A&A 388, 741-757 (2002)
P. Valageas1 - R. Schaeffer1 - J. Silk2
1 - Service de Physique Théorique, CEN Saclay, 91191 Gif-sur-Yvette, France
2 - Astrophysics, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
Received 28 November 2001 / Accepted 5 April 2002
We investigate the behaviour of cosmological baryons at low redshifts after reionization, through analytic means. In particular, we study the density-temperature phase-diagram that describes the history of the gas. We show how the location of the matter in this diagram expresses the various constraints implied by usual hierarchical scenarios. This yields robust model-independent results that agree with numerical simulations. The IGM is seen to be formed via two phases: a "cool'' photo-ionized component and a "warm'' component governed by shock-heating. We also briefly describe how the remainder of the matter is distributed over galaxies, groups and clusters. We recover the fraction of matter and the spatial clustering computed by numerical simulations. We also check that the soft X-ray background due to the "warm'' IGM component is consistent with observations. We find in the present universe a baryon fraction of 7% in hot gas, 24% in the warm IGM, 38% in the cool IGM, 9% within star-like objects and, as a still un-observed component, 22% of dark baryons associated with collapsed structures, with a relative uncertainty no larger than 30% on these numbers.
Key words: cosmology: theory - large-scale structure of Universe - galaxies: intergalactic medium
The latter conclusion was reached from numerical simulations. In this article, we reconsider this problem in order to derive the properties of the IGM by analytic means. In particular, we wish to investigate whether one can understand this behaviour in a quantitative manner from robust, model-independent, arguments.
First, in Sect. 2 we study the phase-diagram of cosmological baryons. While the Lyman- forest is described by a well-defined Equation of State the "warm'' IGM component shows a broad scatter (e.g., Dave et al. 1999; Dave et al. 2001) since its temperature depends through shock-heating on the neighbouring gravitational potential which is a stochastic field. Nevertheless, we show that it is constrained to lie in a well-defined domain in the plane, and determine its average location in this plane, which may be considered as the "Equation of State'' of the "warm'' IGM. We also give the location in this diagram of galaxies, groups and clusters.
Next, in Sect. 3 we use our results to compute the redshift evolution of the fraction of matter enclosed within the different phases. Then, in Sect. 4 we estimate the two-point correlation function and the clumping factor of the "warm'' IGM. Finally, in Sect. 5 we check that the X-ray background emitted by the "warm'' component agrees with observations.
In this article, all our numerical results are obtained with the following cosmological parameters. We consider a low-density flat universe with and . The baryonic density parameter is and the Hubble constant is H0=65 km s-1/Mpc. The CDM power-spectrum of the linear density fluctuations is normalized by .
Second, the gas temperature may be set by an external heating source, like a UV background radiation. Then, the temperature is no longer given by Eq. (1) since it is not fixed by gravitational energy. However, the pressure of the gas erases the baryonic density fluctuations over a scale R given by:
There is a characteristic density at which shock heating starts to play a role, that is where Eq. (1) is to be used in place of Eq. (2). It is expected to correspond to a density contrast of a few units at least. The detailed discussion of the relation between the two regimes around may be found in Sect. 2.4 below.
|J21 = 0.05
We assume that the baryon density
scales as the dark matter density through
Substituting Eq. (1) into the relation derived in Eq. (A.8) we obtain a curve . The subscript "loc'' in the overdensity cutoff refers to the fact that this is a "local'' heating process. It is due to the gravitational interaction with neighbouring structures. This high-density cutoff corresponds to the dot-dashed curve shown in the diagrams in Fig. 1 which runs from up to and which crosses the cooling curve.
Substituting Eq. (2) into the relation obtained from Eq. (A.8) we get a curve . This corresponds in Fig. 1 to the short branch which runs upward from the curve at . Indeed, we note that for the gas is located close to non-linear structures so that local shock-heating must be taken into account. However, for large densities we have since Eq. (2) yields . In this case external heating plays no role. This is why we only plot the curve up to the characteristic density contrast where . At higher densities the threshold becomes irrelevant. In fact, we see in Fig. 1 that the curve due to external heating plays no role since at moderate densities it is repelled to quasi-linear scales (see Sect. 2.2.3).
