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Astron. Astrophys. 317, 1-13 (1997)

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4. What can we learn from the redshift distribution?

As we have already mentioned, structure formation occurs at different rate according to the present mean density of the universe. Evolutionary properties are therefore expected to differ with [FORMULA]. Such a difference is expected to appear from various observations, and in particular from the redshift distribution that we will now investigate in detail. In order to put in light this difference, we will compare the predictions given by two different models which fit equally well the data at zero redshift. These models are those which give the best-fit to HA in the case [FORMULA] ( n= -1.85 and b=1.65 , see OBB) and in the case [FORMULA] (see Table 1). Hereafter, these two models are referred to as the critical model and the open ( [FORMULA] ) model respectively.

[FIGURE]Fig. 2. The redshift distribution of clusters in all the sky, in the case of the critical universe. Two models are represented: a model with the spectral index, n, equal to -2 (thick lines), and a model with n equal to -1 (thin lines). In both cases, the bias parameter b is equal to 1.7 . The different lines correspond to different apparent temperatures: [FORMULA] (dashed line), [FORMULA] (solid line) and [FORMULA] (dotted-dashed line)

4.1. Insensitivity to the spectrum

The evolution of the X-ray cluster population has been discussed by Kaiser (1986) in his pioneering work. In the case [FORMULA] and using scaling arguments, he concluded that the density evolution of X-ray selected clusters should reflect the index of the initial power spectrum. This can be seen in the following way. For power-law initial fluctuation spectra, Kaiser (1986) showed that the evolution of the statistical properties of clusters in the critical universe is self-similar. This means that, by applying an appropriate scaling to some cluster property, for instance the temperature, one can recover the distribution function at any epoch from the observed distribution function at z=0 . The only physical scale in the problem is the characteristic mass which is going non-linear and which varies as [FORMULA]. The higher the value of n, the slower the decrease of the characteristic mass. Kaiser (1986) used this argument to claim that the evolution of the comoving density of galaxy clusters depends on the value of the initial power spectrum index. This has been illustrated by Pierre (1991), who applying the above scaling to the local ( z = 0 ) temperature distribution function and converting it into a luminosity function, found a conspicuous difference in the redshift distribution between models with values n=-1 and n=-2 of the power spectrum index. However, the above arguments contain a loophole and the implicit assumptions which have been made are not self-consistent. Let us clarify this point by explicitly using the general form of the mass function. As we mentioned in Sect. 1, the mass function can be written as:

[EQUATION]

[FORMULA] being [FORMULA]. Numerical simulations suggest that the function F is well fitted by [FORMULA] ( [FORMULA] ), independently of the power spectrum. Therefore, if the present-day number is fitted on some mass scale, the amplitude of the fluctuations, [FORMULA] is fixed in a way which does not depend on the power spectrum (neglecting the change in the coefficient [FORMULA] ). Since [FORMULA] in the critical universe, the number density can be evaluated at any redshift:

[EQUATION]

This clearly does not depend on the power spectrum. For theoretical models normalized to the scale [FORMULA] which corresponds to clusters of [FORMULA] 4.5 keV, the redshift distribution of [FORMULA] 4.5 keV apparent temperature clusters (the apparent temperature corresponds to the same mass whatever the redshift) will be the same for different spectra. If the selection procedure is based on some other quantity (like the luminosity), the same argument applies provided that the relation of the quantity to the mass is independent of the spectrum. Actually, only measurements on two different mass scales can lead to information on the power spectrum. When one compares the redshift distribution for different masses, there is some difference because one is sampling different parts of the function F, but this difference is weak. Furthermore, such difference will also occur by changing the normalization. This is illustrated in Fig. 2. Of course, the total number of objects is not the same for the two power spectra, reflecting the fact that the local properties are different. We conclude that the redshift distribution does not carry useful information on the power spectrum. On the contrary, the redshift distribution depends on the cosmological parameters ( [FORMULA], [FORMULA] ). In original Kaiser's scaling argument (1986) it was assumed that whatever the power spectrum is, the local observed distribution function is reproduced. This is of course not true, since different spectra should lead to different distribution functions which cannot fit simultaneously all the observed local distributions.

