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Reionization can be thought of as the second major change in the ionization state of hydrogen (and helium) in the universe (the first being the recombination occurring at z approx 1100). The process is of immense importance in the study of structure formation since, on one hand, it is a direct consequence of the formation of first structures and luminous sources while, on the other, it affects subsequent structure formation. In this article, we attempt to review the basic physical processes related to the reionization with particular emphasis on the link between theory and observations.

The study of reionization consists of two broad areas, namely the properties of the intergalactic medium (IGM) and the formation of sources. Once the first sources produce photons capable of ionizing the surrounding IGM, the process of reionization (and reheating) can be thought of having begun, thus changing the thermal, ionization and chemical properties of the IGM. This change in the nature of the IGM affects the formation of next generation of sources (like metal enrichment changing the initial mass function of stars). The subject area of formation of sources is quite involved in itself dealing with formation of non-linear structures (haloes and filaments), gas cooling, accretion and generation of radiation (stars and quasars). This is somewhat beyond the scope of this review, and we will mostly concentrate on the effects of sources on the IGM.

Historically the study of reionization of the IGM has been closely linked with the observations related to spectra of distant quasars, in particular to the Lyalpha forest, though it was not obvious in the beginning whether the Lyalpha forest traces baryons of cosmological significance. In particular, models in which the Lyalpha forest arises from some kind of "confined clouds" predicted that the amount of baryons within the forest may not be of cosmological significance and hence may not have any substantial connection to the cosmic reionization as we understand it today. To stress this in slightly more detail, let us briefly review some of the major ideas in the development of this field. Of course, the literature of the Lyalpha forest has been reviewed various times, (see for example [1]); nevertheless it might be appropriate to review some the major ideas from the point of view of cosmic reionization.

1.1. Initial models based on pressure and gravitational confinement

In a classic paper, Gunn & Peterson [2] showed that the hydrogen in a diffuse uniform IGM must have been highly ionized at z approx 2 in order to avoid complete absorption of the transmitted flux at wavelengths bluewards of the Lyalpha emission line of the QSO; this is now commonly known as the Gunn-Peterson (GP) effect. Following that, it was proposed [3] that this GP effect can be used to probe the ionization state of hydrogen within the IGM at various redshifts (and also for other elements). The GP effect has remained one of the most stringent tests of the ionization state of the IGM to date.

At the same time, it was also realized that gas which was not uniformly distributed would produce discrete Lyalpha absorption lines. In the beginning, the most natural structures considered were gas clumped into groups of galaxies [4] or low mass protogalaxies [5]. However, these were soon found to be unrealistic when different groups [6, 7] discovered a large number of discrete absorption lines in the QSO spectra, which are usually known as the "Lyalpha forest". It was shown that these forest lines could not be associated with galaxy clusters, rather they have an intergalactic origin and arise in discrete intergalactic clouds at various cosmological redshifts along the line of sight (for reviews see [7, 8, 9]). Various arguments (like the apparent lack of rapid evolution in the properties of the forest, the short relaxation time scales for electrons and protons and short mean free paths) led to the notion that the clouds were "self-contained entities in equilibrium" [7]. A two-phase medium was postulated, with the diffuse, very hot, intercloud medium (ICM) in pressure equilibrium with the cooler and denser Lyalpha clouds. In this two-phase scenario, the ICM was identified with the IGM, while the Lyalpha clouds were treated as separate entities.

According to the pressure confinement model [7, 10, 11], the Lyalpha clouds are supposed to be in photoionization equilibrium with an ionizing ultra-violet (UV) background. The gas is heated by photoionization and cools via thermal bremsstrahlung, Compton cooling, and the usual recombination and collisional excitation processes. Since the ICM is highly ionized, the photoheating is not efficient and hence the medium cools adiabatically through cosmic expansion. The denser clouds embedded in the hot ICM have a nearly constant temperature fixed by thermal ionization equilibrium (~ 3 × 104 K) [10, 11]. The available range of cloud masses is constrained by the requirement that the clouds must be small enough not to be Jeans-unstable but large enough not to be evaporated rapidly when heated by thermal conduction from the ambient ICM [7, 10]. According to such constraints, clouds formed at high redshifts would survive down to observed redshifts only if their masses range between 105 - 1010 Modot.

