Galaxy cluster angularsize data constraints on dark energy
^{1} Department of Astronomy, Beijing Normal University, 100875, Beijing, PR China
email: chenyun@mail.bnu.edu.cn
^{2} Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA
email: ratra@phys.ksu.edu
Received: 25 August 2011
Accepted: 20 April 2012
We use angular size versus redshift data for galaxy clusters provided by Bonamente and collaborators to place constraints on model parameters of constant and timeevolving dark energy cosmological models. These constraints are compatible with those from other recent data, but are not very restrictive. A joint analysis of the galaxy cluster angularsize data with the more restrictive baryon acoustic oscillation peak length scale and supernova Type Ia apparentmagnitude data, favors a spatially flat cosmological model currently dominated by a timeindependent cosmological constant, but does not exclude timevarying dark energy.
Key words: cosmology: miscellaneous / cosmology: theory / dark energy
© ESO, 2012
1. Introduction
A number of lines of observational evidence support a “standard” model of cosmology with an energy budget that is significantly dominated by dark energy. Dark energy is most simply characterized as a negativepressure substance that powers the observed accelerated cosmological expansion. It can evolve slowly in time and vary weakly in space, although current data are consistent with it being equivalent to Einstein’s cosmological constant. On the other hand, some studies argue that the observed accelerated expansion should instead be viewed as an indication that general relativity does not accurately describe gravitational physics on large cosmological length scales. For relevant reviews, we refer to Frieman (2008), Ratra & Vogeley (2008), Caldwell & Kamionkowski (2009), Sami (2009), Bartelmann (2010), Cai et al. (2010), and Brax (2009). In what follows, we assume that general relativity provides an accurate description of gravitation on cosmological length scales.
There are many dark energy models under discussion. For recent discussions, we cite the studies of Wei (2011), Jamil & Saridakis (2010), Maggiore (2011), Dutta & Scherrer (2010), Shao & Chen (2010), Lepe & Peña (2010), Sloth (2010), Liu (2010), Honorez et al. (2010), and references therein. Perhaps the most economical – and the current “standard” model – is the ΛCDM model (Peebles 1984), where the accelerated cosmological expansion is powered by Einstein’s cosmological constant, Λ, a spatially homogeneous fluid with equation of state parameter ω_{Λ} = p_{Λ}/ρ_{Λ} = −1 (with p_{Λ} and ρ_{Λ} being the fluid pressure and energy density). In this model, the cosmological energy budget is dominated by ρ_{Λ}, with cold dark matter (CDM) being the second largest contributor. The ΛCDM model provides a reasonable fit to most observational constraints, although the “standard” CDM structure formation model might be in some observational trouble (see, e.g., Peebles & Ratra 2003; Perivolaropoulos 2010).
The ΛCDM model has a few apparent puzzles. Prominent among these is that the needed Λ energy density scale is of order an meV, very small compared to the higher (cutoff) value suggested by a perhaps naive application of quantum mechanics. Another puzzle is that we happen to be observing at a time not very different from when the Λ energy density started dominating the cosmological energy budget (this is the “coincidence” puzzle).
If the dark energy density – that responsible for powering the accelerated cosmological expansion – slowly decreased in time (rather than remaining constant like ρ_{Λ}), the energy densities of dark energy and nonrelativistic matter (CDM and baryons) would remain comparable for a longer period of time, and so alleviate the coincidence puzzle. In addition, a slowly decreasing dark energy density that is based on more fundamental physics at an energy density scale much higher than an meV, would result in a current dark energy density scale of an meV through gradual decrease over the long lifetime of the Universe. Thus, a slowly decreasing dark energy density could resolve some of the puzzles of the ΛCDM model (Ratra & Peebles 1988).
The XCDM parametrization is often used to describe a slowly decreasing dark energy density. In this parametrization, the dark energy is modeled as a spatially homogeneous (X) fluid with an equation of state parameter w_{X} = p_{X}/ρ_{X}, where w_{X} < − 1/3 is an arbitrary constant and p_{X} and ρ_{X} are the pressure and energy density of the Xfluid. When w_{X} = −1, the XCDM parametrization reduces to the ΛCDM model, which is a complete and consistent model. For any other value of w_{X}(< − 1/3), the XCDM parametrization is incomplete as it cannot describe spatial inhomogeneities (see, e.g., Ratra 1991). For computational simplicity, we study the XCDM parametrization only in the spatially flat cosmological case.
If the dark energy density evolves in time, physics demands that it also be spatially inhomogeneous. The φCDM model – in which dark energy is modeled as a scalar field φ with a gradually decreasing (in φ) potentialenergy density V(φ) – is the simplest complete and consistent model of a slowly decreasing (in time) dark energy density. Here we focus on an inverse powerlaw potentialenergy density V(φ) ∝ φ^{ − α}, where α is a nonnegative constant (Peebles & Ratra 1988; Ratra & Peebles 1988). When α = 0, the φCDM model reduces to the corresponding ΛCDM case. Here we only consider the spatially flat φCDM cosmological model.
It has been known for some time that a spatially flat ΛCDM model with a current energy budget dominated by a constant Λ is largely consistent with most observational constraints (see, e.g., Jassal et al. 2010; Wilson et al. 2006; Davis et al. 2007; Allen et al. 2008). Supernovae Type Ia (SNeIa) apparentmagnitude measurements (e.g., Riess et al. 1998; Perlmutter et al. 1999; Shafieloo et al. 2009; Holsclaw et al. 2010), in conjunction with cosmic microwave background (CMB) anisotropy data (e.g., Ratra et al. 1999; Podariu et al. 2001b; Spergel et al. 2003; Komatsu et al. 2009, 2011) combined with low estimates of the cosmological mass density (e.g., Chen & Ratra 2003b), as well as baryon acoustic oscillation (BAO) peak length scale estimates (e.g., Percival et al. 2007; Gaztañaga et al. 2009; Samushia & Ratra 2009b; Wang 2009) strongly suggest that we live in a spatially flat ΛCDM model with nonrelativistic matter contributing a little less than 30% of the current cosmological energy budget, with the remaining slightly more than 70% being contributed by a cosmological constant. These three sets of data carry by far the most weight when determining constraints on models and cosmological parameters.
Future data from space missions will tighten the constraints (see, e.g., Podariu et al. 2001a; Samushia et al. 2011; Wang et al. 2010). However, at present, it is of great importance to consider independent constraints that can be derived from other presently available data sets. While these data do not yet carry as much statistical weight as the SNeIa, CMB, and BAO data, they can potentially reassure us (if they provide constraints consistent with those from the better known data), or if the two sets of constraints are inconsistent this might lead to the discovery of hidden systematic errors or rule out the cosmological model under consideration.
