2. PHYSICAL PROPERTIES OF THE WHIM

2.1. WHIM ionisation conditions

The occurrence and characteristics of the WHIM absorption signatures in the FUV and X-ray band are determined to a high degree by the ionisation conditions in the gas. We briefly discuss the WHIM ionisation properties, as this is crucial for interpretation of the WHIM absorption lines in FUV and X-ray spectra that arise in such warm-hot gas. Generally, there are two processes that determine the ionisation state of warm-hot gas in the intergalactic medium: collisional ionisation caused by the high temperature of the gas in collapsed structures and photoionisation by the cosmic FUV background.

2.1.1 Hydrogen

By far most of the mass of the WHIM is in the form of ionised hydrogen. Therefore, understanding the processes that lead to the ionisation of hydrogen is essential for the interpretation of WHIM absorption lines and for a reliable estimate of the baryon content of warm-hot intergalactic gas. The ionisation potential of neutral hydrogen is 13.6 eV and thus both ionisation by particle collisions and ionisation by high-energy photons contribute to the ionisation of H I in warm-hot gas. We start with collisional ionisation, which is believed to dominate the ionisation of hydrogen at temperatures > 10⁵ K.

In collisional ionisation equilibrium (CIE) - the most simple approach to characterise the ionisation conditions in low-density, high-temperature plasmas - the ionisation fraction depends only on the gas temperature. If we ignore any charge-exchange reactions (which is justified in case of hydrogen), the neutral hydrogen fraction in CIE is simply the ratio between the recombination coefficient _H(T) and the collisional ionisation coefficient _H(T):

(1)

Above gas temperatures of ~ 1.5 × 10⁴ K collisions by thermal electrons efficiently ionise hydrogen to a high degree, and already at T ~ 3 × 10⁴ K the neutral hydrogen fraction in the gas is less then one percent. For the temperature range that is characteristic for the WHIM, T = 10⁵ - 10⁷ K, one can approximate the ionisation fraction in a collisional ionisation equilibrium in the way

(2)

where T is in units K (Richter et al. 2006a; Sutherland & Dopita 1993). Thus, for WHIM gas with T = 10⁶ K the neutral hydrogen fraction in the gas in CIE is only ~ 2.4 × 10^-7.

Next to particle collisions, photons with energies > 13.6 eV contribute to the ionisation of the WHIM, in particular in the low-temperature WHIM tail at ~ 10⁵ K and below. Such ionising photons in intergalactic space are indeed provided by the metagalactic ultraviolet (UV) background, originating from the hard radiation emitted by QSOs and AGN. Fig. 1 shows the spectral shape of the UV background at z = 0 (left panel) and the redshift-dependence of the hydrogen photoionisation rate from the UV background (right panel) based on the models by Haardt & Madau (1996).

Figure 1. Left panel: Spectral shape of the metagalactic UV background at z = 0 (from Haardt & Madau 1996). Plotted is the flux of photons (F = 4 J) against the frequency . The hydrogen ionisation edge is indicated with a dashed line. Right panel: Redshift-dependence of the hydrogen photoionisation rate from the UV background for the range z = 0 to z = 5. Adapted from Haardt & Madau (1996).

Considering photoionisation, one generally can write for the neutral-hydrogen fraction in the gas:

(3)

where _H(T) denotes the temperature-dependent recombination rate of hydrogen, n_e is the electron density, and _{H I} is the photoionisation rate. _{H I} depends on the ambient ionising radiation field J (in units erg cm^-2 s^-1 Hz^-1 sr^-1) in the WHIM provided by the metagalactic UV background (see Fig. 1):

(4)

Here, _L is the frequency at the Lyman limit and denotes the photoionisation cross section of hydrogen, which scales with ^-3 for frequencies larger that _L (see Kaastra et al. 2008 - Chapter 9, this volume). We have introduced the dimensionless scaling factor J_-23 which gives the metagalactic UV radiation intensity at the Lyman limit in units 10^-23 erg cm^-2 s^-1 Hz^-1 sr^-1. For z = 0 we have J_-23 ~ 1-2, while for z = 3 the value for J_-23 is ~ 80, thus significantly higher (Haardt & Madau 1996). Assuming n_e = n_H and inserting a proper function for _H(T), we finally can write for the logarithmic neutral hydrogen fraction in a purely photoionised WHIM plasma

