Zong-Hong Zhu ^{1,2} - Masa-Katsu Fujimoto ^{2} - Xiang-Tao He ^{1}
1 - Department of Astronomy, Beijing Normal University,
Beijing 100875, China
2 -
National Astronomical Observatory,
2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan
Received 15 September 2003 / Accepted 16 December 2003
Abstract
Using recent measurements of angular size of high-z milliarcsecond
compact radio sources compiled by Gurvits et al. (1999) and
X-ray gas mass fraction of galaxy clusters published by Allen et al.
(2002, 2003), we explore their bounds on the equation of
state,
,
of the dark energy,
whose existence has been congruously suggested by various cosmological
observations.
We relax the usual constraint
,
and find that combining
the two databases yields a nontrivial lower bound on .
Under the assumption of a flat universe, we obtain a bound
at 95.4% confidence level.
The 95.4% confidence bound goes to
when the
constraint
is imposed.
Key words: cosmology: cosmological parameters - cosmology: theory - cosmology: distance scale - galaxies: active - radio continuum: galaxies - X-ray: galaxies: clusters
One of the most remarkable cosmological findings of recent years is, in additional to the cold dark matter (CDM), the existence of a component of dark energy (DE) with negative pressure in our universe. It is motivated to explain the acceleration of the universe discovered by distant type Ia supernova (SNeIa) observations (Perlmutter et al. 1998, 1999a; Riess et al. 1998, 2001), and to offset the deficiency of a flat universe, favoured by the measurements of the anisotropy of CMB (de Bernardis et al. 2000; Balbi et al. 2000; Durrer et al. 2003; Bennett et al. 2003; Spergel et al. 2003), but with a subcritical matter density parameter , obtained from dynamical estimates or X-ray and gravitational lensing observations of clusters of galaxies (for a recent summary, see Turner 2002). While a cosmological constant with is the simplest candidate for DE, it suffers from the difficulties in understanding of the observed value in the framework of modern quantum field theory (Weinberg 1989; Carroll et al. 1992) and the "coincidence problem'', the issue of explaining the initial conditions necessary to yield the near-coincidence of the densities of matter and the cosmological constant component today. In this case, quintessence (a dynamical form of DE with generally negative pressure) has been invoked (Ratra & Peebles 1988; Wetterich 1988; Caldwell et al. 1998; Zlatev et al. 1998; Gong 2002; Sahni et al. 2002; Alam et al. 2003a). One of the important characteristics of quintessence models is that their equation of state, , varies with cosmic time whilst the cosmological constant remains a constant . Determination of values of and its possible cosmic evolution plays a central role to distinguish various DE models. Such a challenging has triggered a wave of interest aiming to constrain using various cosmological databases, such as SNeIa (Garnavich et al. 1998; Tonry et al. 2003; Barris et al. 2003; Knop et al. 2003; Zhu & Fujimoto 2003; Alam et al. 2003b; Gong 2004); old high redshift objects (Lima & Alcaniz 2000a); angular size of compact radio sources (Lima & Alcaniz 2002); gravitational lensing (Zhu 2000a,b; Chae et al. 2002; Sereno 2002; Dev et al. 2003; Huterer & Ma 2003); SNeIa plus Large Scale Structure (LSS) (Perlmutter et al. 1999b); SNeIa plus gravitational lensing (Waga & miceli 1999); SNeIa plus X-ray galaxy clusters (Schuecker et al. 2003); CMB plus SNeIa (Efstathiou 1999; Bean & Melchiorri 2002; Hannestad & Mörtsell 2002; Melchiorri et al. 2003); CMB plus stellar ages (Jimenez et al. 2003); and combinations of various databases (Kujat et al. 2002). Other potential methods for the determination of have also widely discussed in the literature, such as the proposed SNAP satellite^{} (Huterer & Turner 1999; Weller & Albrecht 2001; Weller & Albrecht 2002); advanced gravitational wave detectors (Zhu et al. 2001; Biesiada 2001); future SZ galaxy cluster surveys (Haiman et al. 2001); and gamma ray bursts (Choubey & King 2003; Takahashi et al. 2003).
In this work, we shall consider the observational constraints on the DE equation of state parameterized by a redshift independent pressure-to-density ratio arising from the latest observations of angular size of high-z milliarcsecond compact radio sources compiled by Gurvits et al. (1999) and the X-ray gas mass fraction data of clusters of galaxies published by Allen et al. (2002, 2003). The basics of a constant assumption are twofold: on the one hand, the angular diameter distance used in this work is not sensitive to variations of with redshift because it depends on through multiple integrals (Maor et al. 2001; Maor et al. 2002; Wasserman 2002); on the other hand, for a wide class of quintessence models (particularly, those with tracking solutions), both and vary very slowly (Zlatev et al. 1999; Steinhardt et al. 1999; Efstathiou 1999), and an effective equation of state, is a good approximation for analysis (Wang et al. 2000). We relax the usual constraint , because in recent years there have been several models which predict a DE component with (Parker & Raval 1999; Schulz & White 2001; Caldwell 2002; Maor et al. 2002; Frampton 2003) and also we hope to explore its effects on the determination. The confidence region on the (, ) plane obtained through a combined analysis of the two databases suggests at 95.4% confidence level, which goes to when the constraint is imposed.
