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3.1. Cosmological Jeans Mass

The Jeans length lambdaJ was originally defined (Jeans 1928) in Newtonian gravity as the critical wavelength that separates oscillatory and exponentially-growing density perturbations in an infinite, uniform, and stationary distribution of gas. On scales ell smaller than lambdaJ, the sound crossing time, ell / cs is shorter than the gravitational free-fall time, (Grho)-1/2, allowing the build-up of a pressure force that counteracts gravity. On larger scales, the pressure gradient force is too slow to react to a build-up of the attractive gravitational force. The Jeans mass is defined as the mass within a sphere of radius lambdaJ / 2, MJ = (4pi / 3) rho(lambdaJ / 2)3. In a perturbation with a mass greater than MJ, the self-gravity cannot be supported by the pressure gradient, and so the gas is unstable to gravitational collapse. The Newtonian derivation of the Jeans instability suffers from a conceptual inconsistency, as the unperturbed gravitational force of the uniform background must induce bulk motions (compare Binney & Tremaine 1987). However, this inconsistency is remedied when the analysis is done in an expanding universe.

The perturbative derivation of the Jeans instability criterion can be carried out in a cosmological setting by considering a sinusoidal perturbation superposed on a uniformly expanding background. Here, as in the Newtonian limit, there is a critical wavelength lambdaJ that separates oscillatory and growing modes. Although the expansion of the background slows down the exponential growth of the amplitude to a power-law growth, the fundamental concept of a minimum mass that can collapse at any given time remains the same (see, e.g. Kolb & Turner 1990; Peebles 1993).

We consider a mixture of dark matter and baryons with density parameters Omegadmz = rhobardm / rhoc and Omegabz = rhobarb / rhoc, where rhobardm is the average dark matter density, rhobarb is the average baryonic density, rhoc is the critical density, and Omegadmz + Omegabz = Omegamz is given by equation (23). We also assume spatial fluctuations in the gas and dark matter densities with the form of a single spherical Fourier mode on a scale much smaller than the horizon,

Equation 33   (33)
Equation 34   (34)

where rhobardm(t) and rhobarb(t) are the background densities of the dark matter and baryons, deltadm(t) and deltab(t) are the dark matter and baryon overdensity amplitudes, r is the comoving radial coordinate, and k is the comoving perturbation wavenumber. We adopt an ideal gas equation-of-state for the baryons with a specific heat ratio gamma = 5/3. Initially, at time t = ti, the gas temperature is uniform Tb(r, ti) = Ti, and the perturbation amplitudes are small deltadm,i, deltab,i << 1. We define the region inside the first zero of sin(kr) / (kr), namely 0 < kr < pi, as the collapsing ``object''.

The evolution of the temperature of the baryons Tb(r, t) in the linear regime is determined by the coupling of their free electrons to the Cosmic Microwave Background (CMB) through Compton scattering, and by the adiabatic expansion of the gas. Hence, Tb(r, t) is generally somewhere between the CMB temperature, Tgamma propto (1 + z)-1 and the adiabatically-scaled temperature Tad propto (1 + z)-2. In the limit of tight coupling to Tgamma, the gas temperature remains uniform. On the other hand, in the adiabatic limit, the temperature develops a gradient according to the relation

Equation 35   (35)

The evolution of dark matter overdensity, deltadm(t), in the linear regime is described by the equation (see Section 9.3.2 of Kolb & Turner 1990),

Equation 36   (36)

whereas the evolution of the overdensity of the baryons, deltab(t), is described by

Equation 37   (37)

Here, H(t) = adot / a is the Hubble parameter at a cosmological time t, and µ = 1.22 is the mean molecular weight of the neutral primordial gas in atomic units. The parameter beta distinguishes between the two limits for the evolution of the gas temperature. In the adiabatic limit beta = 1, and when the baryon temperature is uniform and locked to the background radiation, beta = 0. The last term on the right hand side (in square brackets) takes into account the extra pressure gradient force in del(rhobT) = (T del rhob + rhob delT), arising from the temperature gradient which develops in the adiabatic limit. The Jeans wavelength lambdaJ = 2pi / kJ is obtained by setting the right-hand side of equation (37) to zero, and solving for the critical wavenumber kJ. As can be seen from equation (37), the critical wavelength lambdaJ (and therefore the mass MJ) is in general time-dependent. We infer from equation (37) that as time proceeds, perturbations with increasingly smaller initial wavelengths stop oscillating and start to grow.

