Journal of Cosmology and
Astroparticle Physics
Constraining H
0
in general dark energy models
from Sunyaev-Zeldovich/X-ray technique and
complementary probes
To cite this article: R.F.L. Holanda et al JCAP02(2012)035
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ournal of Cosmology and Astroparticle Physics
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J
Constraining H
0
in general dark
energy models from
Sunyaev-Zeldovich/X-ray technique
and complementary probes
R.F.L. Holanda,
aJ.V. Cunha,
b,cL. Marassi
dand J.A.S. Lima
aaDepartamento de Astronomia (IAGUSP), Universidade de S˜ao Paulo,
Rua do Mat˜ao 1226, 05508-900, S˜ao Paulo, SP, Brazil
bCentro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC,
Rua Santa Ad´elia 166, 09210-170, Santo Andr´e, SP, Brazil cFaculdade de F´ısica, Universidade Federal do Par´a,
Par´a, Bel´em, Brazil
dEscola de Ciˆencia e Tecnologia, UFRN, 59072-970, Natal, RN, Brasil
E-mail: holanda@astro.iag.usp.br,jvcunha@ufpa.br,luciomarassi@ect.ufrn.br,
limajas@astro.iag.usp.br
Received November 5, 2011 Revised January 16, 2012 Accepted January 31, 2012 Published February 24, 2012
Abstract. In accelerating dark energy models, the estimates of the Hubble constant, H0, from Sunyaev-Zel’dovich effect (SZE) and X-ray surface brightness of galaxy clusters may depend on the matter content (ΩM), the curvature (ΩK) and the equation of state parameter (ω). In this article, by using a sample of 25 angular diameter distances of galaxy clusters described by the elliptical β model obtained through the SZE/X-ray technique, we constrain
H0 in the framework of a general ΛCDM model (arbitrary curvature) and a flat XCDM
model with a constant equation of state parameter ω = px/ρx. In order to avoid the use of priors in the cosmological parameters, we apply a joint analysis involving the baryon acoustic oscillations (BAO) and the CMB Shift Parameter signature. By taking into account the statistical and systematic errors of the SZE/X-ray technique we obtain for nonflat ΛCDM model H0= 74+5.0−4.0 km s−1 Mpc−1(1σ) whereas for a flat universe with constant equation of state parameter we find H0 = 72+5.5−4.0km s−1Mpc−1(1σ). By assuming that galaxy clusters are described by a spherical β model these results change to H0 = 62+8.0−7.0and H0= 59+9.0−6.0km s−1 Mpc−1(1σ), respectively. The results from elliptical description are in good agreement with independent studies from the Hubble Space Telescope key project and recent estimates based
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on the Wilkinson Microwave Anisotropy Probe, thereby suggesting that the combination of these three independent phenomena provides an interesting method to constrain the Hubble constant. As an extra bonus, the adoption of the elliptical description is revealed to be a quite realistic assumption. Finally, by comparing these results with a recent determination for a flat ΛCDM model using only the SZE/X-ray technique and BAO, we see that the geometry has a very weak influence on H0 estimates for this combination of data.
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Contents
1 Introduction 1
2 Basic equations and models 3
3 Theoretical method and samples 4
3.1 SZE/X-ray technique 4
3.2 Baryon Acoustic Oscillations (BAO) 6
3.3 CMB — Shift parameter 7
4 Analysis and results 7
4.1 ΛCDM model 8
4.2 Flat XCDM model 8
5 H0 constraints: the influence of the morphology 10
6 Conclusions 12
1 Introduction
Clusters of galaxies are the largest gravitationally bound structures in the universe and they can be regarded as being representative of the universe as a whole. An interesting phenomenon occurring in galaxy clusters is the Sunyaev-Zel’dovich effect (SZE), a small distortion of the Cosmic Microwave Background (CMB) spectrum, provoked by the inverse
Compton scattering of the CMB photons passing through a population of hot electrons [1–
7]. The SZE is quite independent of the redshift and it provides an useful tool for studies of intermediate and high redshift galaxy clusters being also an important cosmological probe (for reviews see [8,9]).
When the X-ray emission of the intracluster medium (ICM) is combined with the SZE,
it is possible to estimate the angular diameter distance (ADD) DA of galaxy clusters. In
other words, the SZE/X-ray technique distances to galaxy clusters and consequently can be
used to measure the Hubble constant, H0. The main advantages of this method to estimate
H0 are: i) it does not rely on the extragalactic distance ladder, being fully independent of any local calibrator [9, 10]; ii) it is a cross-checking for local direct estimates; and iii) it is free from Hubble bubbles, since the galaxy clusters are into the Hubble flow. As the main disadvantages, we could point out that: i) this method is dependent of galaxy cluster properties assumptions; ii) it may present a cosmological model dependence in the analysis; and iii) it is not yet competitive with traditional methods [11]. The importance to access the distance scale by different methods, and, more important yet, granting independence of any distance calibrators, has already been pointed out in the literature [12].