The plan of the paper is as follows. In the next section, we provide the bounds on from the angular size-redshift data. Constraints from the X-ray gas mass fraction of galaxy clusters are discussed in Sect. 3. Finally, we present a combined analysis, our concluding remarks and discussion in Sect. 4. Throughout the paper, we assume a flat universe which is suggested by the measurements of the anisotropy of CMB and favoured by inflation scenario.
Figure 1: Diagram of angular size vs redshift data for 145 compact radio sources (binned into 12 bins) of Gurvits et al. (1999). We assume the charateristic linear size l = 22.64 h^{-1} pc for theoretical curves. The solid curve corresponds to our best fit with and , while the dashed and dot-dashed curves correspond to a -dominated universe and the standard cold dark matter (SCDM) model respectively. | |
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We begin by evaluating the angular diameter distance
as a function of redshift z.
The redshift-dependent Hubble parameter can be written as
H(z) = H_{0} E(z), where
H_{0}=100 h km s^{-1} Mpc^{-1} is
the Hubble constant at the present time.
For a flat universe that contains (baryonic and cold dark) matter and
dark energy with a constant
(we ignore the radiation components in the universe that are not
important for the cosmological tests considered in this work),
we get (Turner & White 1997; Chiba et al. 1997; Zhu 1998)
As pointed out by the authors of previous analyses on databases of angular
size-redshift
(Jackson & Dodgson 1997;
Gurvits et al. 1999;
Vishwakarma 2001;
Alcaniz 2002;
Zhu & Fujimoto 2002;
Jain et al. 2003;
Chen & Ratra 2003;
Jackson 2003),
when one uses the angular size data to constrain the
cosmological parameters, the results will be strongly dependent on the
characteristic length l.
Therefore, instead of assuming a specific value for l, we have worked on
the interval
l = 15 h^{-1} - 30 h^{-1} pc.
To make the analysis independent of the choice of the characteristic
length l, we also minimize Eq. (2) for l,
and
simultaneously, which gives
l=22.64 h^{-1} pc,
and
as the best fit.
Figure 2 displays the 68.3% and 95.4% confidence level contours in the
(
,
)
plane using the lower shaded and the lower plus
darker shaded areas respectively.
It is clear from the figure that
is poorly constrained from
the angular size-redshift data alone, which only gives
at a 95.4% confidence level.
However, as we shall see in Sect. 4, when we combine this test with the X-ray
gas mass fraction test, we could get fairly stringent constraints on both
and
.
Figure 2: Confidence region plot of the best fit to the database of the angular size-redshift data compiled by Gurvits et al. (1999) - see the text for a detailed description of the method. The 68% and 95% confidence levels in the ( , ) plane are shown in lower shaded and lower + darker shaded areas respectively. | |
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Clusters of galaxies are the largest virialized systems in the universe, and their masses can be estimated by X-ray and optical observations, as well as gravitational lensing measurements. A comparison of the gas mass fraction, , as inferred from X-ray observations of clusters of galaxies with the cosmic baryon fraction can provide a direct constraint on the density parameter of the universe (White et al. 1993). Moreover, assuming the gas mass fraction is constant in cosmic time, Sasaki (1996) show that the data of clusters of galaxies at different redshifts also provide an efficient way to constrain other cosmological parameters decribing the geometry of the universe. This is based on the fact that the measured values for each cluster of galaxies depend on the assumed angular diameter distances to the sources as . The underlying cosmology should be the one which make these measured values invariant with redshift (Sasaki 1996; Allen et al. 2003).
Using the Chandra observational data, Allen et al. (2002, 2003) obtained
the
profiles for the 10 relaxed clusters.
Except for Abell 963, the
profiles of the other 9 clusters
appear to have converged or be close to converging with a canonical radius
r_{2500}, which is defined as the radius within which the mean mass
density is 2500 times the critical density of the universe at the redshift
of the cluster (Allen et al. 2002, 2003).
The gas mass fraction values of these nine clusters at r_{2500} (or at the
outermost radii studied for PKS0745-191 and Abell 478)
were shown in Fig. 5 of Allen et al. (2003).
We will use this database to constrain the equation of state of the dark energy
component, .
Our analysis of the present data is very similar to the one performed by
Lima et al. (2003).