Other data that have been used to constrain cosmological parameters include galaxy cluster gasmass fraction (e.g., Allen et al. 2008; Samushia & Ratra 2008; Ettori et al. 2009), gammaray burst luminosity distance (e.g., Schaefer 2007; Liang & Zhang 2008; Wang 2008; Samushia & Ratra 2010), largescale structure (e.g., Courtin et al. 2011; Baldi 2011; Basilakos et al. 2010), strong gravitational lensing (e.g., Chae et al. 2002, 2004; Lee & Ng 2007; Yashar et al. 2009), and lookback time (e.g., Capozziello et al. 2004; Simon et al. 2005; Samushia et al. 2010; Dantas et al. 2011) or Hubble parameter (Samushia & Ratra 2006; Samushia et al. 2007; FernandezMartinez & Verde 2008; Yang & Zhang 2010) data. While the constraints provided by these data are much less restrictive than those derived from the SNeIa, CMB, and BAO data, both types of data result in largely compatible constraints that generally support a currently accelerating cosmological expansion. This gives us confidence that the broad outlines of the “standard” cosmological model are now in place.
Angularsize data from compact radio sources have also been used to constrain cosmological parameters (see, e.g., Gurvits et al. 1999; Guerra et al. 2000; Lima & Alcaniz 2000; Lima & Alcaniz 2002; Chen & Ratra 2003a; Podariu et al. 2003; Santos & Lima 2008). There are two very recent samples of angular size versus redshift data from galaxy clusters obtained by combining their SunyaevZeldovich effect (SZE) and Xray surface brightness observations. These are the 25 data points from De Filippis et al. (2005) and the 38 data points from Bonamente et al. (2006). De Filippis et al. (2005) obtained their angularsize data by using an isothermal elliptical model for the galaxy clusters, while Bonamente et al. (2006) derived their data by assuming a nonisothermal spherical model. The sample of De Filippis et al. (2005) was used to constrain H_{0}, result in good agreement with the independent studies of the Hubble Space Telescope key project and the estimates of WMAP (Cunha et al. 2007). The sample of Bonamente et al. (2006) has been previously used to constrain some cosmological parameters and to test the distance duality relationship of metric gravity models (see, e.g., De Bernardis et al. 2006; Holanda et al. 2012a; Lima et al. 2010; Cao & Liang 2011; Liang et al. 2011). In some literature, these two samples have been adopted to test the distance duality relationship, and it turns out that the sample of De Filippis et al. (2005) is more in accordance with no violation of the duality relation (Holanda et al. 2010, 2011, 2012b; Li et al. 2011; Meng et al. 2012).
In this paper, we use the newer and larger sample of galaxy cluster angular size versus redshift data from Bonamante et al. (2006, hereafter B06) to constrain cosmological models that have not been previously considered, and to constrain other cosmological parameters in models that have been previously considered. We show that these constraints are compatible with those derived using other data. We also perform a joint analysis of these angularsize data and SNeIa and BAO measurements and show that including the angular size data in the analysis affects the constraints, although not greatly so as the angularsize data do not yet have sufficient statistical weight.
Our paper is organized as follows. In Sect. 2, we present the basic equations of the three dark energy models we study. Constraints from the B06 angular diameter distances of galaxy clusters are derived in Sect. 3. In Sect. 4, the BAO data and the SNeIa measurements are used to constrain the dark energy models. In Sect. 5, we determine joint constraints on the dark energy parameters from different combinations of the data sets. Finally, we summarize our main conclusions in Sect. 6.
2. Basic equations of the dark energy models
The Friedmann equation of the ΛCDM model with spatial curvature can be written as (1)where z is the redshift, E(z) = H(z)/H_{0} is the dimensionless Hubble parameter where H_{0} is the Hubble constant, and the modelparameter set is p = (Ω_{m0},Ω_{Λ}) where Ω_{m0} is the nonrelativistic (baryonic and cold dark) matter density parameter and Ω_{Λ} that of the cosmological constant. Throughout, the subscript 0 denotes the value of a quantity today. In this paper, the subscripts Λ, X, and φ represent the corresponding quantities of the dark energy component in the ΛCDM, XCDM, and φCDM models.
In this work, for computational simplicity, the spatial curvature is set to zero in the XCDM and φCDM cases. The Friedmann equation for the XCDM parametrization is then (2)where the modelparameter set is p = (Ω_{m0},w_{X}).
In the φCDM model, the inverse powerlaw potentialenergy density of the scalar field adopted in this paper is , where m_{p} is the Planck mass, and α and κ are nonnegative constants (Peebles & Ratra 1988). In the spatially flat case, the Friedmann equation of the φCDM model is (3)where H(z) = ȧ/a is the Hubble parameter, and a(t) is the cosmological scale factor and an overdot denotes a time derivative. The energy densities of the matter and the scalar field are (4)and (5)respectively. According to the definition of the dimensionless density parameter, one has (6)The scalar field φ obeys the differential equation (7)Using Eqs. (3) and (7), as well as the initial conditions described in Peebles & Ratra (1988), one can numerically compute the Hubble parameter H(z). In this case, the modelparameter set is p = (Ω_{m0},α).
To use observational data to constrain cosmological models, we need various distance expressions. The coordinate distance is (8)where Ω_{k} is the spatial curvature density parameter and c is the speed of light, and (9)The luminosity distance d_{L} and the angular diameter distance d_{A} are simply related to the coordinate distance as (10)and (11)
3. Constraints from the angularsize data
Xray observations of the intracluster medium combined with radio observations of the galaxy cluster SunyaevZeldovich effect allow an estimate to be made of the angular diameter distance (ADD) d_{A} of galaxy clusters. Here we use the 38 ADDs of B06 to constrain cosmological parameters. These data can be found in Tables 1 and 2 of B06. For convenience, we recollect them in Table 1.
Angular diameter distances of galaxy clusters from B06.
There are three sources of uncertainty in the measurement of d_{A}: the cluster modeling error σ_{mod}, the statistical error σ_{stat}, and the systematic error σ_{sys}. The modeling errors are shown in Table 1 and the statistical and systematic errors are presented in Table 3 of B06. In our analysis here, we combine these errors in quadrature. Thus, the total uncertainty σ_{tot} satisfies .
We constrain cosmological parameters by minimizing (12)Here z_{i} is the redshift of the observed galaxy cluster, is the predicted value of the ADD in the cosmological model under consideration and is the measured value. From , we compute the likelihood function L(H_{0},p). We then treat H_{0} as a nuisance parameter and marginalize over it using a Gaussian prior with H_{0} = 68 ± 3.5 km s^{1} Mpc^{1} (Chen et al. 2003) to get a likelihood function L(p) that is a function only of the cosmological parameters of interest. The bestfit parameter values p ∗ are those that maximize the likelihood function and the 1, 2, and 3σ constraint contours are the set of cosmological parameters (centered on p∗) that enclose the 68.27%, 95.45% and 99.73% confidence levels, respectively, of the probability under the likelihood function.
Fig. 1 We display the 1, 2, and 3σ constraint contours for the ΛCDM model from the ADD data. The dashed diagonal line corresponds to spatially flat models and the shaded area in the upper lefthand corner is the region for which there is no big bang. The star marks the bestfit pair (Ω_{m0},Ω_{Λ}) = (0.19, − 0.62) with . 