It should be stressed that the determination of the Hubble constant has a practical
and theoretical importance to many astrophysical and cosmological observations [13]. In
2001, Freedman et al. [11] presented the final results of the Hubble Space Telescope (HST) key project by combining five different methods to measure distances: Tully-Fisher relation,
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72 ± 8 km/s/Mpc (statistical + systematic errors). On the other hand, by using Cepheid-calibrated luminosity of Type Ia supernovae, Sandage and collaborators [14] announced the
results from their HST programme, H0 = 62.3 ± 1.3 km/s/Mpc (statistical). However,
since the effect of metal abundance on the Cepheid P-L relation is yet controversial, Riess
et al. using the same data found H0 = 73 ± 4 km/s/Mpc (statistical). Later on, Van
Leeuwen et al. [15] by revising Hipparcos parallaxes for Cepheid distance obtained values higher than previous results advocated by Sandage et al. [14] (H0 = 70 ± 5) and Freedman et al. (H0 = 76 ± 8) [11]. These last results are insensitive to Cepheid metallicity corrections.
Recently, Riess et al. [16] reported results of a program to determine the Hubble constant to ≈ 5% precision from a refurbished distance ladder based on extensive use of differential
measurements. They obtained H0 = 74.2 ± 3.6 km/s/Mpc, a 4.8% uncertainty including
both statistical and systematic errors. Indeed, some estimates of H0 have yielded ≃ 74
km/s/Mpc [11,16,17].
Komatsu et al. [17] have shown that only CMB analysis can not supply strong
con-straints on the H0 value. This problem occurs due to the degeneracy on the parameter space and may be circumvented only by using a joint analysis involving complementary probes, such as: independent measurements of H0, type Ia supernovae and baryon acoustic oscilla-tion (BAO). Moreover, there is a significant tension in the H0 value predicted from CMB +
BAO and CMB+high-z SNe Ia for a large class of dark energy models [16, 18]. Amendola
et al. [18] have suggested that a proper treatment of weak-lensing effects may improve the concordance of the supernova data with the CMB and the BAO thereby strengthen the case for the standard ΛCDM model. In this connection, complementary studies based on the time delay from gravitational lens have also allowed different groups to estimate H0 in a flat ΛCDM framework. In particular, Fadely et al. [19], by adding constraints from stellar popu-lation synthesis models, obtained H0 = 79.3+6.7−8.5 km/s/Mpc (1σ, without systematic errors), and Suyu et al. [20], in combination with WMAP obtained H0 = 69.7+4.9−5.0 km/s/Mpc (1σ, without systematic errors).
Cunha, Marassi and Lima [21] (henceforth CML), by assuming a flat ΛCDM framework,
also derived new constraints on the Hubble constant and matter density parameter based on the so-called SZE/X-ray technique. By considering a sample of 25 galaxy clusters compiled by De Filippis et al. (2005) [22], the degeneracy on the Ωm parameter was broken through a joint analysis combining the SZE/X-ray data with the recent measurements of the baryon
acoustic oscillation (BAO) signature from SDSS catalog [23, 24]. The main advantage of
the method is that one does not need to adopt a fixed cosmological concordance model in the analysis, as usually done in the literature [10, 25–31]. For a flat ΛCDM the joint analysis yielded Ωm = 0.27+0.03−0.02 and H0 = 73.8+4.2−3.3 km/s/Mpc (1σ, neglecting systematic uncertainties).
On the other hand, astronomical observations in the last decade have suggested that our world behaves like a spatially flat scenario, dominated by cold dark matter (CDM) plus an exotic component endowed with large negative pressure, usually named dark energy [32–35]. In the framework of general relativity, besides the cosmological constant, there are several candidates for dark energy, such as: a vacuum decaying energy density, or a time varying Λ(t) [36], the so-called “X-matter” [37], a relic scalar field [38], a Chaplygin gas [39, 40], and cosmologies proposed to reduce the dark sector, among them, models with creation of cold dark matter particles [41]. For a scalar field component and “X-matter” scenarios, the equation of state parameter may be a function of the redshift (see, for example, [42]) or still, as has been discussed by many authors, it may violate the null energy condition [43].