To estimate the Jeans wavelength, we equate the right-hand-side of equation (37) to zero. We further approximate deltab ~ deltadm, and consider sufficiently high redshifts at which the universe is matter-dominated and flat (equations (9) and (10) in Section 2.1). We also assume Omegab << Omegam, where Omegam = Omegadm + Omegab is the total matter density parameter. Following cosmological recombination at z approx 103, the residual ionization of the cosmic gas keeps its temperature locked to the CMB temperature (via Compton scattering) down to a redshift of (p. 179 of Peebles 1993)

Equation 38   (38)

In the redshift range between recombination and zt, beta = 0 and

Equation 39   (39)

so that the Jeans mass is therefore redshift independent and obtains the value (for the total mass of baryons and dark matter)

Equation 40   (40)

Based on the similarity of MJ to the mass of a globular cluster, Peebles & Dicke (1968) suggested that globular clusters form as the first generation of baryonic objects shortly after cosmological recombination. Peebles & Dicke assumed a baryonic universe, with a nonlinear fluctuation amplitude on small scales at z ~ 103, a model which has by now been ruled out. The lack of a dominant mass of dark matter inside globular clusters (Moore 1996; Heggie & Hut 1996) makes it unlikely that they formed through direct cosmological collapse, and more likely that they resulted from fragmentation during the process of galaxy formation. Furthermore, globular clusters have been observed to form in galaxy mergers (e.g., Miller et al. 1997).

At z ltapprox zt, the gas temperature declines adiabatically as [(1 + z) / (1 + zt)]2 (i.e., beta = 1) and the total Jeans mass obtains the value,

Equation 41   (41)

Note that we have neglected Compton drag, i.e., the radiation force which suppresses gravitational growth of structure in the baryon fluid as long as the electron abundance is sufficiently high to keep the baryons dynamically coupled to the photons. After cosmological recombination, the net friction force on the predominantly neutral fluid decreases dramatically, allowing the baryons to fall into dark matter potential wells, and essentially erasing the memory of Compton drag by z ~ 100 (e.g., Section 5.3.1. of Hu 1995).

It is not clear how the value of the Jeans mass derived above relates to the mass of collapsed, bound objects. The above analysis is perturbative (Eqs. [36] and [37] are valid only as long as deltab and deltadm are much smaller than unity), and thus can only describe the initial phase of the collapse. As deltab and deltadm grow and become larger than unity, the density profiles start to evolve and dark matter shells may cross baryonic shells (Haiman, Thoul, & Loeb 1996) due to their different dynamics. Hence the amount of mass enclosed within a given baryonic shell may increase with time, until eventually the dark matter pulls the baryons with it and causes their collapse even for objects below the Jeans mass.

Even within linear theory, the Jeans mass is related only to the evolution of perturbations at a given time. When the Jeans mass itself varies with time, the overall suppression of the growth of perturbations depends on a time-averaged Jeans mass. Gnedin & Hui (1998) showed that the correct time-averaged mass is the filtering mass MF = (4pi / 3) rhobar (2pi a / kF)3, in terms of the comoving wavenumber kF associated with the ``filtering scale''. The wavenumber kF is related to the Jeans wavenumber kJ by

Equation 42   (42)

where D(t) is the linear growth factor (Section 2.2). At high redshift (where Omegamz -> 1), this relation simplifies to (Gnedin 2000b)

Equation 43   (43)

Then the relationship between the linear overdensity of the dark matter deltadm and the linear overdensity of the baryons deltab, in the limit of small k, can be written as (Gnedin & Hui 1998)

Equation 44   (44)

Linear theory specifies whether an initial perturbation, characterized by the parameters k, deltadm,i, deltab,i and ti, begins to grow. To determine the minimum mass of nonlinear baryonic objects resulting from the shell-crossing and virialization of the dark matter, we must use a different model which examines the response of the gas to the gravitational potential of a virialized dark matter halo.