Then, the region to the upper-right of the curves and in the plane shown in Fig. 1 corresponds to rare high-density fluctuations which are located in the tail of the pdf . Therefore, there should be very few particles beyond these lines. Thus, this defines a second exclusion region.
Note that at z=0 this constraint yields an upper bound K keV. Of course, there exist some halos with a larger temperature: massive X-ray clusters. However, these are rare objects which only contain a small fraction of the baryonic matter content of the universe (typically ) and they indeed correspond to the high-mass tail of the mass function. This is obviously consistent with the description of the baryonic matter which is worked out in this article, see Sect. 2.7.
As shown in Hui & Gnedin (1997) the low-density photo-ionized IGM exhibits such an equation of State as the gas follows a specific relation
with a rather small scatter. This was derived in Hui & Gnedin (1997) from the Zel'dovich approximation (Zel'dovich 1970) which applies up to the moderately non-linear regime (
). Here we reconsider this problem and we show that this Equation of State is rather robust with respect to the past history of the gas and applies independently of the validity of the Zel'dovich approximation. First, we assume photo-ionization equilibrium (we restrict ourselves to
after reionization) and we only take into account hydrogen. Therefore, the ionization equilibrium reads:
Note that Eq. (21) is independent of the normalization of the UV flux . Indeed, the efficiency of radiative heating is proportional to but the density of neutral hydrogen scales as (at ionization equilibrium for almost fully ionized gas) so that cancels out. Therefore, the result (21) is quite robust since it does not depend on the value of the UV flux. In particular, Eq. (21) still holds even if the UV background is inhomogeneous: we only need to assume local ionization equilibrium. This explains why there is only a very small scatter around the equation of State (21) since the actual physical conditions within these small clouds are almost independent of the actual history of each fluid element (i.e. the evolution of its density and local UV flux). This indeed agrees with the results of numerical simulations (e.g., Dave et al. 1999).
We display in Fig. 1 the equation of State (21) as the solid line which runs through K at . Beyond the overdensity defined in Eq. (22) below, we plot this line as a dashed-line, until it enters the cooling region described in Sect. 2.2.1. Indeed, as we explained in Sect. 2.2.2 at large densities the "virial temperature'' becomes larger than the temperature due to some external energy source (here photo-ionization heating by the UV background radiation). This means that for these regions, which have already reached the non-linear regime as (see Eq. (22)), shock-heating due to the gravitational dynamics can no longer be neglected and it actually becomes dominant. Therefore, for high densities with the gas should no longer fall onto the curve (21). Nevertheless, since shock-heating can only increase the temperature of the gas the relation (21) now provides a lower bound to the temperature T. Hence, the region below the curve (21) is excluded in the diagram. This holds until we enter the cooling region discussed in Sect. 2.2.1. On the low-density side, as described in Sect. 2.2.4 we are constrained by the low-density cutoff .
Therefore, we predict that we should have two phases for the IGM. A first "cool'' phase is described by the Equation of State (21) with intermediate densities
It is photo-ionized gas heated up to
K by the background UV flux. This corresponds to the moderate density fluctuations which form the Lyman-
forest. A second "warm'' phase is made of higher-density regions which have already experienced some shock-heating due to the building of gravitational structures but which have not entered the cooling region yet. These particles should be located in the
plane above the curve (21) and within the constraints described in the previous sections.
|Figure 1: The phase-diagram of the IGM from z=3 down to z=0. The straight solid line shows the Equation of State of the "cool'' IGM (Lyman- forest). The curved solid line shows, as a mean trend, the "Equation of State'' of the "warm'' IGM which is shock-heated through the building of non-linear gravitational structures. The dashed curves draw exclusion regions around the allowed domain for this "warm'' IGM. In counter-clockwise order, starting from the right side, they correspond to 1) a fast-cooling region where the gas cannot remain over a Hubble time, 2) a high-density and high-temperature domain within the exponential tail of the pdf which only contains very rare massive halos, 3) large linear scales where gravitational shock-heating has not appeared yet, 4) a low-density region within the tail of the pdf which is associated with very rare voids and 5) a lower-bound for the temperature set by radiative heating from the UV background. The vertical dotted line is the density threshold of just-virialized halos. The points show the results of numerical simulations from Dave et al. (1999) (Fig. 11).