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In this work, we relax the flat geometry condition of the cosmic concordance model
(ΛCDM). This procedure will prove the robustness of the previous H0 estimate using the
SZE/X-ray technique. In addition, in order to test the real dependence of the method with the equation of state parameter, we compare the predictions of the general ΛCDM model and
“X-matter” cosmologies [37]. For the “X-matter” model we assume a flat XCDM cosmology
with constant equation of state (EoS) parameter. In the present study we consider the 25 ADD of galaxy clusters from De Filippis et al. sample [22]. In order to broke the degeneracy on the basic cosmological parameters, we apply a joint analysis using BAO [23, 24] and the CMB probe known as shift parameter [46–49].
The paper is organized as follows. In section 2, we present the basic equations describing the angular diameter distance from galaxy clusters and the cosmological models used in our analysis. In section 3, we give a short description of the SZE/X-ray technique and the observational data used in our joint analysis. The corresponding constraints on the cosmological parameters are investigated in section 4. In section 5 we discuss the sensitivity of the H0 constraints with respect to the assumed morphology of galaxy clusters. The article is ended with a summary of the main results in the conclusion section.
2 Basic equations and models
Let us now assume that the Universe is well described by an homogeneous and isotropic geometry (in our units the speed of light c = 1)
ds2= dt2− a2(t) dr2 1 − kr2 + r 2dθ2+ r2sin2θdφ2 , (2.1)
where a(t) is the scale factor and k = 0, ±1 is the curvature parameter.
In such a background, the angular diameter distance DAreads [22,44,53]
DA= 3000h−1 (1 + z)p|Ωk| Sk p|Ωk| Z z 0 dz′ E(z′) Mpc, (2.2)
where h = H0/100 km s−1 Mpc−1 (henceforth we use this notation for our Hubble constant estimates), the function E(z) = H(z)/H0is the normalized Hubble constant (which is defined by the specific cosmology adopted), Ωk is the curvature parameter, and Sk(x) = sin x, x, sinh x for k = +1, 0, −1, respectively.
In this paper, we consider that the Universe is driven by nonrelativistic matter (cold
dark matter plus baryons) with density parameter ΩM plus a dark energy density parameter
(Ωx), with constant EoS parameter (ω ≡ p/ρ). Following standard lines, we list below the functions E(z) describing the cosmological models adopted in this paper:
(i) ΛCDM model. By allowing deviations from flatness, the normalized Hubble parameter is given by
E2(z) = ΩM(1 + z)3+ ΩΛ+ Ωk(1 + z)2, (2.3)
where Ωk = 1 − ΩΛ− ΩM.
(ii) Flat Parametric Dark Energy model (XCDM). In this case, the normalized Hubble parameter reads:
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For ω = −1, the above XCDM expression reduces to the previous ΛCDM case (Ωx ≡
ΩΛ). When the EoS parameter of dark energy is restricted on the interval −1 ≤ ω ≤ 0, it may be represented by a scalar field (quintessence), and when ω ≤ −1 (thereby violating the null energy condition) we have the case of phantom dark energy [43]. Given the above expressions, we see clearly that DA is a function of z and h, as well as, of the assumed cosmological model.
3 Theoretical method and samples
It should be recalled that the basic aim here is to constrain the Hubble constant and other cosmological parameters using the above quoted models. The method adopted is primarily
based on the angular diameter distances, DA, furnished by the SZE/X-ray technique. The
degeneracies on the remaining cosmological parameters will be broken by a joint analysis involving BAO and CMB signature (shift parameter).
3.1 SZE/X-ray technique
An important phenomenon occurring in clusters is the Sunyaev-Zel’dovich effect (SZE), a small distortion of the Cosmic Microwave Background (CMB) spectrum provoked by the in-verse Compton scattering of the CMB photons passing through a population of hot electrons. The SZE is proportional to the electron pressure integrated along the line of sight (l.o.s.), i.e. to the first power of the plasma density. The measured temperature decrement ∆TSZ of the CMB is given by [22]: ∆TSZ TCMB = f (ν, Te) σTkB mec2 Z l.o.s. neTedl, (3.1)
where Teis the temperature of the intra cluster medium, kBthe Boltzmann constant, TCMB=
2.728◦K is the temperature of the CMB, σ
T the Thompson cross section, me the electron
mass and f (ν, Te) accounts for frequency shift and relativistic corrections.
{Another interesting physical phenomenon occurring in galaxy clusters is the observed X-ray spectrum associated to the thermal bremsstrahlung (free-free) emission due to diffuse intracluster gas [52]. The X-ray surface brightness, SX, is proportional to the integral along the line of sight of the square of the electron density [22,52]:
SX = 1 4π(1 + zc)4 Z l.o.s. n2eΛeHdl, (3.2)
where ΛeH is the X-ray cooling function of the intracluster medium (ICM) in the cluster
rest frame. Note that expressions (5) and (6) mean that the SZE and X-ray emission are explicitly dependent on the ICM properties through the pair of parameters (ne, Te).
