Figure 3.14. The variance of the matter-density field versus mass M, for several different cosmologies, all based on WMAP results. Solid curve: 3-year-WMAP-only best-fit model. Dotted curve: AWDM with a particle mass mX = 2 keV, otherwise the same as before. Dashed curve: 3-year-WMAP best-fit running spectral index model. Dash-dotted curve: 1-year-WMAP-only best-fit tilted model. Here n refers to the spectral index at k = 0.05 Mpc-1. The horizontal line at the top of the figure shows the value of the extrapolated critical collapse over-density Sc(z) at the reionization redshift z = 11.

as a function of wavenumber k (Spergel et al. 2003). In the 3-year WMAP data such a "running spectral index" leads to a marginal improvement in the fit. Models with either n < 1 or dn/dln k < 0 predict a significantly lower amplitude of fluctuations on small scales than does standard ACDM. The suppression of small-scale power has the advantage of reducing the amount of substructure in galactic halos and makes small halos form later (when the Universe was less dense) (Zentner & Bullock 2002).

Figure 3.14 shows the linearly extrapolated (to z = 0) variance of the mass-density field for a range of cosmological parameters. Note that the new WMAP results prefer a low value for a8, the rms mass fluctuation in an 8^-1-Mpc sphere. This is consistent with a normalization by the z = 0 X-ray-cluster abundance (Reiprich & Bohringer 2002). For comparison we have also included a model with a higher normalization, a8 = 0.9, from the best-fit model to 1-year WMAP data. In the CDM paradigm structure formation proceeds bottom-up, so it then follows that the loss of small-scale power modifies structure formation most severely at the highest redshifts, significantly reducing the number of self-gravitating objects then. This, of course, will make it more difficult to reionize the Universe early enough.

It has been argued, for example, that one popular modification of the CDM paradigm, warm dark matter (WDM), has so little structure at high redshift that it is unable to explain the WMAP observations of an early epoch of reionization (Barkana et al. 2001). Yet the WMAP running-index model may suffer from a similar problem. A look at Figure 3.14 shows that 106M0 halos will collapse at z =11 from 2.4a fluctuations in a tilted ACDM model with n = 0.951 and a8 = 0.74 (best-fit 3-year WMAP model), but from much rarer 3.7a and 3.6a fluctuations in the WDM and running-index models, respectively. The problem is that scenarios with increasingly rarer halos at early

Figure 3.15. The mass fraction in all collapsed halos above the cosmological filtering (Jeans) mass as a function of redshift, for various power spectra. Curves are the same as in Figure 3.14. Left panel: filtering mass computed in the absence of reionization. Right panel: filtering mass computed assuming that the Universe is reionized and reheated to Te = 104 K by UV radiation at ^ ~ 11.

Figure 3.15. The mass fraction in all collapsed halos above the cosmological filtering (Jeans) mass as a function of redshift, for various power spectra. Curves are the same as in Figure 3.14. Left panel: filtering mass computed in the absence of reionization. Right panel: filtering mass computed assuming that the Universe is reionized and reheated to Te = 104 K by UV radiation at ^ ~ 11.

times require even-more-extreme assumptions (i.e. higher star-formation efficiencies and UV-photon-production rates) in order to be able to reionize the Universe suitably early (e.g. Somerville et al. 2003; Wyithe & Loeb 2003; Ciardi et al. 2003; Cen 2003). Figure 3.15 depicts the mass fraction in all collapsed halos with masses above the cosmological filtering mass for a case without reionization and one with reionization occurring at z ~ 11. At early epochs this quantity appears to vary by orders of magnitude among models.

The study of the non-linear regime for the baryons is far more complicated than that of the dark matter because of the need to take into account pressure gradients and radiative processes. As a dark-matter halo grows and virializes above the cosmological Jeans mass through merging and accretion, baryonic material will be shock-heated to the effective virial temperature of the host and compressed to the same fractional over-density as the dark matter. The subsequent behavior of gas in a dark-matter halo depends on the efficiency with which it can cool. It is useful here to identify two mass scales for the host halos: (1) a molecular cooling mass MH2 above which gas can cool via roto-vibrational levels of H2 and contract, Mh2 ~ 105[(1 + z)/10]~3/2M0 (virial temperature above 200K); and (2) an atomic cooling mass MH above which gas can cool efficiently and fragment via excitation of hydrogen Lya, MH « 108[(1 + z)/10]~3/2M0 (virial temperature above 104 K). Figure 3.16 shows the cooling mechanisms at various temperatures for primordial gas. Figure 3.17 shows the fraction of the total mass in the Universe that is in collapsed dark-matter halos with masses greater than MH2 and MH at various epochs.

High-resolution hydrodynamics simulations of early structure formation are a powerful tool with which to track in detail the thermal and ionization history of a clumpy IGM and guide studies of primordial star formation and reheating. Such simulations performed in the context of ACDM cosmologies have shown that the first stars (the so-called "Population III") in the Universe formed out of metal-free gas in dark-matter minihalos of mass above a few times 105M0 (Abel et al. 2000; Fuller & Couchman 2000; Yoshida et al. 2003; Reed et al. 2005) condensing from the rare high-^c peaks of the primordial density

Figure 3.16. Cooling rates for Bremsstrahlung (dotted line), H (dashed line), and He (dash-dotted line) line cooling, and H2 (circles) cooling. The e-, H II, He ii, and He iii abundances were computed assuming collisional equilibrium, and the H2 fractional abundance was fixed at 3 x 10-4, which is typical for early objects. At temperatures between 100 and 10000K, the H2 molecule is the most-effective coolant. From Fuller & Couchman (2000).

