Issue 
A&A
Volume 573, January 2015



Article Number  A88  
Number of page(s)  5  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201424429  
Published online  23 December 2014 
Constraints on the duality relation from ACT cluster data
^{1}
Department of Physics, Imperial CollegeBlackett Laboratory,
London
SW7 2AZ,
UK
email:
r.desousagoncalves@imperial.ac.uk
^{2}
Departamento de Astronomia, Observatório Nacional,
20921400
Rio de Janeiro –
RJ,
Brazil
email:
bernui@on.br
^{3}
Departamento de Física, Universidade Estadual da
Paraíba, 58429500
Campina Grande –
PB,
Brazil
email:
holanda@uepb.edu.br
^{4}
Departamento de Física, Universidade Federal de Campina
Grande, 58429900
Campina Grande –
PB,
Brazil
email:
alcaniz@on.br
Received: 18 June 2014
Accepted: 8 August 2014
Aims. The cosmic distanceduality relation, d_{L}(z) /d_{A}(z)(1 + z)^{2} = η, where d_{L}(z) and d_{A}(z) are, respectively, the luminosity and the angular diameter distances and where η = 1 holds as long as the number of photons is conserved and gravity is described by a metric theory. Testing such hypotheses is, therefore, an important task for both cosmology and fundamental physics.
Methods. We estimated the gas mass fraction of 91 galaxy clusters (GC) from current measurements of the GC total mass recently reported by the Atacama Cosmology Telescope collaboration. We use these data, along with type Ia supernovae observations of the Union 2.1 compilation, to test a possible deviation from the value η = 1.
Results. Although they agree with the standard hypothesis, we find that this combination of both data tends to favor values of η less than 1, which might be associated with some physical processes that increase the number of photons and modifying the above relation to d_{L}< (1 + z)^{2}d_{A}.
Key words: cosmology: theory / galaxies: clusters: general / distance scale
© ESO, 2014
1. Introduction
The socalled distance reciprocity law, which was proven long ago by Etherington (1933; see also Ellis 1971, 2007), is a fundamental keystone for interpreting observational data in cosmology. Considering that the number of photons from a given source is conserved, it provides the following relation between the angular (d_{A}) and luminosity (d_{L}) distances (1)This version of the reciprocity relation, also known as the cosmic distanceduality relation (CDDR), is valid for all cosmological models based on Riemannian geometry and only requires that source and observer are connected by null geodesics in a Riemannian spacetime and that the number of photons be conserved. Examples of nonstandard frameworks that violate the CDDR include scenarios where photons do not travel on unique null geodesics (Csaki et al. 2002), models with variations in fundamental constants (Brax et al. 2013), those with photon nonconservation due to coupling to particles beyond the standard model of particle physics (Avgoustidis et al. 2010, 2012) and with absorption by dust (Basset & Kunz 2004), among others (see Uzan et al. 2004, and references therein).
Recently, several authors have explored different techniques for testing the CDDR. For instance, Bassett & Kunz (2004) used type Ia supernovae (SNe Ia) data, such as measurements of d_{L} and estimates of angular distances from FRIIb radio galaxies (Daly & Djorgovski 2003) and from ultracompact radio sources (Gurvitz 1994, 1999; Lima & Alcaniz 2002), to test possible deviations of the CDDR. Perhaps owing to a lensing magnification bias, they found a 2σ violation caused by an excess in brightening of SNIa at z> 0.5. Ellis et al. (2013) propose an interesting test of Eq. (1) using the cosmic microwave background spectrum. From this observable, it was found that the CDDR relation cannot be violated by more than 0.01% between the decoupling era and today. Khedekar & Chakraborti (2011) propose the use of a redshifted 21 cm signal from disk galaxies, where neutral hydrogen (HI) masses are seen to be almost linearly correlated with surface area, to detect a possible violation of the CDDR. More et al. (2009) discuss the constraints on the CDDR using measurements of baryon acoustic feature and SNe Ia data.