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However, the distribution of matter obtained in Dave et al. (1999) from numerical simulations does not fill entirely this allowed region (note indeed that this is not implied by our previous considerations) and it is not uniform. Hence it would be convenient to derive a curve in the
plane which would describe the "mean'' behaviour of this warm phase of the IGM. To this purpose, we first compute the gas temperature associated with a given halo of mass M which forms at redshift z. As noticed above, this does not apply to low-density regions with
but this model should provide a useful estimate for the higher density regions which actually contain most of the mass. Gravitational heating is effective in the collapsing phase of the halo which may be set at
to be of the order of a few units. We set it to
Now, seeking an average location of these warm spots, we estimate the typical mass M0(z) which collapses at each redshift z by:
The mass M0 and the scale R0 characterize the non-linear structures which just turned non-linear. For instance, at z=0 we have Mpc. At virialization when the density contrast reaches the threshold this yields a radius Mpc as for large clusters. On the other hand, the shocked regions, at temperature , correspond to much smaller scales. Their size R can be estimated by using approximate hydrostatic equilibrium within the filaments along the transverse direction. As a consequence, the Eq. (1) still holds, up to factors of order unity, if R is taken to be the thickness of the filament. This is a natural result since the inflow and the shock actually are at the origin of the filament. Also, the thickness R of the filaments turns out, as a rule, to be much smaller than R0. For instance, at z=0 we find it is of order of a few hundred kpc, see Sect. 4 below. Note that the same processes are at work for the dark matter density field, except that shocked regions now correspond to areas where shell-crossing governs the dynamics and builds a large velocity dispersion. Therefore, we can again assume that the gas follows the dark matter density field and the gravitational potential is dominated by the dark matter. Our findings agree with the results of numerical simulations which show that the shocked regions are associated with the filaments which appear in the dark matter density field. Thus, as shown by the simulation map in Fig. 3 in Dave et al. (2001), the "warm'' IGM is an intricate network of filaments which extends over a few Mpc (the scale R0) but the thickness of the filaments (the scale R) is much smaller than this global scale. In Sect. 4 we will see in addition that the scale R of the warm regions shows up as a small-scale cutoff for the "warm'' IGM two-point correlation function. We obtain a length of a few hundred kpc at z=0 which is much smaller than the few Mpc which characterize the underlying global structure which is collapsing. This also agrees with the numerical simulations of Dave et al. (2001).
Therefore, the "warm'' IGM component is described by the curve (23) in the plane, together with Eq. (1) which yields the local size R of the clouds (or the thickness of the filaments). Our model relies on three major points: 1) there is a critical overdensity associated to the "warm'' IGM which reflects the turn-around of patches just going non-linear, 2) shock-heating locally transforms into heat the collective kinetic energy and 3) the scale of the shocked regions is given by local approximate hydrostatic equilibrium at the latter temperature. This yields 1) the locus in parameter space where the "warm'' IGM appears, 2) the size of the non-linear network of shocked regions (filament network) which is the scale R0 just turning non-linear and 3) the thickness R of the filaments which appears to be much smaller than the previous scale.