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Two standarddeviation bounds on cosmological parameters.
Figures 1–3 show the constraints from the ADD data on the three dark energy models we consider. Comparing these results to those shown in Figs. 1–3 of Chen & Ratra (2003a), which were derived using the compact radio source angularsize data of Gurvits et al. (1999), and to Figs. 1, 2 of Podariu et al. (2003), derived using the FRIIb radio galaxy angularsize data from Guerra et al. (2000), we see that the B06 galaxy cluster angularsize data provide approximately comparable constraints on cosmological parameters as those derived from the two earlier angularsize data sets. These ADD constraints are comparable to those from gammaray burst data (Samushia & Ratra 2010, Figs. 1–3 and 7–9), as well as those from Hubble parameter measurements (Samushia et al. 2007, Figs. 1–3).
Fig. 2 We show the 1, 2, and 3 σ constraint contours for the XCDM parametrization from the ADD data. The dashed horizontal line at ω_{X} = −1 corresponds to spatially flat ΛCDM models. The star marks the bestfit pair (Ω_{m0},w_{X}) = (0.01, − 0.12) with . 

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Fig. 3 We perform the 1, 2, and 3 σ constraint contours for the φCDM model from the ADD data. The horizontal axis at α = 0 corresponds to spatially flat ΛCDM models. The star marks the bestfit pair (Ω_{m0},α) = (0.54,5) with . 

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Current ADD data constraints are clearly not very restrictive, although it is encouraging that the ADD constraints on these dark energy models do not disfavor the regions of parameter space that are favored by other data. More importantly, we anticipate that ADD data to be acquired in the near future will provide significantly tighter constraints on cosmological parameters.
4. Constraints from BAO and SNeIa data
The BAO peak length scale can be used as a standard ruler to constrain cosmological parameters. Here we use the BAO data of Percival et al. (2010) to constrain the parameters of the ΛCDM and φCDM models and the XCDM parametrization.
Fig. 4 We show the 1, 2, and 3 σ constraint contours for the φCDM model derived from the BAO data. The horizontal axis at α = 0 corresponds to spatially flat ΛCDM models. The star marks the bestfit pair (Ω_{m0},α) = (0.32,2.01) with . 