3.2. Response of Baryons to Nonlinear Dark Matter Potentials

The dark matter is assumed to be cold and to dominate gravity, and so its collapse and virialization proceeds unimpeded by pressure effects. In order to estimate the minimum mass of baryonic objects, we must go beyond linear perturbation theory and examine the baryonic mass that can accrete into the final gravitational potential well of the dark matter.

For this purpose, we assume that the dark matter had already virialized and produced a gravitational potential phi(r) at a redshift zvir (with phi -> 0 at large distances, and phi < 0 inside the object) and calculate the resulting overdensity in the gas distribution, ignoring cooling (an assumption justified by spherical collapse simulations which indicate that cooling becomes important only after virialization; see Haiman, Thoul, & Loeb 1996).

After the gas settles into the dark matter potential well, it satisfies the hydrostatic equilibrium equation,

Equation 45   (45)

where pb and rhob are the pressure and mass density of the gas. At z ltapprox 100 the gas temperature is decoupled from the CMB, and its pressure evolves adiabatically (ignoring atomic or molecular cooling),

Equation 46   (46)

where a bar denotes the background conditions. We substitute equation (46) into (45) and get the solution,

Equation 47   (47)

where Tbar = pbarb µmp / (kB rhobarb) is the background gas temperature. If we define Tvir = - 1/3 µmp phi / kB as the virial temperature for a potential depth - phi, then the overdensity of the baryons at the virialization redshift is

Equation 48   (48)

This solution is approximate for two reasons: (i) we assumed that the gas is stationary throughout the entire region and ignored the transitions to infall and the Hubble expansion at the interface between the collapsed object and the background intergalactic medium (henceforth IGM), and (ii) we ignored entropy production at the virialization shock surrounding the object. Nevertheless, the result should provide a better estimate for the minimum mass of collapsed baryonic objects than the Jeans mass does, since it incorporates the nonlinear potential of the dark matter.

We may define the threshold for the collapse of baryons by the criterion that their mean overdensity, deltab, exceeds a value of 100, amounting to gtapprox 50% of the baryons that would assemble in the absence of gas pressure, according to the spherical top-hat collapse model (Section 2.3). Equation (48) then implies that Tvir > 17.2 Tbar.

As mentioned before, the gas temperature evolves at z ltapprox 160 according to the relation Tbar approx 170[(1 + z) / 100]2 K. This implies that baryons are overdense by deltab > 100 only inside halos with a virial temperature Tvir gtapprox 2.9 × 103 [(1 + z)/100]2 K. Based on the top-hat model (Section 2.3), this implies a minimum halo mass for baryonic objects of

Equation 49   (49)

where we set µ = 1.22 and consider sufficiently high redshifts so that Omegamz approx 1. This minimum mass is coincidentally almost identical to the naive Jeans mass calculation of linear theory in equation (41) despite the fact that it incorporates shell crossing by the dark matter, which is not accounted for by linear theory. Unlike the Jeans mass, the minimum mass depends on the choice for an overdensity threshold [taken arbitrarily as deltab > 100 in equation (49)]. To estimate the minimum halo mass which produces any significant accretion we set, e.g., deltab = 5, and get a mass which is lower than Mmin by a factor of 27.

Of course, once the first stars and quasars form they heat the surrounding IGM by either outflows or radiation. As a result, the Jeans mass which is relevant for the formation of new objects changes (Ostriker & Gnedin 1996; Gnedin 2000a). The most dramatic change occurs when the IGM is photo-ionized and is consequently heated to a temperature of ~ (1-2) × 104 K. As we discuss in Section 6.5, this heating episode had a dramatic impact on galaxy formation.