The neutral hydrogen within the confining ICM is expected to cause a residual GP absorption trough between the absorption lines (clouds). However, observations at higher spectral resolution [12, 13, 14] revealed no continuous absorption between the discrete lines, placing strong limits on the GP effect, which in turn, puts a strict upper limit on the density of the ICM. The ICM temperature has a lower limit from the absorption line width, while the condition that the cloud must be large enough not to evaporate gives an upper limit on the temperature [10]. Another independent upper limit on the temperature of the ICM comes from the lack of inverse Compton distortions in the spectrum of the cosmic microwave background [15] through the Sunyaev-Zeldovich effect [16]. In fact, the upper limit of the so-called y-parameter [17] is able to rule out any cosmologically distributed component of temperature greater than 106 K. When all the limits are combined, only a relatively small corner of allowed density-temperature parameter space remains for the ICM. It turns out that, according to the pressure-confinement model, the density of the ICM is too small to be cosmologically significant. Hence, during these early days, the connection between the cosmic reionization and the IGM was not at all obvious as most of the baryons was expected to lie somewhere else.

The pressure-confinement model ran into severe problems while trying to match the observed column density distribution [18]. For example, in order to reproduce the low column density systems between, say, 13 < log (NHI / cm-2) < 16 (where NHI is the column density of neutral hydrogen), the mass has to vary by 9 orders of magnitude. On the other hand, the mass is severely constrained in order to ensure cloud survival. Therefore, the only escape route is to invoke pressure inhomogeneities [19]. However, the Lyalpha absorbers are found to be weakly clustered over a large range of scales, which thus excludes any significant pressure fluctuations [20]. Similarly, detailed hydrodynamical simulations [21] show that the small mass range of the clouds leads to a failure in producing the column density distribution at high NHI. In addition, pressure-confinement models predict small cloud sizes which are incompatible with the observations of multiple lines of sight [22]. It was thus concluded that the pure pressure confinement model is unlikely to explain the Lyalpha forest as a whole though it is possible that some lines of sight must go through sites where gas is locally confined by external pressure (say, the galactic haloes, the likely hosts of the dense Lyman limit absorbing clouds).

Even from a theoretical point of view, there are no physical reasons for preferring pressure to gravitational confinement or to no confinement at all. Because of this, self-gravitating baryonic clouds were suggested by [23, 24] as an alternative to the pressure confinement model. In this model, the appearance of the IGM as a forest of lines is because of the variations in the neutral hydrogen density rather than a sharp transition between separate entities. In this sense, there is no real difference between an ICM and the clouds in the gravitational confinement model. This scenario of self-gravitating clouds predicts larger sizes of the absorbing clouds (~ 1 Mpc) compared to the pressure-confinement scenario. However, this model, too, runs into problems while trying to match the observed column density distribution [25] as it predicts larger number of high column density systems than is observed. Secondly, the large absorber sizes seemed to contradict observations. Furthermore, gravitationally confined clouds are difficult to explain theoretically since the mass of such clouds must lie in a restricted range to maintain the gas in equilibrium against free expansion or collapse.

As a further alternative, the properties of gas clouds confined by the gravitational field of dark matter have been investigated [26], more specifically in terms of the "minihalo" model [27, 28]. In this picture, Lyalpha clouds are a natural byproduct of the cold dark matter (CDM) structure formation scenario. Photoionized gas settles in the potential well of an isothermal dark matter halo. The gas is stably confined if the potential is sufficiently shallow to avoid gravitational collapse but deep enough to prevent the warm gas from escaping. CDM minihaloes are more compact than the self-gravitating baryonic clouds of [24] because of the larger dark matter gravity, thus alleviating the size problem. The detailed structure of the halo depends on the relative spatial distribution of baryons and CDM. However, the virial radii of the confining objects (~ 10 kpc) are much lower than the coherence lengths of the Lyalpha systems as obtained from constraints on absorption line observations of lensed or paired QSOs [29, 30]. It was thus natural to extend the minihalo model to non-static systems. A non-static minihalo model was studied by [31], who examined the hydrodynamics of a collapsing spherical top-hat perturbation and suggested that clouds were in a free expansion phase.