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Percival et al. (2010) measure the position of the BAO peak from the SDSS DR7 and 2dFGRS data, determining r_{s}(z_{d})/D_{V}(z = 0.275) = 0.1390 ± 0.0037, where r_{s}(z_{d}) is the comoving sound horizon at the baryon drag epoch, and . By using Ω_{m0}h^{2} = 0.1326 ± 0.0063 and Ω_{b0}h^{2} = 0.02273 ± 0.00061 (where Ω_{b0} is the current value of the baryonic mass density parameter and h is the Hubble constant in units of 100 km s^{1} Mpc^{1}) from WMAP5 (Komatsu et al. 2009), one can get (13)as shown in Eq. (13) of Percival et al. (2010). The error in Ω_{b0}h^{2} is ignored in this work, as the WMAP5 data constrains Ω_{b0}h^{2} to within 0.5%.
Fig. 5 We display the 1, 2, and 3σ constraint contours for the φCDM model from the SNeIa data. The horizontal axis at α = 0 corresponds to spatially flat ΛCDM models. Thin solid lines (best fit at Ω_{m0} = 0.27 and α = 0.0 with , marked by “ × ” ) exclude systematic errors, while thick solid lines (best fit at Ω_{m0} = 0.27 and α = 0.0 with , marked by “♢”) account for systematics. 

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Our results for the ΛCDM model and the XCDM parametrization agree very well with the Percival et al. (2010) results shown in their Fig. 5. Our results for the φCDM model are shown in Fig. 4. The BAO data constrains Ω_{m0} significantly, leaving Ω_{Λ}, w_{X}, and α almost unconstrained (see, e.g., Samushia & Ratra 2009a). The results obtained from the BAO data of Percival et al. (2010) are most directly comparable to those derived from the earlier BAO data of Eisenstein et al. (2005). Comparing to the constraints shown in Figs. 2–4 of Samushia & Ratra (2009a), one sees that the Percival et al. (2010) data lead to slightly more restrictive constraints than the Eisenstein et al. (2005) data.
Type Ia supernovae are standardizable candles that can be used to constrain cosmological parameters. Here we use the Union2 compliation of 557 SNeIa (covering a redshift range 0.015 ≤ z ≤ 1.4) of Amanullah et al. (2010) to constrain parameters of the ΛCDM and φCDM models and the XCDM parametrization.
Cosmological constraints from SNeIa data are obtained by using the distance modulus μ(z). The theoretical (predicted) distance modulus is (14)where μ_{0} = 42.38 − 5log _{10}h and the Hubblefree luminosity distance is given by (15)The bestfit values of cosmological model parameters can be determined by minimizing the χ^{2} function (16)where μ_{obs,i}(z_{i}) is the distance modulus obtained from observations and σ_{μi} is the total uncertainty in the SNeIa data. The zeropoint μ_{0} is treated as a nuisance parameter and marginalized over analytically (Di Pietro & Claeskens 2003; Perivolaropoulos 2005; Nesseris & Perivolaropoulos 2005). The covariance matrix with or without systematic errors can be found on the web^{1}.
Fig. 6 Thick (thin) solid lines are 1, 2, and 3σ constraint contours for the ΛCDM model from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. The cross (“+”) marks the bestfit point determined from the joint sample without the ADD data at Ω_{m0} = 0.28 and Ω_{Λ} = 0.76 with . The star (“ ∗ ”) marks the bestfit point determined from the joint sample with the ADD data at Ω_{m0} = 0.28 and Ω_{Λ} = 0.72 with . The dashed sloping line corresponds to spatially flat models. 

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Our results for the ΛCDM model and the XCDM parametrization agree very well with the Amanullah et al. (2010) results shown in their Figs. 10 and 11. The constraints on φCDM model parameters from these data are shown in Fig. 5. Comparing to Fig. 4 of Samushia & Ratra (2009b), we can see that the constraints of the Amanullah et al. (2010) data with systematic errors are approximately comparable to those of the earlier Kowalski et al. (2008) data for 307 SNeIa without consideration of their systematic errors. As for the BAO data, the SNeIa data constraints are also fairly onedimensional, tightly constraining one combination of the cosmological parameters, while only weakly constraining the “orthogonal” combination.
Fig. 7 Thick (thin) solid lines are 1, 2, and 3σ constraint contours for the XCDM parametrization from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. The cross (“+”) marks the bestfit point determined from the joint sample without the ADD data at Ω_{m0} = 0.28 and ω_{X} = −1.04 with . The star (“ ∗ ”) marks the bestfit point determined from the joint sample with the ADD data at Ω_{m0} = 0.28 and ω_{X} = −0.99 with . The dashed horizontal line at ω_{X} = −1 corresponds to spatially flat ΛCDM models. 

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Fig. 8 Thick (thin) solid lines are 1, 2, and 3σ constraint contours for the φCDM model from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. The cross (“ × ”) marks the bestfit point determined from the joint sample without the ADD data at Ω_{m0} = 0.28 and α = 0 with . The diamond (“♢”) marks the bestfit point determined from the joint sample with the ADD data at Ω_{m0} = 0.28 and α = 0.01 with . The α = 0 horizontal axis corresponds to spatially flat ΛCDM models. 