The calculation begins by constructing a model for the intracluster gas distribution. For instance, by assuming the spherical isothermal β model [54]
ne(r) = 1 +r 2 r2 c −3β/2 , (3.3)
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For the SZE, one obtains
∆TSZ= ∆T0 1 +θ 2 θ2 c 1/2−3β/2 , (3.4)
where θc = rc/DA is the angular core radius and ∆T0 is the central temperature decre-ment which includes all the physical constants and the terms resulting from the line of sight integration ∆T0 ≡ TCMBf (ν, Te)σTkBTe mec2 ne0√πDAθcg (β/2) , (3.5) with: g(α) ≡ Γ [3α − 1/2]Γ [3α] , where Γ(α) is the standard gamma function.
The X-ray surface brightness is given by SX = SX0 1 + θ θ2 c 1/2−3β , (3.6)
where the central surface brightness SX0 reads: SX0≡
ΛeH µe/µH 4√π(1 + zc)4
n2e0DAθc g(β), (3.7)
with µ denoting the molecular weight of each component (µi ≡ ρi/nimp).
Now, by eliminating ne0, one can solve equations (9) and (11) for the angular diameter distance DA = ∆T02 SX0 mec2 kBTe0 2 g (β) g(β/2)2 θ c × ΛeH0 µe/µH 4π3/2f (ν, T e)2 TCMB2 σ2T (1 + zc)4 . (3.8)
Therefore, the SZE/X-ray method is able to provide distances to galaxy clusters, and, con-sequently, it can be used to measure the Hubble constant H0. The advantage of this method for estimating H0 is that it does not rely on the extragalactic distance ladder, being fully independent of any local calibrator, as we can see directly from the above expression.
On the other hand, observations of galaxy clusters based on Chandra and XMM satel-lites have shown that such systems in general exhibit elliptical surface brightness maps. In this line, by using an isothermal elliptical 2-Dimensional β-model to describe the clusters,
De Filippis et al. (2005) [22] reanalyzed and derived ADD measurements for 25 clusters
from two previous compilations, namely: (i) the data sample observed by Reese et al. [27] which is composed by a selection of 18 galaxy clusters distributed over the redshift inter-val 0.14 ≤ z ≤ 0.8, and (ii) the sample of Mason et al. [26] which has 7 clusters from the X-ray limited flux sample of Ebeling et al. (1996) [55]. It is also worth mentioning that for the first sample a high S/N detections of SZE, high S/N X-ray imaging and electron
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temperatures were available, and that for both compilations, a spherical isothermal β model to describe the clusters geometry was adopted. In their analysis, De Filippis et al. (2005) obtained the angular diameter distances by using two different morphologies to describe the galaxy clusters: the elliptical and spherical β models. It should be recalled that our general analysis in section IV will be performed by using the angular diameter distances based on the elliptical assumption while in section V, the influence of the morphological descripton on H0 constraints will be discussed in detail.
In a series of recent papers, Holanda et al. [56, 57] also shown that De Filippis et al. sample is in agreement with the cosmic distance duality relation involving the luminosity distance (DL) and angular diameter distance (DA), namely, DL(1 + z)−2/DA = 1. In
par-ticular, through a model-independent cosmological test involving DA from galaxy clusters
sample and DL from supernovae Ia data provided by the Constitution sample [58], it was
found that the elliptical geometry is consistent (2σ c.l.) with no violation of the cosmic dis-tance duality [57]. They also shown that the Bonamente et al. [30] sample is only marginally compatible with the duality distance relation when a non-isothermal spherical β model was used to describe the clusters.
In the CML paper [21], the Abell 773 cluster was excluded from the statistical analysis due to its large contribution to the χ2. Now, in order to preserve the completeness of the original data set, these statistical cut-offs arguments will be neglected, and, as such, all the 25 clusters from the original De Fillipis et al. [22] sample will be considered. In what follows, the complementary tests used in our joint analysis to determine the Hubble constant with the SZE/X-ray technique it will be presented.
3.2 Baryon Acoustic Oscillations (BAO)
The detection of a peak in the large-scale correlation function of luminous red galaxies selected from the Sloan Digital Sky Survey (SDSS) Main Sample gave rise to a powerful cosmological probe, often referred to as BAO scale.
Basically, the peak detected at the scale of 100 h−1 Mpc, happens due to the
baryon acoustic oscillations in the primordial baryon-photon plasma prior to recombina-tion, and, such a result, provides a suitable “standard ruler” for constraining dark energy models [23,24].