Figure 3.16. Cooling rates for Bremsstrahlung (dotted line), H (dashed line), and He (dash-dotted line) line cooling, and H2 (circles) cooling. The e-, H II, He ii, and He iii abundances were computed assuming collisional equilibrium, and the H2 fractional abundance was fixed at 3 x 10-4, which is typical for early objects. At temperatures between 100 and 10000K, the H2 molecule is the most-effective coolant. From Fuller & Couchman (2000).

fluctuation field at z > 20, and were likely very massive (e.g. Abel et al. 2002; Bromm et al. 2002); see Bromm & Larson (2004) and Ciardi & Ferrara (2005) for recent reviews. In Kuhlen & Madau (2005) we used a modified version of ENZO, an adaptive-mesh-refinement (AMR), grid-based hybrid (hydrodynamic plus A-body) code developed by Bryan & Norman (see http://cosmos.ucsd.edu/enzo/) to solve the cosmological hydrodynamics equations and study the cooling and collapse of primordial gas in the first baryonic structures. The simulation samples the dark-matter density field in a 0.5-Mpc box with a mass resolution of 2000M0 to ensure that halos above the cosmological Jeans

Figure 3.18. A three-dimensional volume rendering of the IGM in the inner 0.5-Mpc simulated box at z = 21 (left panel) and z = 15.5 (right panel). Only gas with over-density 4 < < 10 is shown: the locations of dark-matter minihalos are marked by spheres with sizes proportional to halo mass. At these epochs, the halo-finder algorithm identifies 55 (z = 21) and 262 (z = 15.5) bound clumps within the simulated volume. From Kuhlen & Madau (2005).

Figure 3.18. A three-dimensional volume rendering of the IGM in the inner 0.5-Mpc simulated box at z = 21 (left panel) and z = 15.5 (right panel). Only gas with over-density 4 < < 10 is shown: the locations of dark-matter minihalos are marked by spheres with sizes proportional to halo mass. At these epochs, the halo-finder algorithm identifies 55 (z = 21) and 262 (z = 15.5) bound clumps within the simulated volume. From Kuhlen & Madau (2005).

mass are well resolved at all redshifts z < 20. The AMR technique allows us to home in, with progressively finer resolution, on the densest parts of the "cosmic web." During the evolution from z = 99 to z = 15, refined (child) grids with twice the spatial resolution of the coarser (parent) grid are introduced when a cell reaches a dark-matter over-density (baryonic overdensity) of 2.0 (4.0). Dense regions are allowed to dynamically refine to a maximum resolution of 30 pc (comoving). The code evolves the non-equilibrium rate equations for nine species (H, H+, H-, e, He, He+, He++, H2, and H+) in primordial self-gravitating gas, including radiative losses from atomic- and molecular-line cooling, and Compton cooling by the cosmic background radiation.

The clustered structure around the most-massive peaks of the density field is clearly seen in Figure 3.18, a three-dimensional volume rendering of the simulated volume at redshifts 21 and 15.5. This figure shows gas at 4 < < 10, with the locations of dark-matter minihalos marked by spheres colored and sized according to their mass (the spheres are only markers; the actual shape of the halos is typically non-spherical). Several interleaving filaments are visible, at the intersections of which minihalos are typically found. At z = 21, 55 bound halos are identified in the simulated volume: by z = 17.5 this number has grown to 149, and by z = 15.5 to 262 halos. At this epoch, only four halos have reached the critical virial temperature for atomic cooling, Tvir = 104 K.

The primordial fractional abundance of H2 in the low-density IGM is small, xH2 = [H2/H] ^ 2 x 10-6, since at z > 100 H2 formation is inhibited because the required intermediaries, either H+ or H-, are destroyed by CMB photons. Most of the gas in the simulation therefore cools by adiabatic expansion. Within collapsing minihalos, however, gas densities and temperatures are large enough that H2 formation is catalyzed by H- ions through the associative detachment reaction H + H- —► H2 + e-, and the molecular fraction increases at the rate dxH2/dt <x xenH TVi'88, where xe is the number of electrons per hydrogen atom. For Ivir less than about a few thousand kelvins the virializa-tion shock is not ionizing, the free electrons left over from recombination are depleted in the denser regions, and the production of H2 stalls at a temperature-dependent

Figure 3.19. Left: the fraction of cold and cold plus dense gas within the virial radius of all halos identified at z = 17.5 with Tvir > 400 K, as a function of halo mass. Triangles: fc, the fraction of halo gas with T < 0.5 Tvir and over-density 5b > 1000 that cooled via roto-vibrational transitions of H2. Diamonds: the fcd, fraction of gas with T < 0.5Tvir and p > 1019M0 Mpc~3 that is available for star formation. The straight lines represent mean regression analyses of fc and fcd with the logarithm of halo mass. Right: the mass-weighted mean H2 fraction as a function of virial temperature for all halos at z = 17.5 with Tvir > 400 K and fcd < 0.1 (empty circles) or fcd > 0.1 (filled circles). The straight line marks the scaling of the temperature-dependent asymptotic molecular fraction.

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