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2.1 The H2 Cooling Regime

Detailed studies of structure formation generally focus on some variant of the cold dark matter (CDM) cosmogony - with a specific choice for OmegaCDM, Omegab and Lambda. Even if such a model turns out to be oversimplified, it offers a useful 'template' whose main features apply generically to any `bottom up' model for structure formation. There is no minimum scale for gravitational aggregation of the CDM. However, the baryonic gas does not `feel' the very smallest clumps, which have very small binding energies: pressure opposes condensation of the gas on scales below a (time dependent) Jeans scale - roughly, the size of a comoving sphere whose boundary expands at the sound speed.

The overdense clumps of CDM within which `first light' occurs must provide a deep enough potential well to pull the gas into them . But they must also - a somewhat more stringent requirement - yield, after virialisation, a gas temperature such that radiative cooling is efficient enough to allow the gas to contract further. The dominant coolant for gas of primordial composition is molecular hydrogen. This has been considered by many authors, from the 1960s onwards; see recent discussions by, for instance, Tegmark et al. (1997), Haiman, Rees and Loeb (1996), Haiman, Abel and Rees (1999). In a uniformly expanding universe, only about 10-6 of the post-recombination hydrogen is in the form of H2. However this rises to 10-4 within collapsing regions - high enough to permit cooling at temperatures above a few hundred degrees.

So the first `action' would have occurred within clumps with virial temperatures of a few hundred degrees (corresponding to a virial velocity of 2-3 km/s). Their total mass is of order 105 Msun ; the baryonic mass is smaller by a factor Omegab / OmegaCDM.

The gas falling into such a clump exhibits filamentary substructure: the contraction is almost isothermal, so the Jeans mass decreases as the density rises. Abel, Bryan and Norman (1999) have simulated the collapse, taking account of radiative transfer in the molecular lines, up to 1012 times the turnaround density; by that stage the Jeans mass (and the size of the smallest bound subclumps) has dropped to 50-100 Msun.

There is still a large gap to be bridged between the endpoint of these impressive simulations and the formation of `protostars'. Fragmentation could continue down to smaller masses; on the other hand, there could be no further fragmentation - indeed, as Bromm, Coppi and Larson (1999) argue, infall onto the largest blobs could lead to masses much higher than 100 Msun.

And when even one star has formed, further uncertainties ensue. Radiation or winds may expel uncondensed material from the shallow potential wells, and exert the kind of feedback familiar from studies of giant molecular clouds in our own Galaxy. In addition to this local feedback, there is a non-local effect due to UV radiation. Photons of hnu > 11.18 eV can photodissociate H2, as first calculated by Stecher and Williams (1967). These photons, softer than the Lyman limit, can penetrate a high column density of HI and destroy molecules in virialised and collapsing clouds. H2 cooling would be quenched if there were a UV background able to dissociate the molecules as fast as they form. The effects within clouds have been calculated by Haiman, Abel and Rees (1999) and Ciardi, Ferrara and Abel (1999).

(If the radiation from the first objects had a non-thermal component extending up to KeV energies, as it might if a contribution came from accreting compact objects or supernovae, then there is a counterbalancing positive feedback. X-ray photons penetrate HI, producing photoelectrons (which themselves cause further collisional ionizationwhile being slowed down and thermalised); these electrons then catalyse further H2 formation via H-).

It seems most likely that the negative feedback due to photoionization is dominant. When the UV background gets above a certain threshold, H2 is prevented from forming and molecular cooling is suppressed. Under all plausible assumptions about UV spectral shape, etc, this threshold is reached well before there has been enough UV production to ionize most of the medium. Therefore, only a small fraction of the UV that ionized the IGM can have been produced in systems where star formation was triggered by molecular cooling.

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