4.2. The redshift distribution of galaxy clusters according to the value of [FORMULA]

In this section we investigate in more detail the dependence on [FORMULA] of the characteristics of the X-ray cluster population at high redshifts. We mainly focus on whether this could provide an useful test of the mean density of the universe and distinguish between open and critical density models. In particular, we predict the redshift distribution of galaxy clusters expected to be observed by the ROSAT satellite in both models. The ROSAT satellite performed the first all-sky survey using an imaging X-ray telescope. It covers the energy band between 0.1 and 2.4 keV and its detection limit lies close to [FORMULA] for most of the sky (Böhringer et al., 1992). Around the ecliptic poles and for a region which spans roughly a 10 deg radius area, the flux limit is close to [FORMULA]. Nearly 50000 sources have already been detected within the all-sky survey. A first identification of the galaxy clusters within the southern sky is based on the comparison of the source list with an optical galaxy catalogue, as well as on the extent of the X-ray sources (Guzzo et al., 1995). However, such criteria will probably introduce a bias towards nearby clusters since optical catalogues are complete only at low redshifts and most of the clusters have angular core radii which are below the resolution of the instrument and are therefore classified as point sources and missed as clusters.

Let us now examine the expected redshift distribution of the [FORMULA] and [FORMULA] models within the above described flux limited surveys, in the case where the [FORMULA] correlation does not undergo any evolution. In this case, the luminosity is related to the mass according to the following scaling:

[EQUATION]

Figure 3 shows the expected number of clusters detected per redshift bin in the All-Sky survey (Fig. 3a) and in the north ecliptic pole region (NEP) (Fig. 3b) for both the critical (shaded areas) and the open (solid lines) models. The total number of clusters at the flux limit of the All-Sky survey is different according to the model: while 7200 clusters are expected in the open case, 5300 are expected in the critical case. At low redshifts, below [FORMULA], the cluster distribution is identical in both models. This result was expected since the models were normalized to match the local observed temperature distribution function. In the case of the critical model, strong evolution appears at a redshift as low as 0.2 resulting in a net decrease, and there is no cluster expected beyond [FORMULA]. In the open model, clusters form earlier and their distribution function remains almost constant with redshift. Consequently, the expected number does not decrease as fast as in the critical density case: more than 100 clusters lie beyond redshift 0.4 . However, the high redshift clusters still represent a small fraction of the total number of clusters expected at this flux limit: assuming the absence of any bias in the selection procedure, optical spectroscopic follow up of at least 10% of the survey should be achieved in order to probe the difference significantly. In Fig. 4, we show the expected redshift distribution in the NEP for the two models we have investigated. We recall that the flux limit in the NEP region is close to [FORMULA]. At this sensitivity, clusters are actually probed up to higher redshifts and the redshift distribution is significantly different. This difference mainly occurs because of the larger number of high temperature clusters at high redshifts in the case of the open model. In this model, half of the sources are expected beyond [FORMULA], while the median redshift is only z=0.14 in the case of the critical model. Although the total number of expected clusters is smaller in the NEP region than in the All-Sky survey (the NEP survey is expected to contain less than 15% of the All-Sky survey), our analysis reveals the fact that the redshift distribution of clusters in the NEP offers a better way to differenciate a low density universe from the critical universe. Nevertheless, hotter clusters are more luminous and their detectability should be less affected by the survey flux limit. The redshift distribution of the hottest clusters within a flux limited sample can therefore provide the adequate information. This is illustrated in Fig. 5 showing the redshift distribution of clusters whose apparent temperature is 6 keV. The redshift distribution is given for various flux limits, in the case of the critical model. Clearly when the flux limit is low enough, the redshift distribution should depend only on the density parameter. Therefore, the temperature information allows one to overcome the high flux limit problem. Let us now examine this point quantitatively. As an illustration, we focus our predictions for samples whose characteristics are similar to those of ROSAT. For clusters which apparent temperature is higher than [FORMULA] (although the temperature information cannot be obtained over the All-Sky survey, this temperature limit ensures one that the cutoff of these clusters X-ray spectra is above 2.4 keV which is the higher energy limit of the ROSAT window), the redshift distribution is almost identical for the two models as we can see in Fig. 3a, and is not significantly different than in the case where only a flux selection is used. This is due to the fact that the distribution is dominated by clusters whose temperature is close to [FORMULA]: again, these clusters, owing to their low luminosity are detected essentially at low redshifts. However, there are more clusters of this apparent temperature range in the open model than in the critical model (5411 and 2628 respectively). In Fig. 3b, we give the redshift distribution of clusters with apparent temperatures higher than [FORMULA]. The median redshift is [FORMULA] in the case of the open model and [FORMULA] in the case of the critical model. Clusters of [FORMULA] are enough luminous to be detected even at high redshifts: in the case of the open model there are almost 600 clusters which could be detected between z=0.25 and z=0.35 . Only 100 are expected in the case of the critical model. This large difference can easily be detected with few optical spectroscopic measurements. Of course, the difference in the redshift distribution increases with temperature: in Fig. 3c, we show the case of clusters with [FORMULA]. These objects lie on the exponential cutoff of the temperature distribution function and the median redshifts are noticeably different for the two models: 0.37 and 0.16 for the open model and the critical model respectively. The redshift distribution in this case is a very powerful test, even at a sensitivity limit of [FORMULA]. However, a complete optical identification of the ROSAT sources is needed - at least in a given region of the sky - in order to make the test feasible. Moreover, reliable data on the temperature are also necessary. Although this information can probably not be obtained over the All-Sky survey, it is conceivable that a satellite like AXAF or XMM will make temperature measurements on a subsample of ROSAT selected clusters in order to perform the test we propose.

