 70 − 100M /yr (z = 4 − 2), and 30M /yr (z = 0.5). For galaxies selected at a fixed comoving space den- sity n = 0.003 h3 Mpc−3, a simulation of a 50h−1 Mpc cube predicts a galaxy correlation function (r/5h−1 Mpc)−1.8 in comoving coordinates, essentially independent of redshift from z = 4 to z = 0.5. Different cosmological models predict global histories of star formation that reflect their overall histories of mass clustering, but robust numerical predictions of the comoving space density of star formation are difficult because the simulations miss the contribution from galaxies below their resolution limit. The LCDM model appears to predict a star formation history with roughly the shape inferred from observations, but it produces too many stars at low redshift, predicting Ω ≈ 0.015 at z = 0. We conclude with a brief discussion of this discrepancy and three others that suggest gaps in our current theory of galaxy formation: small disks, steep central halo profiles, and an excess of low mass dark halos. While these problems 1 could fade as the simulations or observations improve, they could also guide us towards a new understanding of galactic scale star formation, the spectrum of primordial fluctuations, or the nature of dark matter.

1. Theoretical Approaches to Galaxy Formation In broad outline, the current theory of galaxy formation is remarkably similar to the one described by White & Rees (1978) two decades ago. Gravitational instability of primordial density fluctuations leads to the collapse of dark matter halos. Gas falls into these potential wells, heats up as it does so, then radiates its energy, loses pressure support, contracts, and eventually forms stars. Dissipation thus leads to the formation of dense baryonic cores, which can survive as distinct entities even if their parent dark halos merge.

Relative to the situation in 1978, we now have much better theoretical models for the origin of the primordial fluctuations, with inflation as the leading candidate. There has also been a major change to the theoretical picture, the idea that the dominant mass component is not baryonic dark matter but some form of non-baryonic, cold dark matter (CDM). No existing models that assume purely baryonic matter can account for the cosmic structure seen today while remaining consistent with the observed low amplitude of cosmic microwave background anisotropies. In inflation+CDM models normalized to the COBE observations, the properties of the initial conditions for structure formation are completely determined by a small number of parameters, principally Ωm, Ωb, and ΩΛ (the density parameters of matter, baryons, and vacuum energy), H0, and the inflationary spectral index n (where n = 1 corresponds to scale-invariant fluctuations).

There have also been substantial developments in the “technology” for modeling galaxy formation theoretically. One line of attack, semi-analytic modeling, descends directly from the approach of White & Rees (1978). These methods use extensions of the Press-Schechter (1974) formalism that describe merger histories of collisionless dark matter halos (Bond et al. 1991, Bower 1991, Lacey & Cole 1994). They adopt an idealized description of gas dynamics and cooling within halos and incorporate parametrized models of star formation from cooling gas and reheating by supernova feedback (e.g., White & Frenk 1991, Kauffmann et al. 1993, Cole et al. 1994, Avila-Reese et al. 1998, Somerville & Primack 1999).

A simpler branch of this work focuses on the properties of galaxy disks, with the halo spin parameter playing a critical role (Fall & Efstathiou 1980, Dalcanton et al. 1997, Mo et al. 1998). The semi-analytic approach allows traditional population synthesis and chemical evolution models to be placed in a far more realistic framework of structure formation. There are numerous free parameters to be set either by normalization to observations or by calibration against numerical simulations, but once these parameters are fixed the models can be tested against many independent observations. The strengths of semi-analytic models lie in their ability to make contact with a wide range of observations, their ability to explore a wide range of theoretical parameter space, and their description of galaxy formation in simple and physically intuitive terms. The semi-analytic approach can also be combined with N-body simulations to sidestep some of 2 the approximations used in the halo merger models and, more importantly, to obtain detailed predictions of galaxy clustering (e.g., Governato et al. 1998, Kauffmann et al. 1999ab).

The other main approach to theoretical modeling of galaxy formation is direct numerical simulation, including hydrodynamics and star formation. There are two broad classes of simulations, those that zoom in on individual objects (e.g., Katz & Gunn 1991, Katz 1992, Navarro & White 1994, Vedel et al. 1994, Steinmetz & M¨ller 1994, Navarro & Steinmetz 1997, Dominguez-Tenreiro et u al. 1998, Kaellender & Hultman 1998) and those that simulate larger, random realizations of cosmological volumes (e.g., Cen & Ostriker 1992, Katz et al.

1992, Evrard et al. 1994, Pearce et al. 1999). The main goal of the first class of simulations is to study the formation mechanisms and properties of individual galaxies, while the main goal of the second is to study spatial clustering of the galaxy population and the statistical distributions of galaxies’ stellar, gas, and total masses. There are two main technical approaches, Eulerian grid codes and smoothed particle hydrodynamics (SPH), with Lagrangian grid codes (Gnedin 1995, Pen 1998) having intermediate properties. Eulerian grid codes can often achieve high mass resolution (i.e., have many grid cells and hence small mass per cell), but the uniformity of the fixed grid means that the spatial resolution in computing hydrodynamic forces is usually rather low (though it can be improved by using multiple grids, as in Anninos & Norman [1996]). For example, the recent simulations of Cen & Ostriker (1999) use 200h−1 kpc grid cells and attempt to identify sites and rates of galactic level star formation based on the gas properties averaged over this scale. SPH simulations compute hydrodynamic forces by smoothing over a fixed number of particles and therefore have higher spatial resolution in denser regions. In SPH simulations that include radiative cooling, the gas in well resolved halos almost invariably has a two-phase structure, with dense, cold lumps embedded in a hot, pressure-supported medium. The cold lumps have sizes and masses roughly comparable to the luminous regions of observed galaxies, and they stand out distinctly from the background, so that there is no ambiguity in identifying the “galaxies” in such simulations.