However, in addition to including new data from Allen et al. (2003),
we also take into account the bias between the baryon fractions in galaxy
clusters and in the universe as a whole.
Following Allen et al. (2002), we have the model function as
(3) |
Again, we determine
and
through
a
minimization method with the same parameter ranges and
steps as in the last section.
We constrain
,
the result from the
primodial nucleosynthesis (O'Meara et al. 2001),
and
,
the final result from the Hubble Key Project by
Freedman et al. (2001).
The
difference between the model function and SCDM data is then
(Allen et al. 2003)
(4) |
Figure 3: Confidence region plot of the best fit to the of 9 clusters published by Allen et al. (2002, 2003) - see the text for a detailed description of the method. The 68% and 95% confidence levels in the - plane are shown in lower shaded and lower + darker shaded areas respectively. | |
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Figure 3 displays the 68.3% and 95.4% confidence level contours in the (, ) plane of our analysis using the lower shaded and the lower plus darker shaded areas respectively. The best fit happans at and . As shown in the figure, although the X-ray gas mass fraction data constrains the density parameter very stringently, it still poorly limits the dark energy equation of state . The situation can be dramatically improved when the two databases are combined in analysis, in particular, a nontrivial lower bound on will be obtained (see below).
Now we present our combined analysis of the constraints from the angular size-redshift data and the X-ray gas mass fraction of galaxy clusters and summarize our results. In Fig. 4, we display the 68.3% and 95.4% confidence level contours in the (, ) plane using the lower shaded and the lower plus darker shaded areas respectively. The best fit happens at and . As is shown, fairly stringent bounds on both and are obtained, with and at the 95.4% confidence level. The bound on goes to when the constraint is imposed.
Figure 4: Confidence region plot of the best fit from a combined analysis for the angular size-redshift data (Gurvits et al. 1999) and the X-ray gas mass fractions of 9 clusters (Allen et al. 2002, 2003). The 68% and 95.4% confidence levels in the - plane are shown in lower shaded and lower + darker shaded areas respectively. The best fit happans at and . | |
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Although precise determinations of and its possible evolution with cosmic time are crucial for deciphering the mystery of DE, currently has not been determined well even with an assumption of being constant (Hannestad & Mörtsell 2002; Spergel et al. 2003; Takahashi et al. 2003). One should determine using a joint analysis. In this paper we have shown that stringent constraints on can be obtained from the combined analysis of the angular size-redshift data and the X-ray mass fraction data of clusters, which is complementary to other joint analyses. We compare our results with other recent determinations of from independent methods. For the usual quintessence model (i.e., the constraint is imposed), Garnavich et al. (1998) found using the SNeIa data from the High-z Supernova Search Team, while Lima & Alcaniz (2002) obtained using the angular size-redshift data from Gurvits et al. (1999) (95% confidence level). Our result of is slightly more stringent than theirs. However Bean & Melchiorri (2002) found an even better constraint, , by analyzing SNeIa data and measurements of LSS and the positions of the acoustic peaks in the CMB spectrum. For the more general dark energy model including either normal XCDM, as well as the extended or phantom energy (i.e., the constraint is relaxed), Hannestad & Mörtsell (2002) combined CMB, LSS and SNeIa data and obtained at a 95.4% confidence level, whose lower and upper bounds are slightly lower than ours ( at a 95.4% confidence level). Recently, Schuecker et al. (2003) combined REFLEX X-ray clusters and SNeIa data to obtain with a statistical significance. From Fig. 4, it is found that our result is , which is comparable with the results of Schuecker et al. (2003). Using the X-ray gas mass fraction of 6 galaxy clusters, Lima et al. (2003) found ( level), which is less stringent than the result presented in this work. This is because we used more X-ray gas mass fraction data of galaxy clusters and combined the angular size-redshift data of compact radio sources. The analysis presented here reinforces the interest in precise measurements of angular size of distant compact radio sources and statistical studies of the intrinsic length distribution of the sources. Our constraints will be improved when more acurate X-ray data from Chandra and XMM-Newton become available in the near future.
Acknowledgements
We would like to thank L. I. Gurvits for sending us the compilation of the angular size-redshift data and helpful explanation of the data, S. Allen for providing us the X-ray mass fraction data and his help with data analysis, J. S. Alcaniz and D. Tatsumi for their helpful discussions. Our thanks go to the anonymouse referee for valuable comments and useful suggestions, which improved this work very much. This work was supported by a Grant-in-Aid for Scientific Research on Priority Areas (No. 14047219) from the Ministry of Education, Culture, Sports, Science and Technology. Z.-H. Zhu acknowledges support from the National Natural Science Foundation of China and the National Major Basic Research Project of China (G2000077602), and he is also grateful to all TAMA300 & LCGT members and the staff of NAOJ for their hospitality and help during his stay. The support of X.-T. He is from the National Natural Science Foundation of China.