[FIGURE] Fig. 3. Expected redshift distribution of clusters in the All-Sky survey of the ROSAT satellite in the case of the critical universe (shaded area) and in the case of the open universe (thick solid line). The critical model corresponds to n=-1.8 and b=1.6 , and the open model corresponds to [FORMULA], n=-1.2 and b=0.7 . In both cases, the observed local [FORMULA] relation is used ( 3), with the assumption that it does not vary with redshift. a The All-Sky survey: the flux limit is close to [FORMULA]. b The North Ecliptic Pole survey: the flux limit is close to [FORMULA]

This analysis shows that the temperature information is critical to actually access to the density parameter, at least for a high flux limited survey. This is because the redshift difference comes essentially through the hottest clusters: while the redshift distribution of low temperature clusters ( [FORMULA] keV) is almost identical in both models, the difference is unambiugous and significant for clusters with T> 5 keV. Since the difference increases as the flux limit lowers, the knowledge of the temperature does not seem crucial in order to distinguish the two models within a low flux limited survey. However, this last result is no more true if we consider the possible evolution of the [FORMULA] relation. Indeed, the redshift distribution within a flux limited survey could be affected by evolution in luminosity. For example, one can imagine that a negative evolution of the [FORMULA] relation within the [FORMULA] model could mimic the NEP redshift distribution of the non-evolving [FORMULA] model. No definitive conclusion could be therefore drawn on the single basis of the redshift distribution and the temperature information is necessary to conclude. On the other hand, the possible evolution of the [FORMULA] correlation is comparable to the effect of a high flux limited survey, and does not affect the redshift distribution of the hottest cluster. It is therefore clear that the determination of the density parameter can be achieved by the redshift distribution test, provided that high redshift temperature measurements exist. The possible evolution increases the complexity of the analysis, but does not represent a intrinsic limitation of the method.

[FIGURE]Fig. 4. The differential redshift distribution of clusters in all the sky in the case of the critical model ( n=-1.8 and b=1.6 ). The different lines correspond to different flux limits: [FORMULA] (solid line) and [FORMULA] (dashed line)

4.3. Comparison with existing data

In the last years there have been several claims for evolution in the X-ray cluster population. Edge et al. (1990) first noticed evolution in the luminosity function. They derived source counts for two subsamples of their data: one with high luminosity ( [FORMULA] ) clusters and the other with low luminosity clusters. They explained the deficit at low fluxes of the high luminosity subsample by a lack of high luminosity clusters at redshifts greater than 0.1 . At the same time Gioia et al. (1990) using 67 X-ray selected clusters from the Einstein Observatory Extended Medium Sensivity Survey (EMSS) derived the luminosity function in three redshift shells: [FORMULA], [FORMULA] and [FORMULA]. They found significantly steeper slopes in the high redshift shells compared to the low redshift shell. More recently, to test the evolution of the X- ray luminosity function at higher redshifts, Castander et al. (1994) analyzed two deep ROSAT pointings containing distant optically-selected clusters of galaxies. Among the five optically rich cluster candidates they selected, they detected X-ray emission only from two clusters. Castander et al. (1994) claimed, that if the probability distribution that a cluster is selected is assumed to be equal to one in the redshift range 0.5<z<0.9 , then their survey volume is [FORMULA], and the luminosity function they derived is in agreement with the decline in the comoving density of clusters with redshift which has been already observed. Using similar search methods, Nichol et al. (1994) reached the same conclusions. Moreover, recent analysis of the EMSS data (Luppino & Gioia, 1995) seems to indicate that the density derived from high redshift X-ray selected clusters is in agreement with the density derived by Castander et al. (1994) and Nichol et al.'s results (1994). In its own, such an evolution favors critical models since no significant evolution of cluster properties is expected in open models.