Measurements of the angular diameter distance from galaxy clusters observations, calculated with the Xray and SunyaevZel’dovich observations, have also been widely used to test the CDDR (see, e.g., Holanda et al. 2012a, and references therein). It is known, however, that the verification of the CDDR validity depends on the assumptions used to describe the galaxy clusters (Holanda et al. 2010, 2012a,b; Nair et al. 2011; Li et al. 2011; Meng et al. 2012; Lima et al. 2012; Xi et al. 2013; Allen et al. 2013). As an example, assuming the CDDR, Holanda et al. (2012b) and Meng et al. (2012) show that the elliptical βmodel constitutes a better geometrical description of the galaxy cluster structure when compared to the spherical βmodel. Following a different approach, Gonçalves et al. (2012) show that measurements of the gas mass fraction (f_{gas}) of galaxy clusters depend on the validity of the CDDR and used a sample of 38 clusters along with SNe Ia observations to test the CDDR. Assuming a parametrization for a possible deviation from the CDDR, η = 1 + η_{0}z/ (1 + z), they find η_{0} = −0.07 ± 0.24 at a 95.4% confidence level.
Recently, the cluster’s mass data from 91 galaxy clusters has been detected via the SunyaevZel’dovich effect (SZ; Sunyaev & Zel’dovich 1970) at 148 GHz, as reported by the Atacama Cosmology Telescope (ACT) survey (Hasselfield et al. 2013). It is important to emphasize that the total mass measurements of galaxy clusters depend on the model assumed for the physics of the intracluster gas. The ACT team adopted four models (see Sect. 2.1.1 for details) in order to estimate the corresponding cluster mass M_{500}, defined as the mass measured within the radius R_{500}, at which the enclosed mean density is 500 times the critical density at the cluster redshift. To obtain f_{gas} estimates (one of the main quantities for our analysis) for each cluster in the sample, we use a semiempirical relation presented by Vikhlinin et al. (2009), where the observed gas fraction, f_{gas}, is a linear function of the logarithm of the cluster total mass, log M, which is f_{gas} = f_{gas}(M). This relation was derived from an observational sample where the total mass and the gas mass of the clusters were obtained by two different approaches. The first one, assuming hydrostatic equilibrium, directly measures the total mass and gas mass from observed ICM parameters (Kravtsov et al. 2006), whereas the second approach consists in measuring the total mass from the average temperature, with the gas mass determined from the Xray image (Mohr et al. 1999). Although both methods rely on different hypotheses, they show close linear relationships between f_{gas} and log M, as can be inferred from the analyses of Vikhlinin et al. (2009), where the scatter of the data appears consistent with the mass measurement uncertainties of each method. This result validates the use of the f_{gas} = f_{gas}(M) relationship for cluster data, such as the ACT mass estimates, which do not assume hydrostatic equilibrium.
Our goals in this paper are not only to derive new bounds on a possible deviation from Eq. (1) using the current ACT and SNe Ia data, but also to verify the robustness of the methods used to infer M_{500}. In other words, we discuss the compatibility of these methods in light of the CDDR and the assumptions behind it. The paper is organized as follows. In Sect. 2 we describe the observational quantities used in this work. The corresponding constraints on the departures of the CDDR are investigated in Sect. 3. We end this paper by summarizing our main results (Sect. 4).
2. Observational test
We combined two observational data sets to perform our CDDR test: the gas mass fraction (f_{gas}), obtained from the M_{500} data sets provided by the ACT galaxy clusters survey, and measurements of distance moduli from the Union 2.1 SNe Ia compilation.
2.1. f_{gas}
The socalled gas mass fraction test (Sasaki 1996; Allen et al. 2002, 2008, 2013; Ettori et al. 2009) assumes that the ratio f_{gas} ≡ M_{gas}/M_{tot} is fairly constant over their recent cosmic history, where M_{gas} is the gas mass and M_{tot} the total mass (including dark matter) of the galaxy cluster.
