On the Gas Surrounding High Redshift Galaxy Clusters¹
Paul J Francis , Greg M. Wilson , Bruce E. Woodgate, PASA, 18 (1), in press.

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Title/Abstract Page: On the Gas Surrounding
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Contents Page: Volume 18, Number 1

The Nature of the Gas

The three QSO sight-lines may be unrepresentative of the space around this cluster. If they are representative, however, this implies that a fraction of order unity of all sight-lines passing within several Mpc of this cluster would intersect a cloud of hydrogen with a neutral column density of

$N_H \sim 10^{19} {\rm cm}^{-2}$ or greater.

In this section we hypothesise that the region surrounding the 2142-4420 cluster is optically thick in neutral hydrogen clouds. We further hypothesise that this cluster is typical of clusters at this redshift. What would be the physical consequences if our hypotheses were correct?

Gas Geometry

The absorption-line gas in cluster 2142-4420 seems to be substantially more extended spatially than the Ly $\alpha$ emitting galaxies, as can be seen in Fig 1. Strong absorption is seen in QSO 2138-4427, whose sight-line passes $\sim 4$ Mpc from the concentration of Ly $\alpha$ emitting galaxies. It is also seen in QSO 2139-4433, at a redshift that places the absorption about 2 Mpc in front of the cluster. We therefore hypothesise that the cluster of Ly $\alpha$ sources is embedded within a much larger structure of absorbing gas. Indeed, it is suggestive that the QSO sight-line passing closest to the three Ly $\alpha$ emitting galaxies shows the lowest neutral hydrogen column density. This might indicate that the central region of the neutral gas structure is hotter and more ionised than the outer regions.

If this neutral gas structure does exist, what is its geometry? With only the QSO absorption to guide us, all we can do is bracket the possibilities with three straw models:

1.: Sheet Model: the gas lies in a sheet, of thickness $\sim 1$ Mpc, width > 4 Mpc and depth (along the line of sight) $\sim 4$ Mpc. This sheet would be edge-on to our sight-line.
2.: Spherical model: The gas lies in a spherical halo, centred on the Ly $\alpha$ emitting galaxies. The halo radius must be at least 4 Mpc.
3.: Filamentary Model: the cluster of galaxies lie at the intersection of a number of gas-filled filaments. Each filament is at least 4 Mpc long, and perhaps around 1 Mpc thick. Filamentary distributions are predicted by many cosmological simulations (eg. Rauch, Haehnelt & Steinmetz 1997).

Is the region around the galaxy cluster completely full of absorbing gas? Let us assume that a fraction f of all randomly chosen site-lines passing within $\sim 5$ proper Mpc of the galaxy cluster would intercept at least one gas cloud with a total hydrogen cross section

$>10^{19}{\rm cm}^{-2}$ . The probability of all three of our QSO sight-lines showing such absorption is thus f³. If we require this probability to be greater than 1%, that implies f> 22%.

The Mass and Density of the Neutral Gas Structure

Let us assume that the cluster really is surrounded by a $\sim 5$ proper Mpc scale structure of gas with $f \sim 1$ and a typical neutral absorption-line column density of

$N_H \sim 10^{19.5}{\rm cm}^{-2}$ . What would be the consequences? The average density of absorption-line gas within the gas structure will then be

$\sim 8 \times 10^{10} M_{\odot }{\rm Mpc}^{-3}$ in proper coordinates. This is $\sim 100$ times greater than the typical density of neutral hydrogen at this redshift (eg. Steidel 1990).

Given this density, the combined gas mass of all the absorption-line systems within the neutral gas structure would be

$> 6 \times 10^{11} M_{\odot }$ for the sheet or filamentary models, and

$> 3 \times 10^{12} M_{\odot }$ for the spherical model.

