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Once the metal pollution has been accounted for in the structure formation simulations, the interpretation of the metal ionised states accessible to the observations to infer the gas properties requires theoretical models for the radiation field maintaining the observed ions. A commonly adopted approach consists in avoiding to solve the full radiative transfer problem and using a simplified model for the UV background. The assumed radiation is then used as energetic input for photo-ionisation codes which compute the metal ionisation states ([184, 186]).

The results of this approach are affected by the uncertainties associated to the assump- tions on the shape and intensity of the radiation field, which typically is not calculated self-consistently through a radiative transfer across the inhomogeneous gas distribution. Many studies suggest in fact that the radiative transfer effects of shadowing, filtering and self-shielding induce deviations in the shape and intensity of the background with respect to models in which the effects of the radiative transfer are neglected ([158] and references therein). Fluctuations in the photo-ionisation rates as well as spatial deviations in the IGM temperature due to the inhomogeneity of the cosmic web support this view at least on scales of few comoving Mpc (see [158, 90, 167] and references therein). On large scales of about 100h−1 _{Mpc comoving, the QSO spatial distribution and their spectral index variability} could be an additional cause of variations in the background ionising field ([166, 295, 296]) below z ∼4.

These fluctuations induced by the radiative transfer could be efficiently recorded in the ionisation balance of the metals because they have a rich electronic structure and their atomic spectrum is more sensitive to the radiation field fluctuations than e.g. hydrogen ([182, 95, 90]).

Numerical schemes which solve the cosmological radiative transfer equation by applying different approximations are now quite mature and well tested ([124, 126] and references therein for an overview of the available codes) and are able to simulate complex scenarios involving large cosmological boxes and number of sources (ex. [125, 266, 162, 54, 52, 11]). Typically these codes are restricted to the hydrogen chemistry, with only a few of them including a self-consistent treatment of the helium component, which is particularly relevant for a correct determination of the gas temperature (see e.g. [49]). None of them though includes the treatment of metal species.

In the ISM community, on the other hand, several photo-ionisation codes are able to simulate the complex physics of galactic HII regions, with great deal of attention to the details of the underlying physics (see [191] for a review). For example, Cloudy [83] and

MAPPINGS III[9] are two codes particularly accurate in simulating the complex physics of the galactic regions largely polluted by heavy atoms and dust grains. At each cosmic scale, the physical properties of the gas (density, temperature) can be derived from the observed ion abundances only under specific assumptions about the spectrum of the radiation field that maintains their ionisation equilibrium; it is then of primary importance to have accur- ate theoretical models reconstructing the shape and intensity of the cosmic UV background as well as the physics of gas photo-ionisation in astrophysical environments. In the follow- ing Chapters I will treat in detail the problem of the radiative feedback on cosmological scales with specific emphasis on the cosmological radiative transfer code CRASH.

In this Chapter the radiative transfer problem is introduced in the context of the Re- ionisation history of the Universe and it is addressed in its general formalism in Section 3.2. The radiative transfer codeCRASHis then introduced in Section 3.3 and it is described in its algorithm and in the new software implementation.

3.1

Ionisation

In Astrophysics many environments can be described in term of gas embedded in the radiation emitted by nearby sources: star forming regions, HII regions, Interstellar Medium and Intergalactic Medium, are just few examples. UV photons with energies E ≥13.6 eV emitted by the sources ionise the neutral hydrogen and helium composing the gas mixture of these environments. Assuming a neutral gas of number densityngas[cm−3] composed by hydrogen and helium, the evolution of the physical state of the gas is described in terms of the following equations:

nHx˙HII =γHI(T)nHIne−αHII(T)nHIIne+ ΓHInHI

nHex˙HeII=γHeI(T)nHeIne−γHeII(T)nHeIIne−αHeII(T)nHeIIne+

αHeIII(T)nHeIIIne+ ΓHeInHeI

nHex˙HeIII=γHeII(T)nHeIIne−αHeIII(T)nHeIIIne+ ΓHeIInHeII ˙

T = 2 3kBp

[kBTp˙+H(T, xA)−Λ (T, xA)]

(3.1)

where T is the gas temperature and the hydrogen and helium ionisation fractions are defined asxHII ≡nHII/nH,xHeII ≡nHeII/nHeandxHeIII ≡nHeIII/nHe. Hereni is the number

density of species i ∈ {H, HII , He, HeII, HeIII }, with nH = fHngas and nHe = fHengas, while fH (fHe) is the fraction of H (He) in number. In the above equations, αI(T) are the gas recombination coefficients for each species (I∈ {HII, HeII, HeIII }) andγ_A(T) are the coefficients of the collisional ionisation processes (A∈ {HI, HeI, HeII}). ΓA defines the time dependent photo-ionisation rate. In the last equation, describing the energy balance

of the system,pis the number of the free particles per unit volume present in the gas andH

and Λ are respectively the gas heating and the cooling functions. Hereafter I will consider the gas in steady state or, equivalently, the time scales of the atomic processes can be considered much faster than the dynamical time scales. In this case, the physical processes which determine Λ include recombination radiation, free-free emission and collisionally excited line radiation. An accurate treatment of such processes can be found in [71]. On a cosmological scale, the cooling function also depends on the redshift z because of the Compton cooling due to the CMB background.

The photo-ionisation process depends on the UV radiation field intensity and shape, as described by the definition:

ΓA(t) = ˆ ∞ νA σA(ν) cuν(t) hν dν, (3.2)

where σA(ν) is the frequency dependent cross section of the species A, as established by the non relativistic description of the quantum mechanical spectrum of their ions; νA indicates the resonant frequency for the photo-ionisation, and the quantity uνdν is the

energy density of the radiation in the frequency interval [ν, ν+dν].

If the dominant ionisation mechanism is photo-ionisation, it is found that the typical temperatures are in the range T = (1·104 −5·104) [K]. On the other hand, if shock heating is present, temperatures exceeding 106_{K can be reached and the ionisation balance} is dominated by the atomic collisions.