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Fig. 9 Onedimensional marginalized distribution probabilities of the cosmological parameters for the LCDM model. Thick (thin) lines are the results from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. 

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Fig. 10 Onedimensional marginalized distribution probabilities of the cosmological parameters for the XCDM parametrization. Thick (thin) lines are the results from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. 

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Fig. 11 Onedimensional marginalized distribution probabilities of the cosmological parameters for the φCDM model. Thick (thin) lines are the results of a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. 

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5. Joint constraints
Figures 6–8 show the constraints provided on the cosmological parameters of both the ΛCDM and φCDM models and the XCDM parametrization by a joint analysis of the BAO and SNeIa data, as well as by a joint analysis of the BAO, SNeIa, and ADD data. With the inclusion of systematic errors in the analysis of the SNeIa data of Amanullah et al. (2010), the new joint BAO and SNeIa constraints (thin solid contours in Figs. 6–8) are similar to the earlier ones shown in Figs. 4–6 of Samushia & Ratra (2010), which made use of the smaller SNeIa data set of Kowalski et al. (2008) that did not include systematic errors.
Figures 9–11 display the onedimensional marginalized distribution probabilities of the cosmological parameters for the three cosmological models considered in this work, derived from a joint analysis of the BAO and SNeIa data, as well as from a joint analysis of the BAO, SNeIa, and ADD data. The marginalized 2σ intervals of the cosmological parameters are presented in Table 2.
The combination of BAO and SNeIa data gives tight constraints on the cosmological parameters. Adding the currently available galaxycluster ADD data to this combination does shift the constraint contours, although the effect is not large. Current ADD data do not have enough statistical weight to significantly affect the combined BAO and SNeIa results. The ADD data have approximately the same weight as currently available gammaray burst luminosity measurements (Samushia & Ratra 2010, Figs. 4–6 and 10–12).
The observational data considered here are clearly very consistent with the predictions of a spatiallyflat cosmological model with an energy budget dominated by a timeindependent cosmological constant. However, the data do not rule out timeevolving dark energy, although they do require that it not vary rapidly.
6. Conclusion
We have shown that the galaxycluster angular size versus redshift data of B06 can also be used to constrain dark energy model cosmological parameters. The resulting constraints are compatible with those derived from other sets of current data, thus strengthening support for the current “standard” cosmological model. The ADD constraints are approximately as restrictive as those that follow from currently available gammaray burst luminosity data, strong gravitationallensing measurements, or lookback time (or Hubble parameter) observations. They are, however, much less restrictive than those that follow from a combined analysis of BAO peak length scale and SNeIa apparentmagnitude data.
The spatially flat ΛCDM model, currently dominated by a constant cosmological constant, provides a good fit to the data that we have studied here. However, these data are not inconsistent with a timeevolving dark energy.
We anticipate that angularsize data to be acquired in the near future will provide significantly tighter constraints on cosmological parameters. In conjunction with other observations, these angularsize data will prove very useful in pinning down parameter values of the “standard” cosmological model.
Acknowledgments
Yun Chen thanks ZongHong Zhu and Lado Samushia for their generous and helpful advice. Y.C. was supported by the Ministry of Science and Technology national basic science program (project 973) under grant No. 2012CB821804. BR was supported by DOE grant DEFG0399EP41093.
References
 Allen, S. W., Rapetti, D. A., Schmidt, R. W., et al. 2008, MNRAS, 383, 879 [NASA ADS] [CrossRef] [Google Scholar]
 Amanullah, R., Lidman, C., Rubin, D., et al. 2010, ApJ, 716, 712 [NASA ADS] [CrossRef] [Google Scholar]
 Baldi, M. 2011, MNRAS, 411, 1077 [NASA ADS] [CrossRef] [Google Scholar]
 Bartelmann, M. 2010, Rev. Mod. Phys., 82, 331 [NASA ADS] [CrossRef] [Google Scholar]
 Basilakos, S., Plionis, M., & Solà, J. 2010, Phys. Rev. D, 82, 083512 [NASA ADS] [CrossRef] [Google Scholar]