We also see that at the beginning of the cloud collapse the shocked regions are predicted to have low temperatures correlated with a rather small spatial extension. As the overall collapse proceeds their temperature and their size increase and the highest temperatures are reached when the cloud is being virialized with scales of the order of the radius of the halo. This is more or less what could be expected to occur. Thus, we model the mean trend of the "warm'' IGM by Eq. (23) which appears as a curved solid line labeled " '' in Fig. 1. It starts from the end-point of the "Cool'' IGM (at ) and it goes towards larger densities and temperatures until it reaches the cooling region (at ) or the density threshold (at low z). Indeed, beyond this point we consider that we have cooled objects (galaxies) or hot virialized halos (clusters) which are not part of the "warm'' IGM.
As seen from Fig. 1, at low z gravitational heating rapidly dominates over the UV heating, for densities close to and temperatures around . On the other hand, the "warm'' IGM clouds are not the same as the UV heated objects which form the "cool'' component. Note that the curve (23) used with the mass (25) only reflects the average trend. Indeed, the stochastic character of the dark matter density field leads to a broad variety of masses which are just collapsing, and hence of trajectories. Moreover, the local properties of the shocks also provide for some additional scatter. This induces a (rather large) dispersion of the points of the "warm'' IGM in the plane, which agrees with numerical simulations (e.g., Dave et al. 2001). However the "warm'' IGM should remain enclosed within the allowed region defined by the constraints discussed in the previous sections. Indeed, we must point out that the validity of Eq. (1) ensures that the constraints obtained in Sects. 2.2.2-2.2.4 still hold.
In the previous sections we have shown that one can distinguish two components in the IGM and we have determined their location in the phase-diagram. Our results are displayed in Fig. 1.
Firstly, there is a "cool'' IGM phase ( K) which corresponds to the Lyman- forest. These are moderate density fluctuations ( ) of photo-ionized gas. They are described by the Equation of State (21) which arises from the heating of the gas by the UV background and the cooling due to the expansion (i.e. pressure work). Thus, this component lies on a well-defined curve in the plane. This curve is bounded towards low densities by the cutoff of the pdf , which expresses the fact that the dark matter density field arising from Gaussian initial conditions exhibits a finite range of densities which occur with a significant probability. Note that this lower bound indeed agrees with the points obtained from numerical simulations shown in Fig. 1. On the other hand, the high-density bound is due to gravitational shock-heating which becomes the dominant energy source for dense regions.
Secondly, there is a "warm'' IGM phase ( K) which describes the gas heated by shocks arising from the gravitational energy of just collapsing objects. Because of the stochastic character of this energy source there is a broad scatter for this component around the "Equation of State'' we have derived. This gas is restricted to a specific allowed region in the phase-diagram. This expresses cooling and heating constraints as well as the properties of the underlying dark matter density field. We nevertheless obtained a curve, Eq. (23), which follows the mean trend of this "warm'' phase. Its low density bound is set by the transition near with the "cool'' IGM phase dominated by radiative heating from the UV background. The high-density bound is given by the intersection with the cooling curve (where bremsstrahlung cooling becomes dominant) or the density threshold (beyond this point we have groups or clusters of galaxies). Note that these results are consistent with the calculations of (Nath & Silk 2001) based on the Zel'dovich approximation.
We can note that our results shown in Fig. 1 agree reasonably well with the outcome of numerical simulations as displayed in Fig. 11 in Dave et al. (1999) (also shown by the points in our Fig. 1) and Fig. 9 in Springel & Hernquist (2001). There is a small offset at low redshift for the normalization of the equation of state (21) of the "cool'' IGM and for the cooling region defined in Eq. (4). Note that the latter could be remedied by adjusting the ratio which we simply set equal to unity in Eq. (4). Similarly, we could obtain a better fit to the numerical results for the "warm'' IGM by tuning the r.h.s. in Eq. (25) which defines the mass M0. However, our goal is not to get the best fit to a specific numerical simulation (which would be of little value) but to explain the physics of the IGM. Moreover, Fig. 1 shows that the simple procedure detailed in the previous sections already provides a good qualitative and quantitative description which should be sufficient for most purposes. Besides, as explained above it should be quite robust. In particular, it could be readily used with any cosmological parameters.
|Figure 2: The phase-diagram of cosmological baryons at z=3 and z=0. As in Fig. 1 the curves at are the equations of state of the "cool'' and "warm'' phases of the IGM. The dashed curve with a vertical part at and a branch towards higher and T corresponds to galaxies. The two upper parts of the vertical line at are groups (solid line at z=0) and rare clusters (dotted line). The upper right dot-dashed line corresponds to the cool cores of groups and clusters where cooling has had time to develop.