In recent simulations Battaglia et al. (2014) have shown that astrophysical processes, such as AGN feedback, contribute to driving cluster gas out of cluster potential wells, but this process is most effective early on during the assembly epoch of the cluster, that is, for z> 2. Instead, these simulations (Battaglia et al. 2014) also show that the gas mass fraction reveals no significant evolution in the interval 0 <z ≲ 1. Because the ACT data we are using here have only five galaxy clusters with z> 1 in a sample of 91 (with the highest redshift z = 1.36), and moreover, because the ACT cluster’s masses are within the constraints used in the above simulations, we expect that f_{gas} does not depend noticeably on redshift in the examined data. In this way, we follow similar studies of this subject (see, e.g., Sasaki 1996; Allen et al. 2002, 2008, 2013; Ettori et al. 2009) and assume that f_{gas} is fairly constant in the redshift and total mass data intervals under analysis.
It is well known that the f_{gas} obtained from Xray measurements is given by (Sasaki 1996) (2)which is valid only when the CDDR is assumed. The general expression for the gas mass fraction is given by (Gonçalves et al. 2012) (3)where the symbol ∗ denotes quantities that were obtained for the fiducial model assumed in the observations, and the parameter N defines the astrophysical modeling of the cluster. Many parametrizations are assumed in the literature (Allen et al. 2002, 2008; Ettori et al. 2009, among others), but because we aim to constrain the CDD relation, N is not an important quantity for our analysis, and we marginalize over it. Along with the assumption of the fairly constant M_{gas}/M_{tot} ratio mentioned above, the distance ratio of Eq. (3) accounts for deviations in the geometry of the universe from the fiducial model, which makes it possible to use f_{gas} data to test different cosmologies (Lima et al. 2003).
2.1.1. f_{gas} from the ACT catalog
As is widely known, properties of galaxy clusters encode information on the growth of structures in the universe. For this, data from optical and Xray cluster surveys are currently used to constrain cosmological parameters. Recently, however, a new window of information regarding galaxy cluster physics has opened with the release of cluster surveys (e.g., Staniszewski et al. 2009; Marriage et al. 2011; Williamson et al. 2011; Planck Collaboration VIII 2011; Reichardt et al. 2013) that make use of the SZ effect, which is a signature that does not diminish with luminosity distance because it is nearly independent of redshift. In fact, thermal SZ signal is a tracer of the total thermal energy of the hot (~10^{7} K) intracluster gas, therefore it is correlated to the mass of the cluster.
Fig. 1 Distribution of the number of clusters according to their gas mass fraction f_{gas}. The f_{gas} values were inferred from the M_{500} data (Hasselfield et al. 2013) according to the four methods discussed in the text. The dashed line, depicted slightly to the right for illustration, is the cosmic mean value for f_{gas} = 0.10 ± 0.02 (David et al. 2012). 

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However, a model for the physics of the intracluster gas has to be assumed. Recently, the ACT team has investigated a new approach to obtaining the cluster mass from cluster signals in filtered SZ maps. They first adopt a oneparameter family of universal pressure profiles (UPP) as a baseline model for the intracluster gas pressure profile and apply it to their cluster sample, in order to estimate the corresponding cluster mass (Arnaud et al. 2010). This approach includes a mass dependence in the profile shape, which has been calibrated to Xray observations using local clusters (z< 0.2). In this way, following this UPP approach, the cluster mass of the sample is measured within a characteristic radius with respect to the critical density such that, for instance, M_{500} is defined as the mass measured within the radius, R_{500}, at which the enclosed mean density is 500 times the critical density at the cluster redshift. In this case, the cluster’s mass obtained using this method are termed .