1.2. IGM as a fluctuating density field

Following the non-static models, it was realized that an IGM with the density fluctuation variance of the order of unity could also produce line-like absorptions in quasar spectra [32, 33]. According to such models, the IGM becomes clumpy and acquires peculiar motions under the influence of gravity, and so the Lyalpha (or GP) optical depth should vary even at the lowest column densities [24, 32, 33, 34, 35]. In a CDM-dominated structure formation scenario, the accumulation of matter in overdense regions reduces the optical depth for Lyalpha absorption considerably below the average in most of the volume of the universe, leading to what has been called the fluctuating GP phenomenon. Traditional searches for the GP effect that try to measure the amount of matter between the absorption lines were no longer meaningful, as they were merely detecting absorption from matter left over in the most underdense regions. If this is not taken into account, the amount of ionizing radiation necessary to keep the neutral hydrogen GP absorption below the detection limits can be overestimated, which would then have severe implications for reionization studies. In this scenario, the density, temperature and thermal pressure of the medium were described as continuous fields and could not be attributed simply to gravitational confinement or pressure confinement. These studies led to a shift in the paradigm of IGM theories, especially since they implied that the IGM contains most of the baryons at high redshifts, thus making it cosmologically significant and hence quite relevant to cosmic reionization.

The actual fluctuation picture can be derived from cosmological N-body and hydrodynamical simulations. It was possible to solve hydrodynamical equations from first principles and set up an evolutionary picture of the IGM in these simulations [36, 37, 38, 39]. Although different techniques and cosmological models were used by different groups, all the simulations indicate a fluctuating IGM instead of discrete clouds.

Since in this new paradigm, the Lyalpha forest arises from a median-fluctuated quasi-linear IGM, it is possible to ignore the high non-linearities. This made it possible to study the IGM through semi-analytical techniques too [33, 40, 41, 42, 43]. The issue of dealing with quasi-linear densities were dealt in two ways. In the first method, it was showed that a quasi-linear density field, described by a lognormal distribution, can reproduce almost all the observed properties of the Lyalpha forest [33, 42]. In fact, this was motivated by earlier ideas of [44] for dark matter distribution. In an alternate method, it was also possible to obtain the density distribution of baryons from simulations which could then be used for semi-analytical calculations [45]. Given the baryonic distribution, the neutral hydrogen fraction was calculated assuming photoionization equilibrium between the baryons and the ionizing radiation field. It was also realized that the equilibrium between photoheating and adiabatic cooling implies a tight relation between the temperature and density of the gas, described by a power-law equation of state [46], which was used for determining the temperature of the gas. Given such simplifying and reasonable assumptions, it was possible to make detailed predictions about the Lyalpha forest. For example, a relation between column density peaks ("absorption lines") and the statistics of density peaks was proposed [41, 43], and analytical expressions for the dependence of the shape of the column density distribution on cosmological parameters were obtained.

The simulations and the semi-analytical calculations both have been quite successful in matching the overall observed properties of the absorption systems. The shape of the column density distribution and the Doppler parameter distribution are reasonably well reproduced by the simulations [36, 38, 39, 37, 47, 48] as well as semi-analytical calculations [43, 49] over a wide redshift range. The large transverse sizes of the absorbers seen against background paired and lensed QSOs are well explained by the coherence length of the sheets and filaments [39, 50, 51]. In addition, the probability distribution function and power spectrum of the transmitted flux in the Lyalpha forest is reproduced very well by the models [52, 53]. The Lyalpha optical depth fluctuations were used for recovering the power spectrum of matter density fluctuations at small scales [54, 55] and also to obtain various quantities related to the IGM [53, 56].