(5)

where n_H is the hydrogen volume density in units cm^-3 and T₄ is the temperature in units 10⁴ K. Thus, for purely photoionised intergalactic gas at z = 0 with n = 5 × 10^-6 and T = 10⁶ K we find that the neutral hydrogen fraction is f_{H I, photo} ~ 2.4 × 10^-6. This is ten times higher than for CIE, indicating that collisions dominate the ionisation fraction of hydrogen in intermediate and high-temperature WHIM regions. However, note that at lower temperatures near T = 10⁵ K at the same density we have f_{H I, photo} ~ f_{H I, coll}. Since this is the WHIM temperature regime preferentially detected by UV absorption features (e.g., O VI and broad Ly), photoionisation is important and needs to be accounted for when it comes to the interpretation of WHIM absorbers observed in the FUV. From a WHIM simulation at z = 0 including both collisional ionisation and photoionisation (Richter et al. 2006b; see Fig. 2) find the following empirical relation between neutral hydrogen fraction and gas temperature for a WHIM density range between log n_H = -5.3 and -5.6:

(6)

Figure 2. The neutral hydrogen fraction, log fH I = log (nH I / nH), in a WHIM simulation (photoionisation+collisional ionisation), is plotted as a function of the gas temperature, log T. The light gray shaded indicates cells in the density range log nH = -5 to -7. The dark gray shaded area refers to cells that have log nH = -5.3 to -5.6, thus a density range that is characteristic for WHIM absorbers. Adapted from Richter et al. (2006b).

This equation may serve as a thumb rule to estimate ionisation fractions in WHIM absorbers at z = 0 if the gas temperature is known (e.g., from measurements of the line widths; see Sect. 2.2.1).

Figure 3. CIE ion fractions of selected high ions of oxygen (O VI, O VII, O VIII; left panel) and neon (Ne VIII, Ne IX; right panel) in the WHIM temperature range log (T/K) = 4.5 - 7.0, based on calculations by Sutherland & Dopita (1993).

2.1.2 Oxygen and other metals

While hydrogen provides most of the mass in the WHIM, the most important diagnostic lines to study this gas phase are from highly ionised metals such as oxygen, neon, carbon, magnesium, and others. Therefore, the understanding of the ionisation properties of the observed high ions of these elements is as important as for hydrogen. As for hydrogen, both collisional ionisation and photoionisation need to be considered. With its single electron, hydrogen can only be either neutral or fully ionised. Heavy elements, in contrast, have several electrons available and are - even at very high temperatures - usually only partly ionised. Thus, electronic transitions exist for such highly-ionised metals ("high ions") in warm-hot gas. Of particular importance for observations of the WHIM are the high ionisation states of oxygen, O VI, O VII, and O VIII, as they have strong electronic transitions in the UV (O VI) and at X-ray wavelengths (O VII & O VIII) and oxygen is a relatively abundant element. Another important metal for observing warm-hot gas in the UV and X-ray band is neon (Ne VII, Ne VIII, Ne IX, Ne X). In collisional ionisation equilibrium, the ionisation state of these elements is determined solely by the temperature of the gas. For each element, the ionisation fractions of the ionisation states (e.g., four-times vs. five-times ionised) then are characterised by the respective ionisation potentials (IPs) of the individual ionisation levels. For instance, at T ~ 1-3 × 10⁵ K, a significant fraction of the oxygen is five-times ionised (O⁺⁵ or O VI, IP = 138 eV). Six-times ionised oxygen (O⁺⁶ or O VII, IP = 739 eV) and seven-times ionised oxygen (O⁺⁷ or O VIII, IP = 871 eV) predominantly exist at higher temperatures in the range 3 × 10⁵ - 3 × 10⁶ K and 3 × 10⁶ - 10⁷ K, respectively. Fig. 3 shows the ionisation fractions of the most important high ions of oxygen and neon, based on the CIE calculations of Sutherland & Dopita (1993); see also Kaastra et al. 2008 - Chapter 9, this volume. High ions of other elements such as carbon, nitrogen, silicon and magnesium are less important for WHIM observations as their observable transitions trace lower temperature gas (e.g., C IV, Si IV) or the abundance of these elements in the intergalactic medium are too low. It is important to note at this point, that the discussed relation between ionisation state/fraction and gas temperature explicitly assumes that the gas is in an ionisation equilibrium. This may not be generally the case in the WHIM, however, as the densities are generally very low. For instance, under particular non-equilibrium conditions the timescales for cooling, recombination, and ion/electron equilibration may differ significantly from each other (see for instance Bykov et al. 2008 - Chapter 8, this volume). In such a case, the presence of high ions such as O VI and/or measured high-ion ratios would not serve as a reliable "thermometer" for the WHIM gas. In addition, WHIM filaments most likely neither are isothermal nor do they have a constant particle density. In fact, as WHIM simulations demonstrate, WHIM absorbers seem to represent a mix of cooler photoionised and hotter collisionally ionised gas with a substantial intrinsic density range. The absorption features from high ions arising in such a multi-phase medium therefore are generally difficult to interpret in terms of physical conditions and baryon budget.