As mentioned earlier, it is possible to obtain f_{gas} from the total mass by using the semiempirical relation discussed by Vikhlinin et al. (2009). Such a relation can be written as (4)where M_{15} is the cluster total mass M_{500} in units of 10^{15}h^{1}M_{⊙}. It is worth mentioning that this relation does not assume the validity of the CDDR in their derivation.
Besides this UPP model procedure, three other approaches have been adopted by the ACT team to study the scaling relations that allow us to estimate M_{500} from the cluster SZ signal strength. These are based on (i) structure formation simulations (Bode et al. 2012), where the density and temperature of the intracluster are modeled as a virialized ideal gas (); (ii) a nonthermal pressure and adiabatic model for the gas (; Trac et al. 2011); and (iii) a dynamical estimate of the cluster mass using the galaxy velocity dispersions (; Sifon et al. 2013). It is worth mentioning that, when assuming the socalled concordance cosmology (a flat ΛCDM model with Ω_{m} = 0.3), the scaling from UPP is nearly identical to the adiabatic nonthermal model, while a model incorporating nonthermal pressure is in better agreement with dynamical mass measurements (Hasselfield et al. 2013). Considering these four methods for estimating M_{500}, in Fig. 1 we show the distribution of the number of clusters according to their gas mass fraction inferred from the ACT sample.
The linear relationship between f_{gas} and log M in Eq. (4) was derived using lowredshift (i.e., z< 0.3) Xray data. It is natural to expect that, owing to galaxy cluster’s evolution, such linear dependence would change for high redshifts. However, Vikhlinin et al. (2009) performed simulations and found that, although the redshiftdependence exists, it is almost completely removed, thereby escalating the cluster’s masses again by the characteristic nonlinear mass scale M_{nl}, which corresponds to a linear fluctuation amplitude of 1.686: σ(M_{nl}) = 1.686. In other words, these simulations indicate that f_{gas}(M/M_{nl}) is almost redshiftindependent, at least at z ~ 1 (see Vikhlinin et al. 2009, Sect. 4.2.2). We performed analyses that reescalate the galaxy cluster’s masses and found that the largest discrepancy in the χ^{2} value is less than 2.7% (UPP model).
2.2. SNe Ia
To obtain measurements of d_{L}, we use the distance moduli (μ) obtained from current SNe Ia observations. This quantity is related to the luminosity distance by (5)The data set used for SNe Ia is the Union 2.1 compilation (Suzuki et al. 2011), which contains 580 events distributed in the redshift range 0.01 <z< 1.41. The SNe Ia redshifts were carefully chosen to coincide with the ones of the associated galaxy cluster sample with Δz ≡  z_{cluster} − z_{SNe Ia}  ≤ 0.01 (Fig. 2). For more details on SNe Ia analysis, we refer the reader to Suzuki et al. (2011).
Fig. 2 Redshift difference between SNe Ia and galaxy clusters used in the analysis. 

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3. Analysis and results
To analyze the validity of the CDDR, we rewrite Eq. (1) as d_{L}/d_{A}(1 + z)^{2} = η(z), where η(z) quantifies a possible deviation from the CDDR. In our analysis we assume η(z) = 1 + η_{0}z, which is compatible with the cosmographic limit η(z ≪ 1) = 1. Combining the above equation with Eqs. (3) and (5), we obtain (Gonçalves et al. 2012) (6)which is the observed quantity built from measurements of μ and f_{gas}. Thus, the likelihood estimator is determined by a χ^{2} statistic , where takes the propagation of the statistical errors into account in Eq. (4). As mentioned earlier, the normalization factor N (see Eq. (2)) is taken as a nuisance parameter so that we marginalize over it.
Bounds on η_{0} from f_{gas} obtained using four different methods, as explained in the text.