3.3. Molecular Chemistry, Photo-Dissociation, and Cooling

Before metals are produced, the primary molecule which acquires sufficient abundance to affect the thermal state of the pristine cosmic gas is molecular hydrogen, H2. The dominant H2 formation process is

Equation 50   (50)
Equation 51   (51)

where free electrons act as catalysts. The complete set of chemical reactions leading to the formation of H2 is summarized in Table 1, together with the associated rate coefficients (see also Haiman, Thoul, & Loeb 1996; Abel et al. 1997; Galli & Palla 1998; and the review by Abel & Haiman 2000). Table 2 shows the same for deuterium mediated reactions. Due to the low gas density, the chemical reactions are slow and the molecular abundance is far from its value in chemical equilibrium. After cosmological recombination the fractional H2 abundance is small, ~ 10-6 relative to hydrogen by number (Lepp & Shull 1984; Shapiro, Giroux & Babul 1994). At redshifts z << 100, the gas temperature in most regions is too low for collisional ionization to be effective, and free electrons (over and above the residual electron fraction) are mostly produced through photoionization of neutral hydrogen by UV or X-ray radiation.

Table 1. Reaction Rates for Hydrogen Species
Table 1
References. - (1) Haiman, Thoul, & Loeb 1996; (2) Abel, et al. 1997.

Table 2. Reaction Rates for Deuterium Species
Table 2
References. - (1) Haiman, Thoul, & Loeb 1996; (3) Gali & Palla 1998.

In objects with baryonic masses gtapprox 3 × 104 Msun, gravity dominates and results in the bottom-up hierarchy of structure formation characteristic of CDM cosmologies; at lower masses, gas pressure delays the collapse. The first objects to collapse are those at the mass scale that separates these two regimes. Such objects reach virial temperatures of several hundred degrees and can fragment into stars only through cooling by molecular hydrogen (e.g., Abel 1995; Tegmark et al. 1997). In other words, there are two independent minimum mass thresholds for star formation: the Jeans mass (related to accretion) and the cooling mass. For the very first objects, the cooling threshold is somewhat higher and sets a lower limit on the halo mass of ~ 5 × 104 Msun at z ~ 20.

However, molecular hydrogen (H2) is fragile and can easily be photo-dissociated by photons with energies of 11.26-13.6eV, to which the IGM is transparent even before it is ionized. The photo-dissociation occurs through a two-step process, first suggested by Solomon in 1965 (compare Field et al. 1966) and later analyzed quantitatively by Stecher & Williams (1967). Haiman, Rees, & Loeb (1997) evaluated the average cross-section for this process between 11.26eV and 13.6eV, by summing the oscillator strengths for the Lyman and Werner bands of H2, and obtained a value of 3.71 × 10-18 cm2. They showed that the UV flux capable of dissociating H2 throughout the collapsed environments in the universe is lower by more than two orders of magnitude than the minimum flux necessary to ionize the universe. The inevitable conclusion is that soon after trace amounts of stars form, the formation of additional stars due to H2 cooling is suppressed. Further fragmentation is possible only through atomic line cooling, which is effective in objects with much higher virial temperatures, Tvir gtapprox 104K. Such objects correspond to a total mass gtapprox 108 Msun[(1 + z) / 10]-3/2. Figure 4 illustrates this sequence of events by describing two classes of objects: those with Tvir < 104K (small dots) and those with Tvir > 104K (large dots). In the first stage (top panel), some low-mass objects collapse, form stars, and create ionized hydrogen (H II) bubbles around them. Once the UV background between 11.2-13.6eV reaches a specific critical level, H2 is photo-dissociated throughout the universe and the formation of new stars is delayed until objects with Tvir gtapprox 104K collapse (Haiman, Abel, & Rees 2000; Ciardi, Ferrara, & Abel 2000; Ciardi et al. 2000). Machacek, Bryan & Abel (2000) have confirmed that the soft UV background can delay the cooling and collapse of low-mass halos (~ 106 Msun) based on analytical arguments and three-dimensional hydrodynamic simulations; they also determined the halo mass threshold for collapse for a range of UV fluxes. Nishi & Omukai (1999; see also Silk 1977) have argued that the photo-dissociation of H2 could be even more effective due to a small number of stars embedded within the gas clouds themselves.