 Bonamente, M., Chapman, S. C., Ibata, R. A., et al. 2006, ApJ, 647, 25 (B06) [NASA ADS] [CrossRef] [Google Scholar]
 Brax, P. 2009 [arXiv:0912.3610] [Google Scholar]
 Cai, Y.F., Saridakis, E. N., Setare, M. R., et al. 2010, Phys. Rept., 493, 1 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Caldwell, R. R., & Kamionkowski, M. 2009, Ann. Rev. Nucl. Part. Sci., 59, 397 [NASA ADS] [CrossRef] [Google Scholar]
 Cao, S., & Liang, N. 2011, RA&A, 11, 1199 [NASA ADS] [CrossRef] [Google Scholar]
 Capozziello, S., Cardone, V. F., Funaro, M., et al. 2004, Phys. Rev. D, 70, 123501 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Chae, K.H., Biggs, A. D., Blandford, R. D., et al. 2002, Phys. Rev. Lett., 89, 151301 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Chae, K.H., Chen, G., Ratra, B., et al. 2004, ApJ, 607, L71 [NASA ADS] [CrossRef] [Google Scholar]
 Chen, G., & Ratra, B. 2003a, ApJ, 582, 586 [NASA ADS] [CrossRef] [Google Scholar]
 Chen, G., & Ratra, B. 2003b, PASP, 115, 1143 [NASA ADS] [CrossRef] [Google Scholar]
 Chen, G., Gott, J. R., & Ratra, B. 2003, PASP, 115, 1269 [NASA ADS] [CrossRef] [Google Scholar]
 Courtin, J., Rasera, Y., Alimi, J.M., et al. 2011, MNRAS, 410, 1911 [NASA ADS] [Google Scholar]
 Cunha, J. V., Marassi, L., & Lima, J. A. S. 2007, MNRAS, 379, L1 [NASA ADS] [CrossRef] [Google Scholar]
 Dantas, M. A., Alcaniz, J. S., Mania, D., et al. 2011, Phys. Lett. B, 699, 239 [NASA ADS] [CrossRef] [Google Scholar]
 Davis, T. M., Mörtsell, E., Sollerman, J., et al. 2007, ApJ, 666, 716 [NASA ADS] [CrossRef] [Google Scholar]
 De Bernardis, F., Giusarma, E., & Melchiorri, A. 2006, Int. J. Mod. Phys. D, 15, 759 [NASA ADS] [CrossRef] [Google Scholar]
 De Filippis, E., Sereno, M., Bautz, M. W., et al. 2005, ApJ, 625, 108 [NASA ADS] [CrossRef] [Google Scholar]
 Di Pietro, E., & Claeskens, J.F. 2003, MNRAS, 341, 1299 [NASA ADS] [CrossRef] [Google Scholar]
 Dutta, S., & Scherrer, R. J. 2010, Phys. Rev. D, 82, 043526 [NASA ADS] [CrossRef] [Google Scholar]
 Eisenstein, D. J., Zehavi, I., Hogg, D. W., et al. 2005, ApJ, 633, 560 [NASA ADS] [CrossRef] [Google Scholar]
 Ettori, S., Morandi, A., Tozzi, P., et al. 2009, A&A, 501, 61 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 FernandezMartinez, E., & Verde, L. 2008, J. Cosmol. Astropart. Phys., 0808, 023 [NASA ADS] [CrossRef] [Google Scholar]
 Frieman, J. A. 2008, AIP Conf. Proc., 1057, 87 [NASA ADS] [CrossRef] [Google Scholar]
 Gaztañaga, E., Cabré, A., & Hui, L. 2009, MNRAS, 399, 1663 [NASA ADS] [CrossRef] [Google Scholar]
 Guerra, E. J., Daly, R. A., & Wan, L. 2000, ApJ, 544, 659 [NASA ADS] [CrossRef] [Google Scholar]
 Gurvits, L. I., Kellermann, K. I., & Frey, S. 1999, A&A, 342, 378 [NASA ADS] [Google Scholar]
 Holanda, R. F. L., Lima, J. A. S., & Ribeiro, M. B. 2010, ApJ, 722, L233 [NASA ADS] [CrossRef] [Google Scholar]
 Holanda, R. F. L., Lima, J. A. S., & Ribeiro, M. B., 2011, A&A, 528, L14 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Holanda, R. F. L., Cunha, J. V., & Lima, J. A. S. 2012a, Gen. Relativ. Gravit., 44, 501 [NASA ADS] [CrossRef] [Google Scholar]
 Holanda, R. F. L., Lima, J. A. S., & Ribeiro, M. B. 2012b, A&A, 538, A131 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Holsclaw, T., Alam, U., Sansó, B., et al. 2010, Phys. Rev. Lett., 105, 241302 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Honorez, L. L., Reid, B. A., Mena, O., Verde, L., & Jimenez, R. 2010, J. Cosmol. Astropart. Phys., 1009, 029 [CrossRef] [Google Scholar]
 Jamil, M., & Saridakis, E. N. 2010, J. Cosmol. Astropart. Phys., 1007, 028 [NASA ADS] [CrossRef] [Google Scholar]
 Jassal, H. K., Bagla, J. S., & Padmanabhan, T. 2010, MNRAS, 405, 2639 [NASA ADS] [Google Scholar]
 Komatsu, E., Dunkley, J., Nolta, M. R., et al. 2009, ApJS, 180, 330 [NASA ADS] [CrossRef] [Google Scholar]
 Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18 [NASA ADS] [CrossRef] [Google Scholar]
 Kowalski, M., Rubin, D., Aldering, G., et al. 2008, ApJ, 686, 749 [NASA ADS] [CrossRef] [Google Scholar]
 Lee, S., & Ng, K.W. 2007, Phys. Rev. D, 76, 043518 [NASA ADS] [CrossRef] [Google Scholar]
 Lepe, S., & Peña, F. 2010, Eur. Phys. J. C, 69, 575 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Li, Z., Wu, P., & Yu, H. 2011, ApJ, 729, L14 [NASA ADS] [CrossRef] [Google Scholar]
 Liang, N., & Zhang, L. N. 2008, AIP Conf. Proc., 1065, 367 [NASA ADS] [CrossRef] [Google Scholar]
 Liang, N., Cao, S., Liao, K., & Zhu, Z.H. 2011 [arXiv:1104.2497] [Google Scholar]
 Lima, J. A. S., & Alcaniz, J. S. 2000, A&A, 357, 393 [NASA ADS] [Google Scholar]
 Lima, J. A. S., & Alcaniz, J. S. 2002, ApJ, 566, 15 [NASA ADS] [CrossRef] [Google Scholar]