The first results of this analysis are presented in Table 1. All methods considered are compatible with the CDDR at the 2σ level. For comparison, we also show the result of the reduced χ^{2} (χ^{2}/ν, where ν are the degrees of freedom) and of the pvalue. The latter quantity refers to the probability of obtaining a test at least as extreme as the observed one, under the assumption that the null hypothesis, i.e. η_{0} = 0, is true. For instance, p< 0.05 indicates that the null hypothesis is probably false (Jeffreys 1961; Robert et al. 2009). Clearly, this is not the case for the results presented above because, for the four M_{500}methods considered here, we find pvalues that are higher than 0.7.
In the analysis above, the maximum difference between the redshifts of the clusters and the SNIa is Δz ≤ 0.01. We also performed the same analysis, restricting our data set to a subset where Δz ≤ 0.001. This is an important aspect for analyses of the CDDR involving different types of observables since the reciprocity theorem is valid for sources at the very same redshift. The number of points in this new subset decreases to 55 points, whereas the χ^{2} improves about 15%. The mean values remain much the same, but the validity of CDDR can be verified even when considering 1σ error, as we can see in Table 2.
An important aspect worth mentioning is that the likelihoods for η_{0} (regardless of the method considered) have values that are preferably negative, indicating that may be underestimated and favoring a reduction of d_{L} relative to d_{A} (see Eq. (1)). This is in full agreement with the results obtained by Holanda et al. (2012a), who only use measurements of f_{gas} from the SZ and Xray observations. Physically, such a result can be explained by a radiative process that increases the number of photons in a light bundle and therefore increases the apparent luminosity, making the source appear closer. (We refer the reader to Avgoustidis 2010, 2012 for more on possible coupling of exotic particles with photons.) Another explanation would be a possible excess of brightness of the SNe Ia data or a value unsuitable for the galaxy cluster borders (R_{500}), which may underestimate the gas mass fraction calculated in Eq. (4).
Same as in Table 1 considering Δz ≤ 0.001.
4. Conclusions
In this paper, we have investigated whether the four physical models used by the ACT to describe the intracluster gas of 91 galaxy clusters are compatible with the validity of the CDDR, d_{L}/d_{A}(1 + z)^{2} = 1. For this purpose, we alleviated the equality to d_{L}/d_{A}(1 + z)^{2} = η, where η is a timedependent parametrization. In our analysis, we adopted a linear parametrization written as η(z) = 1 + η_{0}z for which values apart from η_{0} = 0 leads to a violation of CDDR. To obtain the observational values for d_{L}, we used 580 measurements of distance moduli from SNe Ia (from Union 2.1, see Suzuki et al. 2012). For d_{A}, we first used a result of Vikhlinin et al. (2009) to obtain 91 gas mass fraction data from the galaxy cluster’s total mass measurements (from the ACT survey, see Marriage et al. 2011; Hasselfield et al. 2013) and then followed Gonçalves et al. (2012) to derive the d_{A} data points.
This statistical analyses showed that the results are almost independent of the method used to obtain M_{gas}. The main result of this analysis was performed by restricting the difference between z_{cluster} and z_{SNe Ia} to Δz ≤ 0.001. In this case, the validity of the CDDR can be probed within a 1σ level irrespective of the M_{500} method adopted. These results reinforce the interest in probing the CDDR using new and more precise techniques, since a clear departure from η_{0} = 0 may be associated either with an exotic gravity theory or with nonconservation of the photon number during the cosmic history.
Acknowledgments
The authors thank CNPq, INCTA, INEspaço, and FAPERJ for the grants under which this work was carried out.
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All Tables
Bounds on η_{0} from f_{gas} obtained using four different methods, as explained in the text.
All Figures
Fig. 1 Distribution of the number of clusters according to their gas mass fraction f_{gas}. The f_{gas} values were inferred from the M_{500} data (Hasselfield et al. 2013) according to the four methods discussed in the text. The dashed line, depicted slightly to the right for illustration, is the cosmic mean value for f_{gas} = 0.10 ± 0.02 (David et al. 2012). 

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In the text 
Fig. 2 Redshift difference between SNe Ia and galaxy clusters used in the analysis. 

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In the text 
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