This overdensity of baryonic matter in the form of hydrogen clouds could be explained in two ways:

1.: The volume surrounding the galaxy cluster is overdense in all forms of matter: the efficiency with which baryons form absorbing gas clouds and red galaxies is the same as elsewhere in the universe.
2.: The volume is not greatly overdense, but the efficiency with which baryons formed neutral gas clouds and red galaxies is enhanced in this region.

Cosmological simulations favour the second explanation. At redshift 2.38, there has not been sufficient time to assemble large matter concentrations. Many authors have used analytic methods or n-body simulations to estimate the mass of matter concentrations in the high redshift universe. In Table 3, we have collected a sample of these results, parameterised as the two-point correlation coefficients $\xi$ for mass on the approximate scale of the gas structure, at redshifts $z \sim 2.38$ .

**Table 3:** Predicted mass two-point correlation coefficients $\xi$ .
Ref.	Model Details	Co-moving Radius	z	$\xi(r_0, z)$

1	SCDM: $\Omega_0=1.0$ , $\Lambda=0$ , h=0.5 (small $\sigma_8$ )	8.81h^-1 Mpc	2.3	0.06
1	SCDM: $\Omega_0=1.0$ , $\Lambda=0$ , h=0.5 (large $\sigma_8$ )	8.81h^-1 Mpc	2.3	0.17
2	$\Omega_0=1.0$ , $\Lambda=0.0$	11.4 Mpc	2.8	0.008
2	$\Omega_0 = 0.2$ , $\Lambda=0.0$	11.4 Mpc	2.8	0.095
2	$\Omega_0 = 0.2$ , $\Lambda=0.8$	11.4 Mpc	2.8	0.07
3	$\Omega_0=1.0$ , $\Lambda=0.0$	10h^-1 Mpc	2.0	0.020
3	$\Omega_0=1.0$ , $\Lambda=0.0$	10h^-1 Mpc	3.0	0.016
4	SCDM (with Zel'dovich Approximation)	10h^-1 Mpc	2.0	0.05
5	SCDM: $\Omega_0=1$ , $\Lambda=0$ , h=0.5 (small $\sigma_10$ )	10h^-1 Mpc	2.4	0.0036
5	SCDM: $\Omega_0=1$ , $\Lambda=0$ , h=0.5 (large $\sigma_10$ )	10h^-1 Mpc	2.4	0.03
5	TCDM: $\Omega_0=1$ , $\Lambda=0$ , h=0.5 (small $\sigma_10$ )	10h^-1 Mpc	2.4	0.0014
5	TCDM: $\Omega_0=1$ , $\Lambda=0$ , h=0.5 (small $\sigma_10$ )	10h^-1 Mpc	2.4	0.0069
5	OCDM: $\Omega_0=0.4$ , $\Lambda=0$ , h=0.65	10h^-1Mpc	2.4	0.016
5	$\Lambda$ CDM: $\Omega_0=0.4$ , $\Lambda=0.6$ , h=0.65	10h^-1 Mpc	2	0.0036
6	SCDM: h=0.5	10h^-1 Mpc	2.4	0.02
7	LCDM:	10h^-1 Mpc	3	<0.005
7	C+HDM:	10h^-1 Mpc	3	<0.005

[1] Brainerd & Villumsen (1994).

[2] Cólin, Carlberg & Couchman (1997).

[3] Matarrese et al. (1997).

[4] Porciani (1997).

[5] Moscardini et al. (1998).

[6] Bagla (1998).

[7] Ma (1999).

Given these values of $\xi$ , we can estimate the root-mean squared (rms) fluctuations $\sigma_R$ between the average densities of various spherical regions of this size in the early universe. We use the approximate relationship

$\begin{displaymath} \sigma_R^2\approx 2.5 \xi(R) , \end{displaymath}$

(1)

which is obtained from the relation

$\sigma^2_R=3J_3(R)/R^3$ where a top hat window function has been assumed and J₃ represents the integrated two point correlation function (Kolb & Turner 1990). Thus

$\begin{displaymath} J_3=\frac{1}{4\pi}\int_0^R\xi(r)d^3r, \end{displaymath}$

(2)

where the two point correlation function is assumed to be of the form

$\xi(r)=(r/r_0)^{-\gamma}$ , with

$\gamma\approx 1.8$ (Groth & Peebles 1977, Davis & Peebles 1983).

