A&A 436, 411-416 (2005)
DOI: 10.1051/0004-6361:20041851
A. Blanchard^{1} - M. Douspis^{2}
1 - LATT, 14 avenue Edouard Belin, 31400 Toulouse, France
2 -
LATT, 14 avenue Edouard Belin, 31400 Toulouse, France
Received 16 August 2004 / Accepted 11 February 2005
Abstract
The abundance of local clusters is a traditional way to
derive the amplitude of matter fluctuations, commonly specified by
,
but which suffers from a systematic uncertainty arising
from the lack of accurate knowledge of the mass temperature
relation. In the present work, by assuming that the observed
baryon content of clusters is representative of the universe, we
show that the mass temperature relation (M-T) can be specified
for any cosmological model. WMAP constraints on the baryonic
content of the Universe and the
relation allows one
further improvement in tightening this M-T relation. This
approach allows one to remove most of the above uncertainty, and to
provide an estimation of
whose uncertainty is
essentially statistical. The values we obtain are fortuitously
almost independent of the matter density of the Universe
(
)
with an accuracy better than 5%. Quite
remarkably, the amplitude of matter fluctuations can be also
tightly constrained to similar accuracy from existing CMB
measurements alone, once the dark matter content is specified.
However, the amplitude inferred in this way in a concordance model
(
)
is significantly larger than the value derived from
the above method based on X-ray clusters. Such a discrepancy would
almost disappear if the actual optical thickness of the Universe
was 0 but could also be alleviated from more exotic solutions: for
instance the existence of a new non-baryonic light dark component
in the Universe as massive neutrinos, with
.
However, recent other indications of
favor
a high normalization. In this case, the assumption that the
baryonic content observed in clusters actually reflects the
primordial value has to be relaxed: either there exists a large
baryonic dark component in the Universe with
or baryons in clusters have
undergone a large depletion during the formation of these
structures. We concluded that the baryon fraction in clusters is
not representative and therefore that an essential piece of the
physics of baryons in clusters is missing in standard structure
formation scenario.
Key words: cosmology: observations - galaxies: clusters: general - cosmological parameters
The amplitude of matter fluctuations in the present-day universe is
an important quantity of cosmological relevance. The abundance of
clusters is an efficient way to evaluate this quantity, commonly
expressed by ,
the rms amplitude of the matter
fluctuations on the
scale. A statistical precision
of a few % on
is possible from existing samples of X-ray
clusters, but in practice the relation between mass and temperature
is needed for such evaluation:
The determination of from the cluster abundance is a standard procedure that has been used by many authors, leading to somewhat dispersed values. Here we use the Sheth and Tormen (1999) mass function and a sample of X-ray selected local clusters ( erg/s/cm^{2} and , Blanchard et al. 2000, updated from BAX; Sadat et al. 2004). The relation between - is presented in Fig. 1 for a flat model with with some other recent measurements, based on ROSAT samples of X-ray clusters and recent analytical mass functions (Sheth & Tormen 1999; Jenkins et al. 2001). We do not include analyses based on HEAO-1 such as Henry (2004) or using the classical Press and Schechter expression as do Ikebe et al. (2002), or other alternative mass functions (del Popolo 2003). A NFW profile (Navarro et al. 1995) with c = 5 was assumed when necessary. This shows that most of the dispersion among different analyses (which used nearly the same clusters) is due to the different values used for the normalization constant . Most of the remaining differences are due to differences in temperatures used (with or without cooling flow correction, or temperature cuts). Notice that the point presenting the largest deviation (Viana et al. 2002) is based on the luminosity function. We converted their mean luminosity to a mean temperature of 2.68 keV to derive an equivalent .