 Lima, J. A. S., Holanda, R. F. L., & Cunha, J. V. 2010, AIP Conf. Proc., 1241, 224 [NASA ADS] [CrossRef] [Google Scholar]
 Liu, D.J. 2010, Phys. Rev. D, 82, 063523 [NASA ADS] [CrossRef] [Google Scholar]
 Maggiore, M. 2011, Phys. Rev. D, 83, 063514 [NASA ADS] [CrossRef] [Google Scholar]
 Meng, X.L., Zhang, T.J., & Zhan, H. 2012, ApJ, 745, 98 [NASA ADS] [CrossRef] [Google Scholar]
 Nesseris, S., & Perivolaropoulos, L. 2005, Phys. Rev. D, 72, 123519 [NASA ADS] [CrossRef] [Google Scholar]
 Peebles, P. J. E. 1984, ApJ, 284, 439 [NASA ADS] [CrossRef] [Google Scholar]
 Peebles, P. J. E., & Ratra, B. 1988, ApJ, 325, L17 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Peebles, P. J. E., & Ratra, B. 2003, Rev. Mod. Phys., 75, 559 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Perivolaropoulos, L. 2010, J. Phys. Conf. Ser., 222, 012024 [NASA ADS] [CrossRef] [Google Scholar]
 Percival, W. J., Cole, S., Eisenstein, D. J., et al. 2007, MNRAS, 381, 1053 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Percival, W. J., Reid, B. A., Eisenstein, D. J., et al. 2010, MNRAS, 401, 2148 [NASA ADS] [CrossRef] [Google Scholar]
 Perivolaropoulos, L. 2005, Phys. Rev. D, 71, 063503 [NASA ADS] [CrossRef] [Google Scholar]
 Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, ApJ, 517, 565 [NASA ADS] [CrossRef] [Google Scholar]
 Podariu, S., Nugent, P., & Ratra, B. 2001a, ApJ, 553, 39 [NASA ADS] [CrossRef] [Google Scholar]
 Podariu, S., Ohyama, Y., Murayama, T., et al. 2001b, ApJ, 559, 9 [NASA ADS] [CrossRef] [Google Scholar]
 Podariu, S., Daly, R. A., Mory, M. P., & Ratra, B. 2003, ApJ, 584, 577 [NASA ADS] [CrossRef] [Google Scholar]
 Ratra, B., & Peebles, P. J. E. 1988, Phys. Rev. D, 37, 3406 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Ratra, B. 1991, Phys. Rev. D, 43, 3802 [NASA ADS] [CrossRef] [Google Scholar]
 Ratra, B., Stompor, R., Ganga, K., et al. 1999, ApJ, 517, 549 [NASA ADS] [CrossRef] [Google Scholar]
 Ratra, B., & Vogeley, M. S. 2008, PASP, 120, 235 [NASA ADS] [CrossRef] [Google Scholar]
 Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, AJ, 116, 1009 [NASA ADS] [CrossRef] [Google Scholar]
 Sami, M. 2009, Curr. Sci., 97, 887 [Google Scholar]
 Samushia, L., & Ratra, B. 2006, ApJ, 650, L5 [NASA ADS] [CrossRef] [Google Scholar]
 Samushia, L., & Ratra, B. 2008, ApJ, 680, L1 [NASA ADS] [CrossRef] [Google Scholar]
 Samushia, L., & Ratra, B. 2009a, ApJ, 701, 1373 [NASA ADS] [CrossRef] [Google Scholar]
 Samushia, L., & Ratra, B. 2009b, ApJ, 703, 1904 [NASA ADS] [CrossRef] [Google Scholar]
 Samushia, L., & Ratra, B. 2010, ApJ, 714, 1347 [NASA ADS] [CrossRef] [Google Scholar]
 Samushia, L., Chen, G., & Ratra, B. 2007 [arXiv:0706.1963] [Google Scholar]
 Samushia, L., Dev, A., Jain, D., et al. 2010, Phys. Lett. B, 693, 509 [NASA ADS] [CrossRef] [Google Scholar]
 Samushia, L., Percival, W. J., Guzzo, L., et al. 2011, MNRAS, 410, 1993 [NASA ADS] [Google Scholar]
 Santos, R. C., & Lima, J. A. S. 2008, Phys. Rev. D, 77, 083505 [NASA ADS] [CrossRef] [Google Scholar]
 Schaefer, B. E. 2007, ApJ, 660, 16 [NASA ADS] [CrossRef] [Google Scholar]
 Shafieloo, A., Sahni, V., & Starobinsky, A. A. 2009, Phys. Rev. D, 80, 101301 [NASA ADS] [CrossRef] [Google Scholar]
 Shao, S.H., & Chen, P. 2010, Phys. Rev. D, 82, 126012 [NASA ADS] [CrossRef] [Google Scholar]
 Simon, J., Verde, L., & Jimenez, R. 2005, Phys. Rev. D, 71, 123001 [NASA ADS] [CrossRef] [Google Scholar]
 Sloth, M. S. 2010, Int. J. Mod. Phys. D, 19, 2259 [NASA ADS] [CrossRef] [Google Scholar]
 Spergel, D. N., Verde, L., Peiris, H. V., et al. 2003, ApJS, 148, 175 [NASA ADS] [CrossRef] [Google Scholar]
 Wang, Y. 2008, Phys. Rev. D, 78, 123532 [NASA ADS] [CrossRef] [Google Scholar]
 Wang, Y. 2010, Mod. Phys. Lett. A, 25, 3093 [NASA ADS] [CrossRef] [Google Scholar]
 Wang, Y., Percival, W., Cimatti, A., et al. 2010, MNRAS, 409, 737 [NASA ADS] [CrossRef] [Google Scholar]
 Wei, H. 2011, Phys. Lett. B, 695, 307 [CrossRef] [Google Scholar]
 Wilson, K. M., Chen, G., & Ratra, B. 2006, Mod. Phys. Lett. A, 21, 2197 [NASA ADS] [CrossRef] [Google Scholar]
 Yang, R.J., & Zhang, S. N. 2010, MNRAS, 407, 1835 [NASA ADS] [CrossRef] [Google Scholar]
 Yashar, M., Bozek, B., Abrahamse, A., et al. 2009, Phys. Rev. D, 79, 103004 [NASA ADS] [CrossRef] [Google Scholar]
All Tables
All Figures
Fig. 1 We display the 1, 2, and 3σ constraint contours for the ΛCDM model from the ADD data. The dashed diagonal line corresponds to spatially flat models and the shaded area in the upper lefthand corner is the region for which there is no big bang. The star marks the bestfit pair (Ω_{m0},Ω_{Λ}) = (0.19, − 0.62) with . 