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Let us first consider the right panel, obtained for z=0. The dashed line at is the "cool'' IGM while the solid curve at is the "warm'' IGM. The equations of state of these two components were derived in the previous sections and these curves are identical to those shown in Fig. 1. For clarity we do not plot in Fig. 2 the boundaries discussed in Fig. 1 which constrain the large scatter of the "warm'' IGM phase.
Next, the vertical line at shows the overall density contrast associated with just-virialized objects. We divide this line into three parts.
The first is in the low temperatures K region. These objects with are located within the cooling region shown in Fig. 1. Hence their cooling time is smaller than the Hubble time , see Eq. (4). Therefore, within these halos the gas undergoes a very efficient cooling which leads to the formation of stars. As a consequence, these objects are small galaxies which are just being formed. This part of the vertical line which is enclosed within the cooling region is shown by the lower vertical dashed line in Fig. 2.
The remaining high temperature region may still be subdivided. The halos at which have a higher temperature do not cool over a Hubble time since they obey (except in their center). Hence they correspond to groups or clusters which still contain hot gas which can be observed through its X-ray emission. One may divide this part into two components: groups versus clusters. We identify groups with the low temperature halos located below the cutoff shown in Fig. 1. Therefore, they correspond to the typical just-virialized halos which form at z=0. On the other hand, we identify clusters with the high temperature halos located above the cutoff . Hence they are rare massive objects which probe the high-density tail of the pdf . In other words, they correspond to the high-mass tail of the mass function. Thus, our distinction between groups and clusters is only based on the abundance of these objects (whether they correspond to rare or typical density fluctuations). However, we shall explain in a future paper (Valageas et al. 2002) that this subdivision also marks the boundary between objects which, depending on the depth of their gravitational potential, are affected or not by a preheating of the gas through the gravitational processes which yield the IGM. Groups (resp. clusters) are shown in Fig. 2 by the vertical solid (resp. dotted) line. Thus, these three parts of the vertical line at describe the location of small galaxies, groups and clusters.
For completeness, we must point out that the lower part of the line at
K does not give all galaxies. Indeed, it is clear that there are some galaxies with a higher temperature. As discussed in Valageas & Schaeffer (1999) these objects must still satisfy a somewhat stronger cooling constraint:
Finally, we note that the upper part of the cooling curve (4) shown in Fig. 1 at describes the cool cores of groups and clusters. Indeed, it corresponds to high-density regions which have had time to cool today. This yields the core radius of present groups and clusters where cooling has just had time to come into play and possibly induce cooling flows. For instance, at T = 107 K we obtain a radius kpc, which agrees with observations, while the virial radius is Mpc. We display this curve as the dot-dashed line in Fig. 2. Note that this assumes that baryons follow the dark matter. However, at low z this is not necessarily the case since some "preheating'' may modify the physics of the gas in low-temperature clusters (i.e. groups), see for instance Valageas & Silk (1999) and Valageas & Schaeffer (2000). This is not important for our purpose here which is mainly to describe the physics of the IGM. A specific study devoted to this problem and the entropy of the gas in the light of the results described in this paper is presented in Valageas et al. (2002).