Thus typical predicted rms mass fluctuations on the scale of this absorption-line structure ( $\sim 10$ co-moving Mpc) are only $\sim 25\%$ . Even for models with the most extreme fluctuations (low density models: Cen 1998), and assuming that the cluster sits in a $5 \sigma$ mass fluctuation, the average density on this scale cannot be more than twice the mean density of the universe. Note that this applies to a roughly spherical volume: if the structure really is sheet-like or filamentary, the overdensity within this structure can be significantly greater. So, the average mass density of the $\sim 10$ co-moving Mpc scale volume including the cluster must be of the same order as that of the universe as a whole at this redshift. If, as seems likely, baryonic and non-baryonic matter trace each other on these large scales, primordial nucleosynthesis thus gives us an approximate upper limit on the average baryon density of the cluster halo. Assuming

$\Omega_{\rm baryon} = 0.016 h_{100}^{-2}$ (eg. Walker et al. 1991), and choosing

$H_0 = 70 {\rm km\ s}^{-1}{\rm Mpc}^{-1}$ , this density is

$1.2 \times 10^{-26} {\rm kg\ m}^{-3}$ (

$\sim 2 \times 10^{11} M_{\odot }{\rm Mpc}^{-3}$ ).

Thus even if the QSO absorption-line column densities are representative of the whole cluster halo, the baryonic mass of the cluster inferred (0.6--3

$\times 10^{12} M_{\odot }$ in the form of absorbing clouds) is physically possible: it does not exceed the predicted baryonic mass within the cluster volume (

$\sim 7 \times 10^{12} M_{\odot }$ ). The puzzle would be the high fraction of these baryons that are incorporated into absorption-line systems: the efficiency of formation of these objects is at least 10% and may well be much higher. This efficiency is far higher than is typical at this or any other redshift (eg. Cen & Ostriker 1999)

The Physical State of the Gas

If the QSO sight-lines are representative of all sight-lines through the region around the cluster, it seems to be embedded in a structure of size $\sim 5$ Mpc, and most sight-lines through this structure intersect a gas cloud with a hydrogen column density

$N_H > 10^{19}{\rm cm}^{-2}$ . Is there any physically plausible structure with these properties?

Argument from the Total Baryon Density

In this section, it is shown that the neutral gas structure is probably in the form of many small dense gas clouds, and a crude upper limit is placed on the size of these clouds. In summary, the argument is this: any given $\sim 10$ Mpc region of the early universe must have a baryon density that is close to the average for the whole universe, as discussed in Section 3.2. If these baryons were spread uniformly throughout the region, they would be highly ionised by the UV background radiation and no absorption would be seen. The baryons must therefore be confined into dense clouds, occupying a small fraction of the total region. The density must be high enough that the recombination rate balances the photoionisation by UV background photons. The region could contain a small number of large dense clouds, or a large number of small dense clouds. Only in the latter case, however, would most QSO sight-lines through the region intercept one of these clouds (smaller clouds having a greater ratio of surface area to volume). We now consider this argument in detail.

As discussed in Section 3.2, the density of gas within the $\sim 10$ Mpc scale volume surrounding the galaxy cluster can at most be comparable to

$\Omega_{\rm baryon}$ . If these baryons were spread uniformly throughout this volume, and exposed to the average UV background at this redshift, they would be highly ionised. Why then do the QSO sight-lines show such large neutral column? The mean free path of UV photons at this redshift is $\sim 500$ Mpc (eg. Haardt & Madau 1996): this is far greater than the average separation of UV sources, implying that the UV background intensity is spatially very uniform (Zuo 1992a,b, Fardal & Shull 1993). In this case, we know that at least five QSOs lie within 500 Mpc of the cluster, so if their emission is isotropic, lack of a UV background cannot explain the neutrality of the gas.