The relevant distance measure is the so-called dilation scale, DV, that can be modeled as the cube root of the radial dilation times the square of the transverse dilation, at the typical redshift of the galaxy sample, z = 0.35 [23]:
DV(z) = [(1 + z)2DA(z)2z/H(z)]1/3, (3.9)
Recalling that the comoving size of the sound horizon at zls, where zls is the redshift of the last scattering surface, is ∼ 1/pΩMH02, it was pointed out that the combination A(z) = DV(z)pΩMH02/z is independent of H0, and this dimensionless combination is also well constrained by the BAO data
A(0.35) = DV(0.35)
pΩMH02
0.35 = 0.469 ± 0.017. (3.10)
The BAO quantity A(0.35) is exactly what we add to the χ2, in the joint statistical analysis of the cosmological models studied here.
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3.3 CMB — Shift parameter
Another interesting cosmological probe to dark energy models is the so-called CMB shift parameter. Such a quantity is encoded in the location lT T1 of the first peak of the angular (CMB) power spectrum [46,47]
θA≡
rs(zls) DA(zls)
, (3.11)
where DA(zls) is the angular diameter distance to the last-scattering surface. The quantity, rs(zls) ∼ 1/pΩMH02, is the comoving size of the sound horizon at zls.
By using the WMAP three years results [17] Davis et al. [48] converted the location of the first peak in a reduced distance to the last-scattering surface
R = s ΩM |Ωk| Sk p|Ωk| Z zls 0 dz E(z) = 1.71 ± 0.03. (3.12)
The robustness of the shift parameter has been tested by Elgaroy & Multam¨aki [49] and compared to fits of the full CMB power spectrum (see also [50]). As a result, it is now widely believed that the shift parameter is an accurate geometric measure even for non-standard cosmological models, and this explains why it has been extensively used in the determination of some cosmic parameters [51]. It is weakly dependent on h, since zls = zls(h, ΩM, Ωb) is a smooth function of h, and the degeneracies that arise from using R rather than fitting the full CMB power spectrum are well constrained by other data such as BAO and SZE/X-ray.
In the following computations for the theoretical shift parameter we will apply the correction for zls suggested in refs. [49, 50]. In addition, independent of the adopted dark energy model, we keep the value R = 1.70 ± 0.03 for a flat universe and R = 1.71 ± 0.03 for nonzero cosmic curvature [50].
4 Analysis and results
In our statistical procedure we apply a maximum likelihood analysis determined by a χ2
statistics χ2(z|p) = X i (DA(zi; p) − DAo,i)2 σ2 DAo,i+ σ 2 sys,i + (A(p) − ABAO) 2 σ2 BAO +(R(p) − R) 2 σ2 R , (4.1)
where DAo,i is the observational angular diameter distance, σDAo,i is the uncertainty in the
individual distance, σsys,i is the systematic contribution, zi is the redshift of i galaxy cluster
in sample and p is the complete set of parameters. For the ΛCDM model p ≡ (h, ΩM, ΩΛ)
and p ≡ (h, ω, ΩM) for the flat XCDM model.
For the galaxy clusters sample the common statistical contributions are: SZE point
sources ±8%, X-ray background ±2%, Galactic NH ≤ ±1%, ±15% for cluster asphericity,
±8% kinetic SZ and for CMB anisotropy ≤ ±2%. Estimates for systematic effects are as follow: SZ calibration ±8%, X-ray flux calibration ±5%, radio halos +3% and X-ray tem-perature calibration ±7.5%. Typical statistical errors amounts for nearly 20% in agreement with other works (Mason et al. 2001; Reese et al. [26–28]), while for systematics we find typical errors around + 12.4% and - 12% (see table 3 in Bonamente et al. 2006 [30]).
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Figure 1. ΛCDM Model. a) The filled contours on the (h, ΩK) plane (marginalized in ΩM) correspond
to 1(red), 2(green) and 3(blue) σ statistical confidence regions using only 25 angular diameter distances from De Fillipis et al. sample [22]. Note that the constraints on h are dependent on the curvature of the Universe. The vertical dashed lines correspond to bounds (1(red), 2(green) and 3(blue) σ c.l.) from BAO + Shift parameter. b) The filled contours at 68.3%, 95.4% and 99.7% c.l. (only statistical errors) in the (h, ΩK) plane are from a joint analysis involving SZE/X-ray + BAO + Shift
Parameter. c) Probability of the h parameter. The solid line represents our analysis using only the statistical errors and in the dashed line we have combined the statistical and systematic errors in quadrature. The horizontal lines are cuts in the probability regions of 68.3% and 95.4%. We obtain h = 0.735+0.037
−0.038(only statistical errors) and h = 0.74 +0.05
−0.04(statistical + systematic errors) in 68% c.l..