Open with DEXTER  
In the text 
Fig. 2 We show the 1, 2, and 3 σ constraint contours for the XCDM parametrization from the ADD data. The dashed horizontal line at ω_{X} = −1 corresponds to spatially flat ΛCDM models. The star marks the bestfit pair (Ω_{m0},w_{X}) = (0.01, − 0.12) with . 

Open with DEXTER  
In the text 
Fig. 3 We perform the 1, 2, and 3 σ constraint contours for the φCDM model from the ADD data. The horizontal axis at α = 0 corresponds to spatially flat ΛCDM models. The star marks the bestfit pair (Ω_{m0},α) = (0.54,5) with . 

Open with DEXTER  
In the text 
Fig. 4 We show the 1, 2, and 3 σ constraint contours for the φCDM model derived from the BAO data. The horizontal axis at α = 0 corresponds to spatially flat ΛCDM models. The star marks the bestfit pair (Ω_{m0},α) = (0.32,2.01) with . 

Open with DEXTER  
In the text 
Fig. 5 We display the 1, 2, and 3σ constraint contours for the φCDM model from the SNeIa data. The horizontal axis at α = 0 corresponds to spatially flat ΛCDM models. Thin solid lines (best fit at Ω_{m0} = 0.27 and α = 0.0 with , marked by “ × ” ) exclude systematic errors, while thick solid lines (best fit at Ω_{m0} = 0.27 and α = 0.0 with , marked by “♢”) account for systematics. 

Open with DEXTER  
In the text 
Fig. 6 Thick (thin) solid lines are 1, 2, and 3σ constraint contours for the ΛCDM model from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. The cross (“+”) marks the bestfit point determined from the joint sample without the ADD data at Ω_{m0} = 0.28 and Ω_{Λ} = 0.76 with . The star (“ ∗ ”) marks the bestfit point determined from the joint sample with the ADD data at Ω_{m0} = 0.28 and Ω_{Λ} = 0.72 with . The dashed sloping line corresponds to spatially flat models. 

Open with DEXTER  
In the text 
Fig. 7 Thick (thin) solid lines are 1, 2, and 3σ constraint contours for the XCDM parametrization from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. The cross (“+”) marks the bestfit point determined from the joint sample without the ADD data at Ω_{m0} = 0.28 and ω_{X} = −1.04 with . The star (“ ∗ ”) marks the bestfit point determined from the joint sample with the ADD data at Ω_{m0} = 0.28 and ω_{X} = −0.99 with . The dashed horizontal line at ω_{X} = −1 corresponds to spatially flat ΛCDM models. 

Open with DEXTER  
In the text 
Fig. 8 Thick (thin) solid lines are 1, 2, and 3σ constraint contours for the φCDM model from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. The cross (“ × ”) marks the bestfit point determined from the joint sample without the ADD data at Ω_{m0} = 0.28 and α = 0 with . The diamond (“♢”) marks the bestfit point determined from the joint sample with the ADD data at Ω_{m0} = 0.28 and α = 0.01 with . The α = 0 horizontal axis corresponds to spatially flat ΛCDM models. 

Open with DEXTER  
In the text 
Fig. 9 Onedimensional marginalized distribution probabilities of the cosmological parameters for the LCDM model. Thick (thin) lines are the results from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. 

Open with DEXTER  
In the text 
Fig. 10 Onedimensional marginalized distribution probabilities of the cosmological parameters for the XCDM parametrization. Thick (thin) lines are the results from a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. 

Open with DEXTER  
In the text 
Fig. 11 Onedimensional marginalized distribution probabilities of the cosmological parameters for the φCDM model. Thick (thin) lines are the results of a joint analysis of the BAO and SNeIa (with systematic errors) data, with (and without) the ADD data. 

Open with DEXTER  
In the text 