Let us therefore consider a model of the neutral gas structure which contains at most this average density of baryons. This mass, instead of being distributed uniformly, is confined into clouds of scale-length r and density $\rho$ . If the clouds are sufficiently large and dense, the gas within them will become neutral.

For any given density, a gas cloud must have a certain minimum size for the hydrogen within it to be neutral. This size was estimated analytically, using standard equilibrium photoionisation, and by using the MAPPINGS II photoionisation code (Sutherland & Dopita 1993). If a UV background with a plausible spectrum and intensity is assumed (the details make little difference to the final result), then the UV background ionises a layer of thickness r on the gas surface. The recombination rate within this surface layer, which is proportional to its density $\rho$ squared, must balance the photoionisation rate: inserting numbers, we find that

$\begin{displaymath} r \sim k_0 \rho^{-2}, \end{displaymath}$

(3)

where

$k_0 \sim 3 {\rm pc\ cm}^{-6}$ . Unless a cloud is thicker than this, its neutral column density will be low (eg. Lanzetta 1991).

This gives one constraint on the size of the absorbing clouds. A second constraint comes from our assumption that the three QSO sight-lines are representative of the region around the galaxy cluster. This assumption requires that a fraction f (of order unity) of all lines of sight passing close to the cluster intersect a cloud. To keep the discussion more general, let us allow for the possibility of sheet-like clouds of thickness r and face-on cross-sectional area A, where A > r², so that most of the incident UV flux enters through their face. If there are N clouds per unit volume throughout the region surrounding the galaxy cluster, and the region has a typical thickness T, then this condition implies that

$\begin{displaymath} NAT \sim 1, \end{displaymath}$

(4)

We can also constrain N, r and $\rho$ by requiring that the total density $N r A \rho$ be at most comparable to

$\rho_{\rm\max}$ , the average baryon density of the universe at this redshift. These constraints can only be met if r is small: solving, we find that A cancels, leaving us with the limits

$\begin{displaymath} r < \frac{\rho^2_{\rm max} T^2}{k_0} \sim 1 {\rm kpc} \end{displaymath}$

(5)

and

$\begin{displaymath} \rho > \frac{k_0}{\rho_{\rm max}T} \sim 0.03 {\rm cm}^{-3} \end{displaymath}$

(6)

If the absorption is caused by approximately spherical clouds (ie. $A \sim r^2$ ) these clouds could have masses of

$\sim 10^6 M_{\odot }$ or less, and a space density of

$N > 10^5 {\rm Mpc}^{-3}$ . If they are flattened, they could be more massive and rarer.

Argument from Metal-line Ratios

Most high column density QSO absorption-line systems, when observed with sufficiently high spectral resolution, break down into multiple components with different ionisation states (eg. Prochaska & Wolfe 1997). The wavelength shifts between different metal-line species seen in our data suggest that the same is true for our Lyman-limit systems.

Our spectra are not of sufficient resolution to resolve this substructure, so we are forced to use single-cloud modelling. Nonetheless, useful constraints can be put on the cloud size. We used the MAPPINGS II photoionisation code, as before, to estimate absorption column densities as a function of cloud density and metallicity.

No useful constraints could be placed on the metallicity of the gas, other than noting that it contains metals. The presence of strong high ionisation lines, particularly C IV and Si III, however, implies that a large part of the cloud mass cannot be at densities significantly higher than

$\sim 0.1 {\rm cm}^{-3}$ . The Si III measurement could be contaminated by Ly $\alpha$ forest lines, but C IV should be reliable.

Given this maximum density, the UV background ionises a layer $\sim 1$ kpc deep into the cloud surface. This places a lower limit on the scale-length r of the absorbing clouds, if the high- and low-ionisation lines come from the same clouds. This limit is only marginally consistent with the upper limit on cloud size placed in Section 3.3.1.