4.1 ΛCDM model
By allowing for deviations from flatness, let us first consider the general ΛCDM model as described by eq. (2.3). In figure 1a, the filled contours on the (h, ΩK) plane (marginalized in ΩM) correspond, respectively, to 1(red), 2(green) and 3(blue) σ statistical confidence regions (only statistical errors) using the 25 angular diameter distances of the De Fillipis et al. sample [22]. Clearly, we see that the possible values for h are yet weakly constrained if
one uses only SZE/X-ray data, but they are weakly dependent on the ΩK parameter. Note
that the h parameter lies on the interval 0.63 ≤ h ≤ 0.93 at 99.7% c.l. (2 free parameter) for ΩK ∈ [−0.3, 0.3]. The vertical dashed lines correspond to bounds [1(red), 2(green) and 3(blue) σ c.l.] from BAO + Shift parameter analysis. It restricts only the curvature parameter which is here constrained by −0.09 ≤ ΩK ≤ 0.08 (3σ c.l.) .
In figure 1b, the filled contours at 68.3%, 95.4% and 99.7% c.l. (only statistical errors) in the (h, ΩK) plane are obtained from a joint analysis involving SZE/X-ray + BAO + Shift Parameter. Taking account only statistical errors in our analysis we obtain from the joint analysis h = 0.735+0.042−0.038, Ωk = −0.010+0.012−0.013 and χ2min = 28.12 at 68.3% (c.l.). Its reduced value, that is, taking into account the associated degrees of freedom is χ2red= 1.12. In figure 1c, we display the likelihood function of the h parameter by using the SZE/X-ray data + BAO + Shift Parameter. The solid line represents our analysis using only the statistical errors and in the dashed line we have combined the statistical and systematic errors in quadrature. The horizontal lines are cuts in the probability regions of 68.3% and 95.4%. We obtain h = 0.735+0.037−0.038 (only statistical errors) and h = 0.74+0.05−0.04 (statistical + systematic errors). This plot is very similar to the figure 4 of the previous CML paper [21] for a flat ΛCDM model. Therefore, it is safe to conclude that the constraints on h derived here are weakly dependent on the geometry of the Universe.
4.2 Flat XCDM model
A large class of dark energy candidates can phenomenologically be described by the barotropic equation of state, p = ωρ, where ω is a negative constant parameter [37]. In the flat case,
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Figure 2. Flat XCDM Model. a) The filled contours on the (h, ω) plane (marginalized in ΩM)
correspond to 1(red), 2(green) and 3(blue) σ confidence regions using only angular diameter distances. The vertical dashed lines correspond to bounds (1 (red), 2 (green) and 3 (blue) σ c.l.) from BAO + Shift parameter. b) The filled color contours on the (ω, h) plane are derived from a joint analysis (SZE/X-Ray, BAO and Shift) (only statistical errors). We also have marginalized over ΩM. Note
that the values beyond the phantom barrier are also allowed (ω ≤ −1). c) Probability of the h parameter. The solid line represents our analysis using only the statistical errors and in the dashed line we have combined the statistical and systematic errors in quadrature. We obtain h = 0.71+0.042−0.037 (only statistical errors) and h = 0.72+0.055
−0.04 (statistical + systematic errors). The horizontal cuts
represent 68.3% and 95.4%. In this case, we have marginalized over ω and ΩM.
the normalized Hubble parameter is readily obtained by taking Ωk= 0 in eq. (2.4). In this context, by relaxing the usual imposition ω ≥ −1 we have investigated some implications for the so-called phantom dark energy [43–45]. The basic idea here was to explore the influence of ω parameter on the Hubble constant determination, thereby enabling us to compare with the ΛCDM model (ω = −1) results. In addition, we have also tested the sensibility of the SZE/X-ray data + BAO + Shift Parameter with respect to constraints on the ω parameter. In figure 2a, the filled contours on the (h, ω) plane (marginalized in ΩM) correspond to 1 (red), 2 (green) and 3 (blue)σ statistical confidence regions using only 25 angular diameter distances of the De Fillips et al. sample [22]. We see clearly a more degeneracy between
h and ω than h and ΩK. The vertical dashed lines correspond to bounds (1(red), 2(green)
and 3(blue) σ c.l.) from BAO + Shift parameter. The ω parameter is constrained in −2.5 ≤ ω ≤ −0.3. The limits on h in the (h, ω) plane is wider than in the ΛCDM models, and the h value has a greater dependence on ω parameter than ΩK (see figure 1a). We stress that a large range of this parameter (−3 ≤ ω ≤ 0.0) is explored.
