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Astron. Astrophys. 347, 1-20 (1999)

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6. Evolution of the IGM

We now turn to the IGM itself. We model the universe at a given redshift z as a uniform medium (the IGM), characterized by a density contrast [FORMULA], a gas temperature [FORMULA] and a background radiation field [FORMULA], which contains some mass condensations recognized as individual objects identified with galaxies or Lyman-[FORMULA] clouds as described above.

Since the gas in the IGM has non-zero temperature [FORMULA], baryonic density fluctuations are erased over scales of order [FORMULA] within shallow potential wells with a virial temperature [FORMULA] or within "voids", with:

[EQUATION]

where [FORMULA] is the sound speed, [FORMULA] the age of the universe, [FORMULA] the proton mass and [FORMULA]. Indeed, the pressure dominates over gravitation for objects such that [FORMULA]. Note that the damping scale [FORMULA] is different from the Jeans scale:

[EQUATION]

Both scales are equal (up to a normalization factor of order unity) if the dark matter density is equal to the mean universe density: [FORMULA]. However, we shall consider underdense regions where [FORMULA] can be as low as [FORMULA], see Fig. 13 below. Indeed, as an increasingly large proportion of the matter content of the universe gets embedded within collapsed objects as time goes on the density of the IGM (the volume between these mass condensations) becomes much smaller than the mean universe density. In this case where [FORMULA] we have [FORMULA]. We use [FORMULA] because of the finite age of the universe: the medium cannot be homogenized over scales larger than those reached by acoustic waves over the time [FORMULA] (the scale [FORMULA] corresponds to the limit of large times). Note that for Lyman-[FORMULA] clouds we also use [FORMULA] as the characteristic scale, with [FORMULA] K, since we consider regions with very low or moderate densities [FORMULA], see Sect. 5 and Valageas et al. (1999a). Then, the density contrast of the IGM is given by:

[EQUATION]

This simply states that at high z (when [FORMULA]) we have [FORMULA] (i.e. the universe is almost exactly a uniform medium on scale [FORMULA]) while at low z we have [FORMULA] since most of the matter is now within overdense bound collapsed objects (clusters, filaments etc.) while most of the volume (which we call the IGM) is formed by underdense regions.

Since the mean density of the universe is [FORMULA] we define a baryonic clumping factor [FORMULA] by:

[EQUATION]

where we used the fact that the volume fraction [FORMULA] occupied by the IGM is very close to unity. Here [FORMULA] and [FORMULA] are the fractions of mass formed by Lyman-[FORMULA] forest clouds (with a density contrast lower than [FORMULA]) and by virialized objects. Note that [FORMULA] somewhat underestimates the actual clumping of the gas since we did not take into account the collapse of baryons due to cooling nor the slope of the density profile within virialized halos. However, these latter characteristics are included in our model for Lyman-[FORMULA] clouds. We also define the mean density due to objects which do not cool as:

[EQUATION]

Before reionization this corresponds to the density field of neutral hydrogen since galactic halos (i.e. massive potential wells with [FORMULA] which can cool) ionize most of their gas because of the radiation emitted by their stars or their central quasar. We obtain the mean square density in a similar fashion:

[EQUATION]

and the corresponding clumping factor is simply:

[EQUATION]

The quantities [FORMULA] and [FORMULA] characterize the density fluctuations of neutral hydrogen within the IGM. Note that most of the volume is occupied by regions which satisfy [FORMULA].

The gas which is within the IGM is heated by the UV background radiation while it cools due to the expansion of the universe and to various radiative cooling processes. Note that we neglect here the possible heating of the IGM by supernovae. However supernova feedback is included in our model for galaxy formation: we simply assume it only affects the immediate neighbourhood of these galaxies (see also McLow & Ferrara 1998). Thus, we write for the evolution of the temperature of the IGM:

[EQUATION]

where [FORMULA] is the scale factor (which enters the term describing adiabatic cooling due to the expansion). The heating time-scale [FORMULA] is given by:

[EQUATION]

where [FORMULA] (HI,HeI,HeII), [FORMULA] is the ionization threshold of the corresponding species, [FORMULA] its number density in the IGM and [FORMULA] the baryon number density. The cooling time-scale [FORMULA] describes collisional excitation, collisional ionization, recombination, molecular hydrogen cooling, bremsstrahlung and Compton cooling or heating (e.g. Anninos et al.1997). Next, we can write the evolution equation for the background radiation field [FORMULA]:

[EQUATION]

The first two terms on the r.h.s. describe the effects of the expansion of the universe, while the last two terms represent the radiation emitted by stars and quasars which we obtained previously. The absorption coefficient [FORMULA] is written as:

[EQUATION]

where [FORMULA] is the opacity at frequency [FORMULA] of the IGM over a physical length of 1 Mpc, while [FORMULA] corresponds to the contribution by "Lyman-[FORMULA]" clouds (i.e. discrete mass condensations as opposed to the uniform component which forms the IGM). Thus we have:

[EQUATION]

Note that in this study we consider the medium as purely absorbing and we neglected the reprocessing of ionizing photons. From the evolution of the IGM temperature and the background radiation field we can also follow the chemistry of the gas within this uniform component. More precisely we consider the following species: HI, HII, H-, H2, H[FORMULA], HeI, HeII, HeIII and e- (see for instance Abel et al.1997 for rate coefficients). Thus we obtain the reionization history of hydrogen and helium together with the spectral shape of the background radiation [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999
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