[FIGURE]Fig. 5. Expected redshift distribution of clusters in the All-Sky survey of the ROSAT satellite in the case of the critical universe (shaded area) and in the case of the open universe (thick solid line). Models are as in Fig. 4. a Clusters whose apparent temperature is higher than 2.4 keV. b Clusters whose apparent temperature is higher than 5 keV. c Clusters which apparent temperature is higher than 8 keV

[FIGURE]Fig. 6. The redshift distribution of clusters, taking into account the sky coverage for different sensitivities in the EMSS. The squares are for the open model and the triangles are for the critical model. The points represent the redshift distribution as given by Gioia & Luppino (1994). The errors are from Poisson statistics. The upper limit at z=0.65 accounts for the fact that no cluster has been observed in the redshift bin centered at this point. a The redshift distribution in the case where the bias due to the cluster extension is neglected. b The redshift distribution including in each redshift bin the mean correction factor for the flux. c The redshift distribution of galaxy clusters in the open model assuming a negative evolution of the [FORMULA] relation (see text)

Recently, Gioia & Luppino (1994) have given the redshift histogram for the entire EMSS cluster sample. We can then compare to these data the expected X-ray cluster redshift distribution in the critical model and in the open model respectively and apply the test we propose. Indeed, this sample should be very well adapted to our test since it is entirely X-ray selected and since almost all the sources in the survey have been identified. The main limitation comes from the fact that the temperature information is not yet available. An other difficulty comes from the extended nature of clusters which prevents the survey to be flux limited for these objects. Indeed, clusters with core radii more extended than the detection cell (2.4' [FORMULA] 2.4') are likely to be missed and there is a strong bias at redshifts smaller than z=0.17 for clusters with core radii greater than 300 kpc. Taking into account the sky coverage for different sensitivities in the EMSS, but neglecting the bias due to the extended nature of clusters, we give in Fig. 6a the theoretical redshift distribution corresponding to the survey. Low redshift bins are not well reproduced by the theoretical predictions. Since our models are normalized to low redshift data, this could not mean that the models are to be rejected, but rather illustrates the effect of the selection function. In order to account for this, we have estimated in each redshift bin the mean correction factor for the flux: this correction factor corresponds to the mean ratio of the actual cluster flux over its flux in the detection cell. The former quantity is given by Gioia & Luppino (1994) for each cluster in the EMSS sample. With the selection function modeled in this way, the redshift distributions are strongly modified at low z. This is illustrated in Fig. 6b. As one can see, the agreement between the theoretical predictions and the observations is quite good. It is likely therefore that our modeling of the selection function is reliable. In addition, since the correction factor is greater at low redshifts, we are confident that our correction for the higher redshift clusters is secure. From these figures it appears that the open model produces a large excess between z=0.2 and z=0.5 , within the redshift interval which contains the bulk of the expected clusters. There are 76 clusters detected within this redshift range, whereas 172 are predicted in the case of the open model. On the contrary, the [FORMULA] model fits the data rather well, although at high redshifts, it slightly underestimates the total expected number.

These results were derived assuming no evolution of the [FORMULA] relation, since as we have already mentioned, there is no sign of significant evolution of this relation out to a redshift of z=0.33 (Henry et al., 1994). However, due to the small photon number of each cluster within Henry et al..'s sample (1994), only the average temperature of clusters was determined. Moreover, the highest redshift bin of their sample is [FORMULA] whereas the redshift distribution of the EMSS sample extends out [FORMULA]. Therefore, there is still room for possible evolution. We have then investigated possible implications of such an evolution by modeling the [FORMULA] relation at high redshifts by a power law with a shape identical to the shape measured at low redshift but with an evolving normalization:

[EQUATION]

where c is the normalisation of the [FORMULA] relation at z=0 (Eq. 3) and [FORMULA] is a free parameter. A chi-square fitting of our models to the EMSS cluster redshift distribution gives [FORMULA] and [FORMULA] for the critical and the open universe respectively (the given errors correspond to the 90% confidence intervals). These parameters were used in Fig. 6c. Obviously, these evolutionary laws allow one to fit the observed distribution and no information can be extracted on the density parameter anymore. In the same way, the X-ray cluster number counts and the contribution to the X-ray background will be identical in open and critical models (OBB).