Given the fact that the Lyalpha can be modelled so accurately, it has become the most useful tool in studying the thermal and ionization history of the universe ever since. Subsequently it was realized that this simple description of the IGM could be coupled to the properties of the ionizing sources and hence it was possible to compute the reheating and reionization history. Since the modelling of the sources is a highly non-linear problem and much more non-trivial to solve that the quasi-linear IGM, it was more natural to make some simple assumptions about the sources, calculate their effect on the IGM and then constrain the properties of the sources themselves.

1.3. Sources of ionization

The classic problem of the propagation of ionization fronts from a point source was studied by [57, 58]. It was shown that the recombination timescale is too large for the ionized region to reach the Strömgren radius. Furthermore, the calculations showed that the ionizing photons from the observed population of QSOs cannot produce enough UV flux to reionize the IGM at z approx 3 [57, 59, 58]. This lead for extensive searches and proposals for other sources of UV ionizing flux. The next most obvious choices for UV radiation were the (early) galaxies and stars. This was studied using observed ionization state of heavy element absorption systems in the spectra of QSOs and model-dependent metal production arguments [60, 61], though no firm conclusions could be drawn because of the fraction of photons which are able to escape the host galaxy is unknown (and that situation remains till date).

The possibility of galaxies contributing to the UV flux was implemented in various analytical calculations [62, 63, 64, 65]. These calculations concentrated on the collapse of dark matter haloes, subsequent cooling (atomic and/or molecular) of gas, star formation formalisms and propagation of ionization fronts. Subsequently, detailed modelling for the reheating and reionization histories of the IGM showed that, under standard assumptions regarding hierarchical CDM model, Press-Schechter theory, cooling within collapsed haloes, star-forming efficiency and observed QSO luminosity function, the reionization of the hydrogen is achieved at z approx 10 [66, 67]. Most of these studies generally incorporated the inhomogeneities in the IGM through a (evolving) clumping factor. A model for reionization for the inhomogeneous IGM was proposed [45] which was able to take into account the fact that dense regions would remain neutral longer (because of their high recombination rate).

Most of these effects were also seen in hydrodynamical simulations, thus confirming the overall picture for reionization by UV sources. Usually the limitation in computing power forced small volumes (say, boxes with sizes of a few Mpcs) to be simulated. It was found [65, 68] that a mass resolution of about 104 Modot was required to resolve early epochs of reheating and reionization, which remains a great challenge even now. A better resolution can be achieved if, for example, high-resolution N-body simulations and semi-analytical models for galaxy and star formation are combined [69] to obtain the thermal history of the IGM.

The picture of reionization by UV sources which emerged form these studies can be summarised as follows: (i) The reionization process by UV sources could be classified into three phases [70]. In the "pre-overlap" phase, the ionized regions of individual sources propagate into the neutral IGM. In the "overlap" phase, the ionized regions start overlapping and subsequently ionize the whole of IGM (except for some high-density peaks). At this stage the universe becomes transparent to UV radiation and hence the mean free path of photons increases dramatically. Finally, there is the ever-continuing "post-overlap" phase where the ionization fronts propagate into the neutral high density regions. (ii) The reheating of the IGM preceded the reionization as a small number of hard photons could heat the medium up to several hundred to thousand Kelvins before complete reionization,

It should be mentioned that though the QSOs and galaxies seem to be the most natural choices as sources for reionization of the IGM, the possibility of other sources cannot be ruled out, at least from observations. Hence various other sources have been studied too, the early ones being the supernova-driven winds [71], hard photons from structure formation [72] early formed massive black holes [73] and more exotic sources like decaying dark matter (or other) particles [74, 75, 76, 77, 78, 79], with the list ever-increasing till date [80, 81].

It is thus clear that the transmission regions in the Lyalpha forest at redshifts z ltapprox 6 conclusively implies that the universe is ionized at lower redshifts, though the exact nature of the ionization process or the sources responsible are not understood at the moment. On the other hand, we can also think of the Lyalpha forest as the leftover of the reionization, i.e., the absorption signatures imply that the sources were not able to fully complete the job.

In the next section, we shall review the current observational situation regarding reionization of the IGM and main conclusions that can be drawn from the data. Section 3 would discuss the physics of cosmic reionization along with description of certain analytical and numerical models. We shall summarize the main predictions and future tests for these models.

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