Note that for these collapsed objects (galaxies, groups, clusters and cool cores) the density contrast and the temperature T shown in Fig. 2 only refer to the mean density contrast and virial temperature of the halo over the relevant radius R (which may be different from the virial radius for galaxies and cool cores). At the center of the halo the gas density is larger. Moreover, for galaxies and cool cores the gas temperature can be significantly smaller since cooling is very efficient (and we should take into account feedback from supernovae). Therefore, contrary to the IGM where the location in the phase-diagram directly gave the properties of the gas, here the plane only shows the overall properties of the dark matter halos associated with each object. Finally, we must point out that these classes of objects are not exclusive of each other. Indeed, cooled cores are obviously embedded within groups and clusters while galaxies can be found within filaments, groups and clusters. Therefore, some of the mass associated with galaxies or cool cores is counted within the matter attached to filaments, groups or clusters (this can be handled using the methods of Valageas & Schaeffer 1997, see Valageas & Schaeffer 1999 for galaxies and Valageas & Schaeffer 2000 for clusters).
Thus, as explained above the diagram shown in Fig. 2 describes the physics of cosmological baryons and their distribution between different phases.
The redshift evolution up to z=3 can be easily derived from Fig. 1 and for illustration we show in the left panel in Fig. 2 our result at z=3. The location of the various curves evolves as explained in Sect. 2.6 but we can also note some qualitative changes. Firstly, with our definitions we see that there are no more groups. This means that the typical objects which collapse at z=3 form galaxies as they exhibit efficient cooling. To be more precise, one can still find clusters along , at high temperatures where cooling is not very efficient. However, they are located within the far tail of the mass function and they correspond to extremely rare events. Secondly, we note that the curve which describes the mean "equation of state'' of the "warm'' IGM stops below . Indeed, as seen in Fig. 1 its high-density end-point is now given by the intersection with the cooling region. This is merely another consequence of the fact that we typically form galaxies and not groups (this feature coincides with the "disappearance'' of groups). At an even higher redshift the "warm'' IGM component almost disappears as the equation of state of the "cool'' IGM extends up to the cooling region. However there still remains a "warm'' phase because the allowed region in the plane has not vanished. At these high redshifts most of the mass is within a roughly uniform "cool'' phase. We still have some rare collapsed halos which correspond to galaxies and some non-linear structures where gravitationally induced shocks heat the gas but these latter regions are severely restricted by the high efficiency of cooling processes.
Let us first discuss the "cool'' IGM phase. As described in Sect. 2.3
this corresponds to the Lyman-
forest, that is moderate density
fluctuations governed by the ionization and the heating due to the UV
background. Since its temperature is not zero this gas probes the dark matter
density field over the scale R defined in Eq. (2), which describes
the length-scale over which pressure can homogenize the baryonic matter
distribution. Note that this length depends on the temperature since
Then, we wish to express the fraction of matter within the
"cool'' IGM in terms of the pdf
over the scales R associated
with these clouds. To do so, we first note that these scales are within the
non-linear regime as shown by the curve
in Fig. 1
which marks the transition to the linear regime. Then, as discussed in
Valageas & Schaeffer (1997), in the highly non-linear regime the pdf
As described in Appendix A.1 our approximation for the pdf applies both to the linear and non-linear regimes. Moreover, this formalism does not imply that these clouds are virialized objects which have reached an equilibrium state. The "extended'' function h(x;R,z) defined in Eq. (30) depends on scale and redshift. It is merely another way to write the pdf
(which goes over to the one relevant for the highly non-linear regime in this limit and is thus used by analogy). An important point of Eq. (29) is that it yields the mass fraction associated with various objects from the pdf
of the non-linear density field. This is a strong advantage since, as described in Valageas & Schaeffer (1997), one can show that in the highly non-linear regime, assuming the stable-clustering ansatz is valid, the cloud-in-cloud problem can be handled in a satisfactory way. We can expect this property to extend to the regime we consider here. Note also that Eq. (29) allows us to count objects which are not necessarily defined by a constant density contrast.
|Figure 3: The distribution of baryonic matter. The dashed line which increases with redshift is the fraction of matter within the "cool'' IGM phase, as computed from Eqs. (29) and (21). The dot-dashed line gives the mass fraction associated with the "warm'' IGM component. The solid line is the fraction of matter within collapsed objects. The filled symbols show the results of the numerical simulations from Dave et al. (2001) (panel D2 in Fig. 1) for the "cool'' IGM (squares), the "warm'' IGM (triangles) and condensed gas (circles). The empty squares show the results of the simulation presented in Dave et al. (1999) for the "cool'' IGM phase (Fig. 12).