Alternatively, the absorption-line systems could be a mixture of these low density clouds, and much smaller, higher density neutral clouds that contribute most of the low ionisation and Ly $\alpha$ absorption.

Argument from the Velocity Dispersions

**Figure 4:** Diagram illustrating our terminology
$\begin{figure} \psfig{file=f5.eps}\end{figure}$

Are the hypothesised absorbing clouds spread uniformly throughout the region around the galaxy cluster, or are they gathered into clumps (Fig 4)? The velocity structure of the absorption-lines suggests that the latter is true. The absorption (Figure 2) consists of a number of discrete components: the velocity width of each component (

$b \sim 100 {\rm km\ s}^{-1}$ ) is much smaller than the velocity dispersion between the different components (

$\sim 6000 {\rm km\ s}^{-1}$ ).

Each of the individual Ly $\alpha$ absorption components can only be well fit if the absorbing gas has a velocity dispersion of

$b \sim 100 {\rm km\ s}^{-1}$ (if a velocity dispersion of

$b < 50 {\rm km\ s}^{-1}$ is used in the fits, the column density has to be very large to match the width of the base of the absorption component, in which case the predicted damping wings are much larger than observed). The velocity dispersions of each component, while smaller than those of the cluster as a whole, are thus greater than the probable thermal velocity widths of the clouds themselves (

$b < 30 {\rm km\ s}^{-1}$ for a hot photoionised phase).

Why are the individual absorption components so broad? As discussed in Section 3.4, each component is probably made up of many small absorbing clouds. If these small clouds are not gravitationally bound, the velocity dispersions could be Hubble-flow redshift differences between the clouds at the front and back of the clump: that would imply clump sizes of

$\sim 100 {\rm kpc}$ . If, however, they are gravitationally bound, we can use the virial theorem to constrain the mass of each clump, and hence put a lower limit on the size of each clump by requiring that the total mass of all the clumps in the cluster not exceed a cosmologically plausible amount.

Assume that each clump has size l and mass M_c. Given the observed velocity dispersions $\sigma_v$ , the total mass of each clump can be estimated from the virial theorem, and is proportional to the clump size:

$\begin{displaymath} M_c \sim \frac{l \sigma_v^2}{G}. \end{displaymath}$

(7)

As in Section 3.3.1, we can constrain the size l and space density N_c of these clumps by requiring that most sight-lines through the cluster must intersect a clump: ie. that $f \sim 1$ . Thus

$\begin{displaymath} N_cl^2T \sim 1. \end{displaymath}$

(8)

In addition, the total mass of all the clumps within a given volume, N_c M_c, can at most be comparable to the critical density of the universe

$\rho_{\rm crit}$ , as discussed in Section 3.2. Thus the total mass in all the clouds must be inversely proportional to the typical cloud size l. Solving, we find that

$\begin{displaymath} l>\frac{\sigma_v^2}{GT\rho_{\rm crit}} \sim 20 {\rm kpc}. \end{displaymath}$

(9)

If the clumps were smaller, each would be less massive, but so many would be needed to intersect most QSO sight-lines that the total mass of all the clumps would be physically implausible.

Thus each absorption component probably consists of many small absorbing clouds gathered into clumps of size > 20 kpc.

Discussion

So, if we assume that most sight-lines passing close to the 2142-4420 cluster intersect a cloud with

$N_H \sim 10^{19} {\rm cm}^{-2}$ , and that this cluster is typical of high redshift clusters, these clusters must be embedded in $\sim 10$ co-moving Mpc structures of neutral gas, made up of large numbers of small dense gas clouds, gathered into clumps.