In view of the high energies required to produce the high ions of oxygen and neon in combination with the spectral shape of the metagalactic background radiation (see Fig. 1), photoionisation of high metal ions in the WHIM is less important than for hydrogen. However, for O VI photoionisation is important at low redshifts in WHIM regions with very low densities or in systems located close to a strong local radiation source (e.g., in O VI systems associated with the background QSO). Note that at high redshift, most of the intervening O VI appears to be photoionised owing to the significantly higher intensity of the metagalactic background radiation in the early Universe (see Sect. 5.2).

2.2. WHIM absorption signatures in the UV and X-ray band

2.2.1 UV absorption

As indicated in the previous subsection, five-times ionised oxygen (O VI) is by far the most important high ion to trace the WHIM at temperatures of T ~ 3 × 10⁵ K in the ultraviolet regime (assuming CIE, see above). Oxygen is a relatively abundant element and the two lithium-like (1s²2s)² S_1/2 (1s²2p)²P_1/2,3/2 electronic transitions of O VI located in the FUV at 1031.9 and 1037.6 Å have large oscillator strengths (f₁₀₃₁ = 0.133, f₁₀₃₇ = 0.066). Next to O VI, Ne VIII traces WHIM gas near T ~ 7 ×10⁵ K (in collisional ionisation equilibrium) and thus is possibly suited to complement the O VI measurements of the WHIM in a higher temperature regime. The two available Ne VIII lines are located in the extreme ultraviolet (EUV) at 770.4 Å (f₇₇₀ = 0.103) and 780.3 Å (f₇₈₀ = 0.051), allowing us to trace high-column density WHIM absorbers at redshifts z > 0.18 with current FUV satellites such as FUSE. However, as the cosmic abundance of Ne VIII is relatively low, Ne VIII is not expected to be a particularly sensitive tracer of the WHIM at the S/N levels achievable with current UV spectrographs. The same argument holds for the high ion Mg X, which has two transitions in the EUV at even lower wavelengths ( 609.8, 624.9 Å). So far, only O VI and in one case Ne VIII has been observed in the WHIM at low redshift (see Sect. 3.2). Note that WHIM absorption features by O VI (and Ne VIII) are mostly unsaturated and the line profiles are fully or nearly resolved by current UV instruments such as FUSE and STIS, which provide spectral resolutions of R = / 20,000 and 45,000, respectively. Table 1 summarises physical parameters of O and Ne high ions and their observable transitions in the UV and X-ray bands.