In figure 2b, we display the (ω, h) plane ( the filled color contours) for the joint analysis
involving SZE/X-ray + BAO + Shift, and marginalizing over the possible values of ΩM.
From the joint analysis we find h = 0.71+0.044−0.034, ω = −0.76+0.19−0.28 and χ2min = 28.35 at 68.3% confidence level ( only statistical errors), whereas its reduced value is χ2red= 1.13. Again, we have an excellent fit.
In figure 2c, we plot the likelihood function for the h parameter by marginalizing in ω and ΩM. The solid line represents our analysis using only the statistical errors and in the dashed line we have combined the statistical and systematic errors. We obtain h = 0.71+0.042−0.037 (only statistical errors) and h = 0.72+0.055−0.04 (statistical + systematic errors). The horizontal lines are cuts of 68.3% and 95.4% probability. The agreement with the results using flat and non-flat ΛCDM models, see figures (1c) (this paper) and (4) (previous work), is an evidence for the robustness of the SZE/X-ray h measurements.
JCAP02(2012)035
5 H0 constraints: the influence of the morphology
In the last few years, several measurements of h obtained through the SZE/X-ray method
combined with a fixed cosmology (flat ΛCDM with ΩM = 0.3 and ΩΛ= 0.7) were discussed
in the literature. Carlstrom et al. (2002) compiled distance determinations to 26 galaxy clusters, and obtained h = 0.60 ± 0.03 [59]; Mason et al. (2001), using 5 clusters, gives h = 0.66+0.14−0.11[26]; Reese and coauthors (2002), using 18 clusters, found h = 0.60 ± 0.04 [27]; Reese (2004), with 41 clusters, obtains h ≈ 0.61 ± 0.03 [28]; Jones et al. (2005), using a sample of 5 clusters, obtained h = 0.66+0.11−0.10[10]; and Schmidt et al. (2004) obtain a best-fit h = 0.69 ± 0.08 [29].
The mean value of h from the SZE/X-Ray measurements above appears systematically lower than those estimated in the present paper and with other methods: e.g. h = 0.72 ±0.08 from the Hubble Space Telescope (HST) Project combining five differents methods to measure distances: Tully-Fisher relation, supernovae Ia, supernovae II, fundamental plane and surface brightness [11]; and h = 0.704 ± 0.013 from the CMB anisotropy from combination of 7-year
data from WMAP, Baryon Acoustic Oscillations and the Hubble constant (without high-redshift Type Ia supernovae) [17].
At this point, it is interesting to emphasize that De Filippis et al. (2005) obtained the angular diameter distances by assuming two different morphologies for describing the same galaxy clusters, namely: the elliptical and spherical β models. As discussed by these authors the choice of circular rather than elliptical β model does not affect the resulting central surface brightness or Sunyaev-Zeldovich decrement. However, values significantly different for the core radius are obtained for slight variations of β. Therefore, since the angular diameter distances obtained by using the isothermal spherical and elliptical β-models are different, it is also interesting to obtain the H0 constraints when the spherical β model is assumed.
In what follows we discuss the sensitivity of the H0 determinations with such hypotheses
describing the underlying morphology.
In figures 3a and 3b, we display the likelihood function of the h parameter by using the SZE/X-ray data + BAO + Shift Parameter for both morphological descriptions in the context of the ΛCDM and XCDM models, respectively. In our analysis, we have added the statistical and systematic errors in quadrature.
In figure 3a, we show the result when a ΛCDM model is adopted. For the spherical description of the galaxy clusters we obtain h = 0.62+0.08−0.07 at 2σ while in figure 3b, the corresponding value is h = 0.59+0.09−0.06 (2σ, XCDM). These values are weakly compatible with the ones from elliptical description and clearly incompatible with the recent estimate of Riess et al. [16]. We also stress that the constraints on the Hubble constant and ΩM derived here using the elliptical description for the galaxy clusters in ΛCDM (free geometry) and the flat XCDM model, are in agreement with the independent measurements provided by the WMAP team [17], the HST Project [11], the work of Bonamente et al. (2006) [30] and, more recently, Riess et al. [16]. Some previous papers [27, 28, 30] argued that a spherical model fit to triaxial X-ray and SZE clusters yields an unbiased estimate of cluster distance when a large ensemble of clusters is used. However, our present analysis show that these different morphological descriptions provide H0 constraints only weakly compatible.