If these clumps exist, what could they be? Each clump could be the halo or extended disk of a galaxy (eg. Prochaska & Wolfe 1997): if they are $\sim 100$ kpc in size, the virial mass of each halo would be

$\sim 10^{11} M_{\odot }$ , and the whole cluster halo would have to contain a few hundred of these galaxy halos. Alternatively, the velocity dispersions could represent infall into the potential wells of protogalaxies (eg. Rauch, Haehnelt & Steinmetz 1997). The properties we infer for the absorbing gas within the cluster are strikingly similar to the theoretical predictions of lumpy gas infalling along filaments into protogalaxies (eg. Mo & Miralda-Escudé 1996).

If the halo model is correct, we should see galaxies associated with the absorption systems. For the Lyman-limit absorber in the spectrum of QSO 2139-4434, we do see a galaxy that could plausibly be surrounded by a halo that is causing the absorption: one of the strong Ly $\alpha$ sources lies at the same redshift as the absorber, 20

$^{\prime \prime}$ ( $\sim 100$ projected kpc) from the QSO sight-line. For the damped systems in QSO 2139-4433 and QSO 2138-4427, however, no galaxies are seen within 500 projected kpc of the sight-lines. This is not however conclusive, as any galaxies could easily lack Ly $\alpha$ emission signatures, or lie below our flux limits.

When observed at high angular resolution, many radio galaxies are seen to be surrounded by numerous kpc-sized knots of emission, sometimes made visible by reprocessing nuclear emission (eg. Stockton, Canalizo & Ridgway 1999, Pascarelle et al. 1996). These knots may be the same compact gas clouds we are hypothesising, based on the absorption.

Could all damped Ly $\alpha$ systems be caused by absorption in galaxy protoclusters? The space density of Lyman-break galaxy concentrations (Steidel et al. 1998) is comparable to that of damped Ly $\alpha$ systems, so if all these concentrations are optically thick throughout in Ly $\alpha$ , then most damped Ly $\alpha$ systems could indeed arise in proto-clusters. Note, however, that our lower limit on the size of these clumps of small absorbing clouds is greater than the tentative upper limit placed on the size of two damped Ly $\alpha$ systems by Møller & Warren (1998) and Fynbo, Møller & Warren (1999).

What could be the physical origin of the absorbing clouds? Thermal instabilities can produce small cold dense clouds within the halos of high redshift galaxies (eg. Viegas, Friaça & Gruenwald 1999) or in collapsing proto-galaxies (Fall & Rees 1985): it is intriguing that our inferred cloud masses are comparable to those of globular clusters. The sound crossing times of these small clouds would be small, so if they are long-lived objects, they would need to be confined in some way.

Given that all sight-lines though a clump intersect multiple absorbing clouds, then mean time between cloud-cloud collisions would be less than the time it takes for clouds to cross the clump: ie. 100kpc/

$100 {\rm km\ s}^{-1} \sim 10^9$ years (cf. McDonald & Miralda-Escudé 1999); further evidence for the transient nature of these clouds. What happens to the gas in the clouds during a collision? The collision time-scale is

$\sim 1 {\rm kpc}/100 {\rm km\ s}^{-1} \sim 10^7$ years: if all the gas were converted into stars, this would imply a star formation rate of

$\sim 0.1 M_{\odot }{\rm yr}^{-1}$ , which is slightly below the detection threshold of most current surveys for high redshift star forming galaxies. Over the neutral gas structure as a whole, however, the integrated effect of all the cloud-cloud collisions, if they all form stars, is a star-formation rate of

$\sim 10^3 M_{\odot }{\rm yr}^{-1}$ , which should produce a diffuse H $\alpha$ flux of

$\sim 3 \times 10^{-19}{\rm erg\ cm}^{-2}{\rm s}^{-1} {\rm arcsec}^{-2}$ (Kennicutt 1983) over the entirety of the cluster, and in the absence of self-absorption, a comparable flux of diffuse Ly $\alpha$ emission. We address the detection of such faint Ly $\alpha$ fluxes in Section 4.