Four-times ionised nitrogen (N V; I.P. is 98 eV) also is believed to trace predominantly collisionally ionised gas at temperatures near T ~ 2 × 10⁵ K, but its lower cosmic abundance together with its deficiency in low metallicity environments due to nucleosynthesis effects (e.g., Pettini et al. 2002) makes it very difficult to detect in the WHIM. Other available strong high-ion transitions in the UV from carbon (C IV 1548.2, 1550.8 Å) and silicon (Si IV 1393.8, 1402.8 Å) are believed to trace mainly photoionised gas at temperatures T < 10⁵ K, but not the shock-heated warm-hot gas at higher temperatures.

Next to high-ion absorption from heavy elements, recent UV observations (Richter et al. 2004; Sembach et al. 2004; Lehner et al. 2007) have indicated that WHIM filaments can be detected in Ly absorption of neutral hydrogen. Although the vast majority of the hydrogen in the WHIM is ionised (by collisional processes and UV radiation), a tiny fraction (f_{H I} < 10^-5, typically) of neutral hydrogen is expected to be present. Depending on the total gas column density of a WHIM absorber and its temperature, weak H I Ly absorption at column densities 12.5 log N(H I) 14.0 may arise from WHIM filaments and could be used to trace the ionised hydrogen component. The Ly absorption from WHIM filaments is very broad due to thermal line broadening, resulting in large Doppler parameters of b > 40 km s^-1. Such lines are generally difficult to detect, as they are broad and shallow. High resolution, high S/N FUV spectra of QSOs with smooth background continua are required to successfully search for broad Ly absorption in the low-redshift WHIM. STIS installed on the HST is the only instrument that has provided such data, but due to the instrumental limitations of space-based observatories, the number of QSO spectra adequate for searching for WHIM broad Ly absorption (in the following abbreviated as "BLA") is very limited.

The b values of the BLAs are assumed to be composed of a thermal component, b_th, and a non-thermal component, b_nt, in the way that

(7)

The non-thermal component may include processes like macroscopic turbulence, unresolved velocity-components, and others (see Richter et al. 2006a for a detailed discussion). The contribution of the thermal component to b depends on the gas temperature:

(8)

where T is in K, k is the Boltzmann constant, m is the particle mass, and A is the atomic weight. For the shock-heated WHIM gas with log T 5 one thus expects b_th 40 km s^-1. The non-thermal broadening mechanisms are expected to contribute to some degree to the total b values in WHIM absorbers (see Richter et al. 2006a), so that the measured b value of a BLA provides only an upper limit for the temperature of the gas.

2.2.2 X-ray absorption

The highest ionisation phase of the WHIM will produce and absorb line radiation primarily in the He- and H-like ions of the low-Z elements (C, N, O, Ne), and possibly in the L-shell ions of Fe. In practice, much of the attention is focused on oxygen, because of its relatively high abundance, and because the strongest resonance lines in He- and H-like O are in a relatively 'clean' wavelength band. For reference, the Ly transitions of C VI, N VII, O VIII, and Ne X occur at 33.7360, 24.7810, 18.9689, and 12.1339 Å, respectively (wavelengths of the 1s - 2p_1/2,3/2 doublet weighted with oscillator strength; Johnson & Soff (1985). The He-like ions C V, N VI, O VII, and Ne IX have their strongest transition, the n = 1-2 resonance line, at 40.2674, 28.7800, 21.6015, and 13.4473 Å (Drake 1988; see also Table 1). Data on the higher order series members can be found in Verner et al. (1996). As far as the Fe L shell ions are concerned, the most likely transition to show up would be the strongest line in Ne-like Fe XVII, n = 2p - 3d 15.014 Å. In addition, all lower ionisation stages of C, N, O, and Ne (with the exception of neutral Ne of course) can also absorb by n = 1-2; the strongest of these transitions would be 1s - 2p in O VI at 22.019 Å (Schmidt et al. 2004). Likewise, the lower ionisation stages of Fe could in principle produce n = 2-3 absorption.