In table 1, we summarize the H0 constraints based on the SZE/X-ray technique. All
estimates, except the last five lines, are obtained in the framework of the flat ΛCDM model
(Ωm = 0.3, ΩΛ = 0.7 - known as “cosmic concordance”). In the 8th and 9th lines, we
JCAP02(2012)035
Figure 3. Morphological dependence. a) Likelihood function of the h parameter by using the SZE/X-ray data + BAO + Shift Parameter for both morphological descriptions in the ΛCDM. b) Likelihood function of the h parameter by using the SZE/X-ray data + BAO + Shift Parameter for both mor-phological descriptions in the XCDM model.
Reference Ωm h (1σ) χ2red. Mason et al. 2001 0.3 0.66+0.14+14−0.11−14 0.35 Carlstrom et al. 2002 0.3 0.60+0.03+0.18−0.03−0.18 — Reese et al. 2002 0.3 0.60+0.04+0.13−0.04−0.18 0.97 Reese 2004 0.3 0.61+0.03+0.18−0.03−0.18 — Jones et al. 2005 0.3 0.66+0.11−0.10 — Schmidt et al. 2004 0.3 0.69+0.08−0.08 — Bonamente et al. 2006 0.3 0.77+0.04+0.10−0.03−0.08 0.83 Cunha et al. 2007 0.15+0.57−0.15 0.75+0.07−0.07 1.06
Cunha et al. 2007 + BAO 0.27+0.04−0.03 0.74+0.04−0.03 1.06 Holanda et al. 2011 + BAO 0.27+0.04−0.03 0.765+0.035−0.035 0.96
This Work (ΛCDM ) 0.272+0.03
−0.02 0.735 +0.037
−0.038 1.12
This Work (flat-XCDM) 0.30+0.05
−0.04 0.71
+0.042
−0.037 1.13
Table 1. Constraints on the “little” Hubble constant h based on SZE/X-ray technique applied to galaxy clusters data (to some references we have shown statistical + systematic errors).
with and without BAO (here labeled as “Cunha et al. 2007” and “Cunha et al. 2007 + BAO”, respectively). In the 10th line, we display the results of Holanda et al. using 38 galaxy clusters (spherical symmetry) + BAO in a flat ΛCDM [61]. In the last two lines we present our results derived through a joint analysis involving SZE/X-ray, BAO and CMB shift parameter.
JCAP02(2012)035
6 Conclusions
In this work we have discussed a determination of the Hubble constant H0 and other relevant cosmological quantities based on the SZE/X-ray distance technique for a sample of 25 clusters compiled by De Filippis et al. [22] where a elliptical β model was used to describe the galaxy clusters. In order to prove the robustness of the H0estimates we relaxed the flatness condition
of the ΛCDM model and a very weak dependence on the ΩK parameter was verified (see
figure 1a for Ωk ∈ [−0.3, 0.3]). By combining the 25 angular diameter distances with BAO
and Shift parameter more restrictive limits in (h, ΩK) plane were obtained. A comparison of the results of this work with the ones found by Cunha et al. [21] (see table 1), which
uses only a flat ΛCDM model and BAO, we clearly see that the present estimates of H0
are only slightly dependent on the geometry of the Universe. By taking into account the statistical and systematic errors in our analysis, we obtain for a nonflat ΛCDM cosmology, H0 = 74+5.0−4.0km s−1 Mpc−1(1σ).
In order to test the real dependence of the method with the adopted cosmology, we have also compared the constraints from ΛCDM model with the flat XCDM. In this case, we
have concluded that the H0 estimates presents a degeneracy on ω parameter. However, by
combining the 25 angular diameter distances with BAO and Shift parameter we obtain H0=
72+5.5−4.0 km s−1 Mpc−1(1σ) for a flat XCDM (taking account the statistical and systematic errors in quadrature).
The constraints on the Hubble constant derived here are also consistent with some recent determinations based on the WMAP observations and the HST Key Project. For completeness, since the assumed cluster shape affects considerably the SZE/X-ray distances, and, therefore, the H0 estimates, we have also performed the analysis by using the same 25 galaxy clusters, however, by assuming the spherical β model. As result, we have obtained that the elliptical description furnishes H0 estimates more compatible with independent probes. Implicitly, such an agreement suggests that the elliptical morphology provides a quite realistic description of galaxy clusters.
Finally, we stress that the combination of these four independent phenomena (SZE, X-Ray, BAO and Shift) is an interesting method to constrain the Hubble constant, and more important, it is independent of any calibrator usually adopted in the determinations of the distance scale.
Acknowledgments
The authors are grateful to an anonymous referee of JCAP for helpful comments on an earlier version of this paper. RFLH is supported by FAPESP (No. 07/52912-2), JASL is partially supported by CNPq (No. 304792/2003) and Fapesp (No. 04/13668-0) while JVC is supported by CAPES.
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