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We show our results in Fig. 3. The Lyman- forest corresponds to the dashed line which grows at larger redshift. Indeed, at higher z gravitational clustering was less advanced so that a smaller fraction of matter had been shock-heated to high temperatures or embedded within collapsed objects through the building of large scale structures. Note that the fractions of matter within collapsed halos and the "warm'' IGM are of the same order. Indeed, both components are related to the formation of non-linear gravitational structures. As noticed in Sect. 2.7 we obtain at z > 4. This actually means that the mass fraction within the "warm'' IGM is very small and within the inaccuracy of our computation.
We can check that our results agree with the outcome of the numerical simulations described in Dave et al. (1999) and Dave et al. (2001). The mass fractions we obtain for the "warm'' IGM and virialized objects are also in reasonable agreement with simulations although it is difficult to make a detailed comparison. Indeed, our separation between different components is not exactly the same as in the simulations and the predictions of the various simulations exhibit a significant scatter as shown by the comparison between the filled and empty squares in Fig. 3 (see also the various panels in Fig. 1 in Dave et al. 2001).
|Figure 4: The temperature distribution of the "warm'' IGM component at redshifts z=2 and z=0 (normalized to unity). The mean temperature grows with time as larger scale structures form.
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We display our result normalized to unity in Fig. 4. Of course, we recover the fact that the mean temperature increases with time as larger non-linear structures build up. This could also be seen from Fig. 1. In particular, the mean trend curve of the "warm'' IGM in Fig. 1 at z=0 is very steep because the characteristic temperature of the non-linear structures which collapse at that time is of order K which is much larger than the temperature reached by radiative heating from the UV background K. Therefore, gravitational shock-heating easily heats the gas up to K. However, note that there is still a significant fraction of the matter at lower temperatures T < 106 K, in agreement with Dave et al. (2001). Although the procedure (32) is only a simple approximation, the comparison of Fig. 4 with Fig. 5 in Dave et al. (2001) shows that it captures the main trend. Of course, as explained in Sect. 2.4 the "warm'' IGM shows a broad scatter in the plane so that the actual boundaries of the temperature distribution are not as sharp as in Fig. 4. On the other hand, note that in our description we also have some matter at T>107 K but we associate this gas with massive X-ray clusters. In any case, our result displayed in Fig. 4 appears to provide a reasonable description of the "warm'' IGM properties.
|Figure 5: The "warm'' IGM two-point correlation at redshift z=0 (dashed line). The solid line shows the two-point correlation of the dark matter density field.
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On the other hand, at small scales kpc the two-point correlation function flattens and it reaches a finite limit at r=0. This expresses the fact that the non-zero temperature of the gas reached by radiative heating and shock-heating (T > 104 K) gives rise to pressure effects which homogenize the baryonic distribution over some scale R. A lower limit to this length R is set by the scale associated with the critical point in the phase-diagram. Indeed, as shown by the curved solid line in Fig. 1, at redshift z=0 the length scales associated with the "warm'' IGM run from up to scales of order R0 which are turning non-linear. This sets the knee of the two-point correlation at kpc at z=0. As noticed in Sect. 2.4, the result displayed in Fig. 5 shows that the characteristic thickness of the filaments which build the "warm'' IGM is of order 300 kpc at z=0. We find this, at a given redshift, to be indeed much smaller than the scale which is just turning non-linear, in agreement with numerical simulations.