The thermal widths of all these transitions will be very small, requiring resolving powers of order R ~ 10000 (C, N, O, Ne) for gas temperatures of order 10⁶ K to be resolved; for Fe, the requirement is even higher, by a factor ~ 2. As we will see, for practical reasons, these requirements exceed the current capabilities of astrophysical X-ray spectroscopy by a large factor. Due to the small Doppler broadening (ignoring turbulent velocity fields for now), the lines will rapidly saturate. For He- and H-like O resonance line absorption, saturation sets in at an equivalent width of order 1 mÅ (Kaastra et al. 2008) - Chapter 9, this volume), or column densities of order a few times 10¹⁴ ions cm^-2. The challenge, therefore, for X-ray spectroscopy presented by the IGM is to detect small equivalent width, near-saturation lines that are unresolved.

2.3. The baryon content of the WHIM as measured by UV and X-ray absorbers

One important result from absorption line measurements of the WHIM in the UV is the observed number density of WHIM absorbers, usually expressed as dN / dz, the number of absorbers per unit redshift. For instance, from recent measurements with FUSE and HST/STIS one finds for O VI absorbers and Broad Ly absorbers at z 0 values of dN / dz(O VI) 20 and dN / dz(BLA) 30 (see Sect. 3.2). Currently, the WHIM absorber density is only measurable in the UV, since in the X-ray band both the observed number of WHIM absorption lines and the available redshift path for WHIM observations is too small to derive statistically significant values of dN / dz(O VII) and dN / dz(O VIII).

A particularly interesting question now is, how the observed number density of high-ion lines or BLAs translates into an estimate of the cosmological baryon mass density of the WHIM, _b(WHIM). To obtain such an estimate of the baryon content of the WHIM from UV and X-ray absorption measurements one has to consider two main steps. First, one needs to transform the observed column densities of the high ions (e.g., O VI, O VII, O VIII) into a total gas column density by modelling the ionisation conditions in the gas. In a second step, one then integrates over the total gas column densities of all observed WHIM absorbers along the given redshift path and from that derives _b(WHIM) for a chosen cosmology. Throughout the paper we will assume a CDM cosmology with H₀ = 70 km s^-1 Mpc^-1, = 0.7, _m = 0.3, and _b = 0.045. For the first step the uncertainty lies in the estimate of the ionisation fraction of hydrogen of the WHIM. For this, it is usually assumed that the WHIM is in collisional ionisation equilibrium, but photoionisation and non-equilibrium conditions may play a significant role. In the case of using metal ions such as O VI the unknown oxygen abundance (O/H) of the gas introduces an additional uncertainty (see below) for the estimate of _b(WHIM). For the second step, it is important to have a large enough sample of WHIM absorption lines and a sufficient total redshift path along different directions in order to handle statistical errors and the problem of cosmic variance. As mentioned earlier, these requirements currently are fulfilled only for the UV absorbers.

The cosmological mass density _b of O VI absorbers (and, similarly, for other high ions) in terms of the current critical density _c can be estimated by

(9)

In this equation, μ=1.3 is the mean molecular weight, m_H = 1.673 × 10^-27 kg is the mass per hydrogen atom, H₀ is the adopted local Hubble constant, and _c = 3H₀² / 8 G is the current critical density. The index i denotes an individual high-ion absorption system along a line of sight j. Each measured high-ion absorption system i is characterised by its measured ion column density (e.g., N(O VI)_ij), the ionisation fraction of the measured ion (e.g., f_{O VI,ij}), and the local abundance of the element measured compared to hydrogen (e.g., the local oxygen-to-hydrogen ratio, by number). Each line of sight j has a characteristic redshift range z in which high-ion absorption may be detected. The corresponding comoving path length X available for the detection of WHIM high-ion absorbers then is given by:

(10)

In analogy, we can write for the cosmological mass density of the BLAs:

(11)

As can be easily seen, the advantage of using BLAs for deriving the WHIM mass density is that the metallicity of the gas is unimportant for the determination of _b. The disadvantage is, however, that the ionisation corrections are very large and uncertain, since they are determined indirectly from the BLA line widths (see Sect. 2.2.1).