When considering the photo-dissociation of H2 before reionization, it is important to incorporate the processed spectrum of the UV background at photon energies below the Lyman limit. Due to the absorption at the Lyman-series resonances this spectrum obtains the sawtooth shape shown in Figure 11. For any photon energy above Lyalpha at a particular redshift, there is a limited redshift interval beyond which no contribution from sources is possible because the corresponding photons are absorbed through one of the Lyman-series resonances along the way. Consider, for example, an energy of 11 eV at an observed redshift z = 10. Photons received at this energy would have to be emitted at the 12.1 eV Lybeta line from z = 11.1. Thus, sources in the redshift interval 10-11.1 could be seen at 11 eV, but radiation emitted by sources at z > 11.1 eV would have passed through the 12.1 eV energy at some intermediate redshift, and would have been absorbed. Thus, an observer viewing the universe at any photon energy above Lyalpha would see sources only out to some horizon, and the size of that horizon would depend on the photon energy. The number of contributing sources, and hence the total background flux at each photon energy, would depend on how far this energy is from the nearest Lyman resonance. Most of the photons absorbed along the way would be re-emitted at Lyalpha and then redshifted to lower energies. The result is a sawtooth spectrum for the UV background before reionization, with an enhancement below the Lyalpha energy (see Haiman et al. 1997 for more details). Unfortunately, the direct detection of the redshifted sawtooth spectrum as a remnant of the reionization epoch is not feasible due to the much higher flux contributed by foreground sources at later cosmic times.

Figure 11

Figure 11. The average spectrum during the initial phase of the reionization epoch. The upper panel shows that absorption by neutral hydrogen and helium suppresses the flux above 13.6eV up to the keV range. The lower panel shows a close-up of the sawtooth modulation due to line absorption below 13.6 eV. A constant comoving density of sources was assumed, with each source emitting a power-law continuum, which would result in the spectrum shown by the dashed lines if absorption were not taken into account.

The radiative feedback on H2 need not be only negative, however. In the dense interiors of gas clouds, the formation rate of H2 could be accelerated through the production of free electrons by X-rays. This effect could counteract the destructive role of H2 photo-dissociation (
Haiman, Rees, & Loeb 1996). Haiman, Abel, & Rees (2000) have shown that if a significant ( gtapprox 10%) fraction of the early UV background is produced by massive black holes (mini-quasars) with hard spectra extending to photon energies ~ 1 keV, then the X-rays will catalyze H2 production and the net radiative feedback will be positive, allowing low mass objects to fragment into stars. These objects may greatly alter the topology of reionization (Section 6.3). However, if such quasars do not exist or if low mass objects are disrupted by supernova-driven winds (see Section 7.2), then most of the stars will form inside objects with virial temperatures gtapprox 104K, where atomic cooling dominates. Figure 12 and Table 3 summarize the cooling rates as a function of gas temperature in high-redshift, metal-free objects.

Figure 12

Figure 12. Cooling rates as a function of temperature for a primordial gas composed of atomic hydrogen and helium, as well as molecular hydrogen, in the absence of any external radiation. We assume a hydrogen number density nH = 0.045 cm-3, corresponding to the mean density of virialized halos at z = 10. The plotted quantity Lambda / nH2 is roughly independent of density (unless nH >> 10 cm-3), where Lambda is the volume cooling rate (in erg/sec/cm3). The solid line shows the cooling curve for an atomic gas, with the characteristic peaks due to collisional excitation of HI and HeII. The dashed line (calculated using the code of Abel available at shows the additional contribution of molecular cooling, assuming a molecular abundance equal to 0.1% of nH.

Table 3. Radiative Cooling Processes in the Primordial Gas
Table 3
Note. - T is the gas temperature in K, T3 = T / 103 K, T5 = T / 105 K, T6 = T / 106 K, ne is the density of free electrons, z is the redshift, and TCMB = 2.73 (1 + z) K is the temperature of the CMB.
References. - (1) Galli & Palla 1998; (2) Flower, Le Bourlot, Pineau des Forêts, & Roueff 2000; (3) Cen 1992; Verner & Ferland 1996; Ferland et al. 1992; Voronov 1997 (4) Ikeuchi & Ostriker 1986.

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