Let us now investigate whether temperature information of high redshifts clusters would allow us to recover this information. For this, we considered the 8 EMSS clusters lying in the redshift range [FORMULA]. Taking into account the incertainties in the X-ray temperature measurements and the intrinsic dispersion of the [FORMULA] correlation measured at z=0 , would the measurement of these cluster temperatures allow us to distinguish between the critical and the open universe? To answer this question we decided to use a simple bootstrap approach. Although the uncertainties are sometimes quite large, the X-ray temperature information is available for a substantial number of nearby clusters. In order to handle an uniform sample of good quality, we restricted ourselves to clusters for which the temperature were measured with the Ginga satellite, 27 in total (see references quoted in Fig. 2 of Arnaud 1994). The uncertainty in the determination of the [FORMULA] correlation within this sample comes mainly from the intrinsic dispersion of the data points. We wanted to estimate the uncertainty in the normalisation of the [FORMULA] correlation at high redhifts, in the case where only a small number of X-ray temperature measurements were made. This were done by the bootstrap resampling technic. Using this sample at low redshift, we created 2 synthetic catalogues at z=0.49 (this corresponds to the mean redshift of the 8 EMSS cluster we considered): we modified the luminosity of each cluster according to the evolution needed to fit the redshift distribution for the critical and the open model respectively. Within these 2 new catalogues, we picked up all the clusters whose fluxes would have been within [FORMULA] and [FORMULA] at z=0.49 . These values are the fluxes of the [FORMULA] EMSS clusters. There are 10 clusters corresponding to this criterion within each of the 2 synthetic samples. From each of these new reduced samples, we draw 10000 times a subsample of 8 clusters (we use the bootstrap technic, so each cluster could be drawn from the sample as many times as it happens), and we fit the normalisation of the data points by assuming that the shape of the relation is the shape measured at z=0 within the original sample. From our simulations we notice that the individual computed incertaintities are 10 times lower than the dispersion resulting from the resampling. This confirms that the uncertainty comes mainly from the intrinsic dispersion in the [FORMULA] relation. In practice, even with accurate temperature measurements, about ten clusters are necessary to infer a robust conclusion on the normalisation of the [FORMULA] correlation. From our simulations, we estimated that the normalisation is equal to [FORMULA] and [FORMULA] at 90% confidence level, for the critical and the open model respectively. It appears that these two intervals do not recover each other, and therefore temperature measurement of about ten [FORMULA] clusters would allow to easily distinguish between the two models.

The evolution of the temperature distribution of X-ray clusters can clearly allow the determination of the value of the density parameter of the universe. A direct observational proof of this quantity is still far from being available, and therefore the application of our new test in its complete version is still prospective. We have shown that the redshift distribution of X-ray selected clusters from the EMSS survey favors a high value of the density parameter. Because of possible evolution of the [FORMULA] relation this is however a quite uncertain indication. This is illustrated by our analysis in which we allowed for a possible evolution of the temperature-luminosity relation (Eq. 6). In such a case, the two models reproduce equally well the data, provided that the evolutionary law is adequatly chosen. It is interesting to notice that the few observed high z clusters are not well fitted by any of the models. This could be the indication that these clusters are spurious in some way. Quite obviously, as we have already emphasized, in a flux limited survey, possible unknown evolution of X-ray clusters prevent any confident conclusion on the density of the universe to be drawn from their past abundance: in order to carry the test we propose it is necessary to have the temperature information. Now, taking into account the information provided by the EMSS survey, it is only necessary to determine the possible evolution of the [FORMULA] relation: in a low density universe clusters of a given temperature should be substantially fainter in the past. Using a bootstrap procedure, we have investigated the possibility of distinguishing the two models by this method: we show that the difference can actually be probed with around ten clusters for which temperature is measured with a realistic accuracy. This demonstrates the possibility of achieving our test with ASCA or XMM temperature measurments of a limited number of clusters.

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