Relative to semi-analytic modeling of galaxy formation, the strength of the numerical simulation approach is its more realistic treatment of gravitational collapse, mergers, and heating and cooling of gas within dark halos. The difference in treatment could be quantitatively important, since the simulations show that gas and galaxies are distributed along filamentary networks that resemble the root system of a tree, making the accretion process very different from the spherically symmetric one envisaged in the semi-analytic calculations. The only free parameters (apart from the physical parameters of the cosmological model being studied) are those related to the treatment of star formation and feedback. Given these parameters, simulations provide straightforward, untunable predictions. However, the simulation approach must contend with the numerical uncertainties caused by finite volume and finite resolution, and computational expense makes it a slow way to explore parameter space.

32. Numerical Simulations of the High-Redshift Galaxy Population

Our group’s approach to hydrodynamic cosmological simulations is described in detail by Katz et al. (1996). We use a cosmological version of the Hernquist & Katz (1989) TreeSPH code; the results shown here are from simulations that use the parallel code developed by Dav´ et al. (1997). In brief, the simulations e use two sets of particles to represent the dark matter and gas components, respectively. Dark matter particles respond only to gravitational forces, which are computed using a hierarchical tree method (Barnes & Hut 1986). Gas particles respond to gravity and to gas dynamical forces computed by SPH. The simulations include heating by shocks, adiabatic compression, and photoionization, and cooling by adiabatic expansion, Compton interaction with the microwave background, and, most importantly, all of the radiative atomic processes that arise in a primordial composition gas.

In simulations without star formation, a fraction of the gas condenses into cold, dense lumps, with typical sizes of one to several kpc, and masses up to a few ×1011 M. Our star formation algorithm is essentially a prescription for turning this cold, dense gas into collisionless stars, returning energy from supernova feedback to the surrounding medium. A gas particle is “eligible” to form stars if it is Jeans unstable, resides in a region of converging flow, and has a physical density exceeding 0.1 hydrogen atoms cm−3. Once a gas particle is eligible to form stars, its star formation rate is given by dρ dρg c ρg =− =, (1) dt dt tg where c = 0.1 is a dimensionless star formation rate parameter, ∗ = 1/3 is the fraction of the particle’s gas mass that will be converted to stellar mass in a single simulation timestep, and the gas flow timescale tg is the maximum of the local gas dynamical time and the local cooling time. Recycled gas and supernova feedback energy are distributed to the particle and its neighbors. Because the surrounding medium is dense and has a short cooling time, the feedback energy is usually radiated away rather quickly, so it has only a modest impact in our simulations.

This description of galactic scale star formation is clearly idealized, but our predictions of galaxy properties are not sensitive to its details. As shown in Katz et al. (1996), changing c by an order of magnitude makes almost no difference to the stellar masses of simulated galaxies. With lower c, gas simply settles to higher density before star formation begins to deplete it, and the dependence of dρ /dt on ρg and tg in equation (1) ensures that star formation cannot get too far out of step with the rate at which gas cools out of the hot halo. One could, however, imagine radically different formulations of galactic scale star formation that would lead to different results, e.g., if cooled gas does not form stars steadily but instead accumulates until an interaction triggers a violent starburst. One could also imagine a picture in which multi-phase substructure in the ISM allows supernova feedback to have a greater impact on the surrounding halo gas.

We have analyzed the clustering of galaxies at z = 2 − 4 in simulations of a variety of CDM models in Katz et al. (1999, hereafter KHW), and we will discuss the star formation properties of the galaxies in those simulations 4 in a forthcoming paper. Here we focus instead on results that cover a wider range of redshifts for a single model: flat cosmology with Ωm = 0.4, ΩΛ = 0.6, Ωb = 0.0473, h = 0.65, n = 0.95, with a COBE-normalized mass fluctuation amplitude σ8 = 0.80 on the scale of 8h−1 Mpc at z = 0. We have been carrying out simulations of this LCDM model with a range of box sizes, particle numbers, and ending redshifts, to address a variety of questions, including the influence of numerical parameters on the predictions of high redshift structure.

Figure 1 presents the result most relevant to those searching for high-z galaxies: the predicted surface density of objects as a function of star formation rate (SFR), from z = 10 to z = 0.5. The SFR can be converted approximately to a UV continuum luminosity using one’s favorite IMF and population synthesis model. For example, the model adopted by Steidel et al. (1996) gives an absolute magnitude MAB = −20.8 at rest-frame wavelength 1500˚ for SFR=10M /yr.

A Each line type in Figure 1 corresponds to a simulation with a different number of particles and/or simulation box size, as indicated by the legends. In general, simulations with higher mass resolution better represent the low end of the luminosity function, while simulations with larger volume better represent the high end. Since limited resolution and limited box size both cause underestimates of galaxy numbers, one can generally take the highest line at a given SFR in a given panel as a lower limit to the predicted surface density of objects. In panels where the curves from different simulations connect up, we can infer that the prediction based on the upper envelope of these curves is reasonably robust to the numerical parameters of the simulations, though of course it may still be sensitive to the cosmological parameters and to the adopted model of star formation.

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