The value at r=0 yields the clumping factor
which we define by:
Therefore, we compute here the soft X-ray background due to the diffuse "warm'' IGM phase, in the 0.1-0.4 keV band. Note that the X-ray background due to resolved sources (AGN, clusters, cooling galaxies) was already studied in Valageas & Schaeffer (2000) from an analytic model similar to the one used here to describe the underlying dark matter density field. The X-ray flux
within this frequency band can be written:
Firstly, we have distinguished a "cool'' IGM phase corresponding to the Lyman- forest. It follows a well-defined equation of state on the plane. We have shown that the properties of this gas do not depend much on its previous history because of the form of the recombination coefficient. This explains why the scatter of this component obtained in numerical simulations is quite small. The fraction of baryons in this components, as well as its evolution with redshift is seen to be consistent with earlier modelling of the frequency of occurrence of the Lyman- lines. We however obtain (consistently with the numerical simulations) 38% of the baryons in the cool IGM at z=0. This may be a little high with a mass fraction closer to 30% being consistent with observations. This must be considered as being within the error bars of our analytical calculation (and the error bars of the simulations!).
Secondly, we find that a "warm'' IGM phase is formed by the gas which has been shock-heated to larger temperatures K as non-linear gravitational structures appear. The dependence on the stochastic gravitational potential entails a broad scatter in the plane. However, we have explained that robust constraints only allow a closed region in the phase-diagram for this component. We have also defined a simple curve which represents its mean behaviour and plays the role of the "warm'' IGM Equation of State. The latter is found to be consistent with the outcome of numerical simulations (Dave et al. 2001), but differs from the simple power-law fit of a mean curve done there. This is because its locus is based on physical considerations, which we have thoroughly justified. The "warm'' IGM is seen to be due to large structures on their way of collapsing under the action of gravity. The quite simple picture behind this model is that the collapse of an object turning non-linear (that is with average overdensity above 5) induces shocks which heat patches of much smaller size, to a temperature of the order of the locally available kinetic energy of the collapse, creating local conditions within hot spots close to hydrostatic equilibrium. We are able to estimate this size, as well as the temperature of the latter, and to show they are in agreement with the simulations. Definitely, these objects are not just Lyman- absorbers at larger densities. The hot gas which lies within clusters and galactic halos, on the other hand, corresponds to the high-density "continuation'' (e.g., ) of the "warm'' IGM curve, in a region of phase-space where virialization insures that all the available gas has been gravitationally heated.
Next, we find two quite different regimes for the "warm'' IGM. At high redshift, some of the "warm'' component enters the cooling region of the plane: this gives rise to galactic disks and stars. Some of the baryonic matter, also, should lie on the low temperature branch of this cooling region, in agreement with the results of numerical simulations (Dave et al. 1999). At low redshift, the "warm'' IGM no longer goes into the cooled region, and it provides the origin of the hot gas in clusters. The evolution with redshift of the baryon phase-diagram provides a natural confirmation that at high z collapsed halos form galaxies while at low redshift they build groups or clusters. The warm phase occupies 24% of the baryon fraction at z=0 in our analytic model.
Then, we have checked that our results for the fraction of matter enclosed within the various phases and the two-point correlation function of the "warm'' IGM component agree with numerical simulations (e.g., Dave et al. 1999 and Dave et al. 2001). This confirms the validity of our analysis. Note however that the latter relies on simple physical considerations and it is independent of the findings of the former. Besides, the soft X-ray background due to the "warm'' IGM is consistent with the upper bound set by observations. Our prediction actually is not far from the observational limit: this offers the prospect of measuring this X-ray emission in future observations.
We predict that approximately 60% of the baryons are accounted for at z=0 in the cool/warm phases of the intergalactic medium. The remaining baryons (i.e. 40%) are in collapsed structures. With a total of , this amonts to . The observed luminous component of the baryons (stars, remnants and gas) is estimated to be about , as noted in the introduction, and the hot gas in clusters represents . Therefore, there remains an "unidentified'' or dark baryon component, . This is at least twice the stellar-like component but only about 1/4 of the "identified'' baryons (IGM, stellar components and very hot gas). In our model this dark baryonic matter corresponds to (cool?) gas within galactic halos and groups, which has not been observed yet.
We thank R. Dave for providing us with some of the data published in Dave et al. 1999.