Clusters are useful cosmological probes in several important
ways. Their baryonic fraction
can be inferred from
observations:
Figure 1: The central area is the amplitude of the matter fluctuations expressed in term of versus the normalization of the mass temperature (Eq. (1)) for a flat model with . Grey areas are our one, two and three sigma level contours. Filled squares are from Vauclair et al. (2003), open squares are Pierpaoli et al. (2001), around and Pierpaoli et al. (2003), symbol is from Evrard et al. (2002), Triangle is from Seljak (2002), inverted triangles are from Viana et al. (2002) and Viana et al. (2003). | |
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While the above relations have been widely used to obtain constraints
on
assuming that the M-T relation is known, here we
follow a different approach. Given the uncertainty in the actual
value of
,
we can use the knowledge of gas (and stars) masses
in clusters and of
to constrain the total mass in
clusters, at least as a function of
and h, and thereby
infer the mass-temperature relation. A slight source of complexity in
gas mass measurements comes from the fact that the gas in clusters
may be clumpy. If so, the gas mass estimation from average radial
profile of the emissivity overestimates the actual gas mass by a
factor C^{1/2}, where C is a measurement of the clumping of the
gas. Sadat & Blanchard (2001) have studied in detail the change in
shape of the gas fraction with radius in clusters: they found that
the gas fraction follows rather well a scaling law, i.e. it is
roughly identical among different clusters when expressed in term of
the radius normalized to the virial radius. Furthermore they found
that in the outer part the shape was close to what has been found in
numerical simulations provided the outer amplitude is corrected for
clumping (the value of
being roughly constant 0.925 for
). This implies that correction for
clumping is indeed important to have an unbiased description of the
internal structure of clusters. Mathiesen et al. (1999) found an
average C^{1/2} of 1.16 at the radius corresponding to a contrast
density
of 500 (they also found that taking only clusters
with no secondary peak at a level of 1% of the global maximum
reduced the average C^{1/2} to 1.093). Because the clumping factor
seems to vary rapidly with radius, it is safe to work on clusters at
a similar radius. We have used the gas mass determination from
Vikhlinin et al. (1999, VFJ99 hereafter), using their most
external radius for mass determination, which is h-dependent, at
the average temperature of 4 keV. VFJ99 provided gas mass
measurements at the radius R1000 where the contrast density in the
gas is 1000 times some fiducial baryon density (
/Mpc^{3}), which corresponds to nearly half of the best
derived from WMAP. Typical density contrasts at our
working radius are in the range 480-625, at which we can directly
apply the above correction for clumping. VFJ99 excluded clusters with
double or very irregular X-ray morphology, a criteria that seems less
demanding than the criteria for regularity used by Mathiesen et al. (1999).
Figure 2: The red line is the central value of which is the normalization of the mass temperature relation Eq. (1). The WMAP relation between H_{0} and has been used, as well as the constraint on the baryon content of the Universe. One and two uncertainties arising from uncertainty on ( ) are shown as blue and yellow areas. Horizontal areas correspond to estimations of from hydrostatic methods (light green) obtained by Roussel et al. (2000) and Markevitch (1998) and from numerical simulations (light blue) obtained from Bryan & Norman (1998) and Evrard et al. (1996). | |
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(4) |
The result is shown in Fig. 3a. As one can see, at a given value of the amplitude of is well constrained. Furthermore to the first order the best is independent of : for . Interestingly this is close to the value obtained by Viana et al. (2002): . Our conclusion appears somewhat surprising as it differs from standard analyses based on a fixed normalization , which cannot simultaneously account for the baryon fraction in a consistent way for arbitrary . Gas masses from the VFJ99 sample present a moderate dispersion of the order of 20% (Sadat et al. 2005), implying rather small uncertainties on our gas fraction estimates of the order of about 4% at our working radii, which will produce an uncertainty on of 2.5%. More important is the correction for clumping. For instance, Voevodkin & Vikhlinin (2004) have estimated from the baryon mass function in a Cold Dark Matter framework. In the case their approach is very close to ours, but they used gas mass estimation at the virial radius and did not correct for clumping. This leads to virial masses which are lower than ours, leading to keV and therefore , in very good agreement with their estimation.
Figure 3: a) Top: the amplitude of matter fluctuations from clusters abundance using the mass-temperature relation found in the present analysis compared to the amplitude of matter fluctuations derived from CMB data (Grainge et al. 2003; Pearson et al. 2003; Ruhl et al. 2003; Kuo et al. 2004). The grey area corresponds to 1, 2, 3 contours on two parameters, dashed lines are contours on one parameter. The one and two sigma amplitudes obtained from an average of recent weak shear measurements are also shown as dashed regions (see text for references). b) Bottom: amplitude of matter fluctuations from CMB data versus optical thickness. | |
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The amplitude of matter fluctuations is strongly constrained by the CMB data. In the following we use the constraint on in a concordance model obtained from the CMB fluctuation analysis including the temperature-polarization cross power spectrum (TE) by the WMAP team (Kogut et al. 2003).
The comparison of the value of from CMB data with the one from clusters reveals a critical discrepancy between the two measurements (Fig. 3a). It is clear that within any model with the amplitude of we derived from clusters, , is significantly smaller than what is expected from the CMB alone ( ), which is close to the recent determination from the combination of WMAP and the Lyman forest (Seljak et al. 2004).
The non-zero optical thickness , which is requested only from the TE spectrum, is a key factor in this discrepancy: the high obtained from CMB data depends critically on the actual amplitude of the optical thickness (see Fig. 3b), although forcing does not entirely remove the discrepancy. An accurate knowledge of is therefore critical to properly evaluate the amplitude of matter fluctuations in the concordance model. One can see from Fig. 3b how much the value of obtained from CMB data depends on the actual value of the optical thickness and remains the main source of uncertainty in establishing the value of . We have also checked that when CMB data are restricted to the range , the above discrepancy remains essentially unchanged. Allowing a non-power law initial power spectrum is therefore not expected to solve this issue.
Here above, we have considered models in which the dark matter is
only made of cold dark matter, the dark energy being a pure
cosmological constant (in terms of the equation of state of vacuum
,
this means w = -1), and that X-ray gas and known stars
are the only existing baryons in clusters. A first possibility to
investigate is to examine whether a different equation of state for
the vacuum, so-called quintessence, might solve this discrepancy. We
have therefore investigated flat models with arbitrary w and
quintessence content
.
Indeed combinations of CMB and
cluster data are known to provide tight constraints on such models
(Douspis et al. 2003). With the approach developed here, models
which were found to match CMB and clusters were found to satisfy the
following constraints:
and
- 0.5 < w < -
0.4. Such models are currently at odds with constraints on
quintessential models (Douspis et al. 2003; Tegmark et al. 2004;
Riess et al. 2004) resulting from the combination of various data
including type Ia supernovae data. We therefore require an
alternative approach to solve the above issue. In the following, we
examine whether the introduction of an additional component of the
dark matter content of the universe would remove the above
discrepancy. Neutrinos are known to exist and to be massive, so
perhaps the most natural massive component of the universe to be
introduced is in the form of a neutrino contribution. This solution
has already been advocated to solve this discrepancy in an Einstein
de Sitter Universe (Elgarøy & Lahav 2003; Blanchard et al. 2003). Indeed, the presence of a light, but non-zero, component
of dark matter significantly modifies the transfer function of
primordial fluctuations which results in a lower amplitude on small
scales. Given existing measurements of mass differences we consider
only the case where the masses are equal. Within a concordance model
(
), by combining the
constraints from CMB and cluster data, and marginalizing on
(
,
H_{0}, n, )
we found that a contribution
of
is preferred with a significance
level well above 3
(see Fig. 4a), improving the
significance of such possible evidence compared to Allen et al. (2003). This confirms that the presence of a small contribution
of neutrinos (with a typical mass of 0.25 eV consistent with recent
possible detection, Klapdor-Kleingrothaus et al. 2004)
to the density of the
universe allows one to reconcile the amplitude of matter fluctuations
from clusters with the one inferred from CMB data. We notice that
such a value is above the upper limit inferred by the WMAP team using
a combination of several sources of astronomical data (Spergel et al. 2003). Finally, weak shear estimations have provided measurements
of the amplitude of matter fluctuations which can be compared to that
obtained from clusters (Refregier 2003). There are some differences
in published values which probably reflect systematic uncertainties
not yet fully identified.
Figure 4: Constraints on and given by the combined analysis of CMB and Cluster data. The dark energy component has been set to 0.7 in a flat cosmology. | |
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The determination of the amplitude of matter fluctuations within pure cold dark matter models using two methods, namely the CMB and the local cluster abundance, leads to two significantly different values. There are several ways to solve this discrepancy, although each represents a noticeable departure from the standard concordance model. The existence of a non-baryonic dark component, like a neutrino contribution, would allow us to solve this discrepancy, although such a solution leads to a low value of which is not favored by other evidence. If the actual value is larger, , and , the unavoidable conclusion is that the baryonic content of clusters at is not representative of the Universe. In this case, an astrophysical solution could be that baryons in clusters could be in a dark form, or at least undetected until now. Alternatively, baryons in clusters could have been severely depleted implying that the actual value is much smaller than the value we used above, the apparent baryon fraction being biased to low values compared to the actual primordial value. Finally, several observations might help to clarify this issue: the above conclusion relies on the actual value of the optical depth found by WMAP. If the actual value was consistent with zero most of the discrepancy would disappear. Confirmation of the actual value of is therefore critical and its better estimation will allow a better estimation of from the CMB. Other sources of information on will also obviously clarify this issue: weak lensing can potentially allow one to directly measure the actual amplitude of matter fluctuations with a similar precision to what has been obtained here with clusters, provided that systematic uncertainties are fully understood; the clusters masses could be measured from their lensing signal providing a direct estimation of the normalization constant , allowing one to distinguish between low and high normalizations. Other direct measurements of the amplitude of matter fluctuations like those derived from the Lyman- forest power spectrum (Croft et al. 1998) could also help to clarify this issue. It is remarkable that some of the observations that are expected in the near future can potentially bring fundamental information on clusters physics or alternatively may reveal the existence of a previously unidentified type of dark matter with as low as 0.01.
Acknowledgements
This research has made use of the X-ray cluster database (BAX) which is operated by the Laboratoire d'Astrophysique de Toulouse-Tarbes (LATT), under contract with the Centre National d'Études Spatiales (CNES). We acknowledge useful comments from the referee which contributed improving the paper.