X-ray surface brightness observations of galaxy clusters, cosmic opacity and the limits on the matter density parameter
aa r X i v : . [ a s t r o - ph . C O ] M a r X-ray surface brightness observations of galaxy clusters, cosmic opacity and thelimits on the matter density parameter
R. F. L. Holanda , , ∗ Kamilla V. R. A. Silva , † and V. C. Busti , ‡ Departamento de F´ısica, Universidade Federal de Sergipe, 49100-000, Aracaju - SE, Brazil, Departamento de F´ısica, Universidade Federal de Campina Grande, 58429-900, Campina Grande - PB, Brazil, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA, Departamento de F´ısica Matem´atica, Universidade de S˜ao Paulo,Rua do Mat˜ao 1371, S˜ao Paulo - SP, 05508-090, Brazil
In this work, we use two gas mass fraction samples of galaxy clusters obtained from their X-raysurface brightness observations jointly with recent H ( z ) data in a flat ΛCDM framework to imposelimits on cosmic opacity. It is assumed that the galaxy clusters are in hydrostatic equilibrium andtheir gas mass fraction measurement is constant with redshift. We show that the current limits onthe matter density parameter obtained from X-ray gas mass fraction test are strongly dependenton the cosmic transparency assumption even for a flat scenario. Our results are consistent witha transparent universe within 1 σ c.l. in full agreement with other analyses which used type Iasupernovae, gamma ray burst and H ( z ) data. I. INTRODUCTION
From a general point of view, cosmic opacity canbe an important systematic error source in several as-tronomical observations. By considering type Ia su-pernova (SNe Ia) observations, for instance, there arefour different sources of opacity by dust absorption:the Milky Way, the hosting galaxy, intervening galax-ies, and the Intergalactic Medium [1–5]. In this con-text, the approach of Ref.[6] considered SNe Ia dataand two different scenarios with cosmic absorption andthe main conclusion was that the description of an ac-celerating Universe powered by dark energy or somealternative gravity theory only must be invoked if thecosmic opacity is fully negligible. Similar studies alsocan be found in Refs.[8–11]. The Ref.[12] investigatedthe luminosity and redshift dependence of the quasarcontinuum and suggested that the reddening observedcould come from cosmic dust extinction. Infrared sur-veys can also be affected by a population of dust grains[13]. By considering the cosmic microwave backgroundradiation, the results of Ref.[14] showed that whereasthe dust emission in galaxies could be taken out, theintergalactic dust emission is diffuse and cannot be re-moved easily from the maps. There is also a moreexotic possibility as opacity source, namely, photons ∗ Electronic address: holandarfl@gmail.com † Electronic address: [email protected] ‡ Electronic address: [email protected] For details please refer to Ref.[7]. turning into unobserved particles beyond the standardmodel due to interaction with extragalactic magneticfields (please refer to [15–18] for details).Other cosmic opacity tests have been performed: theapproaches of Refs.[19, 20] used current measurementsof the expansion rate H ( z ) and SNe Ia data to im-pose cosmological model-independent constraints oncosmic opacity. As a result, a fully transparent uni-verse is in agreement with the data considered (see alsoRef.[15, 16] for analyses in a flat ΛCDM framework).In order to explore a possible presence of an opacity athigher redshifts ( z > H ( z )data and luminosity distances of gamma-ray bursts inthe ΛCDM and ω CDM flat models. More recently, theRef.[22] used 32 old passive galaxies and SNe Ia datato obtain cosmological model-independent constraintson cosmic opacity. No significant opacity was foundin these studies although the results do not completelyrule out the presence of some dimming source and ad-ditional tests are still required. This is exactly thesubject of the present paper, where X-ray gas massfraction samples of galaxy clusters jointly with recent H ( z ) data will be used. It is worth emphasizing thatX-ray astronomy provides an unique opportunity todetect opacity sources that may be missed by tradi-tional detection methods, such as those using the dustreddening of background quasars by foreground galax-ies and associated large scale structure [23–25].The gas mass fraction is defined as f = M gas /M T ot [26], where M T ot is the total mass and it can be ob-tained via hydrostatic equilibrium assumption while M gas (gas mass) is obtained by integrating a gas den-sity model (see next section for details) and using X-ray or Sunyaev-Zel’dovich effect observations. By us-ing hot, massive and relaxed galaxy clusters as labora-tories, gas mass fraction samples have been compiledand used to constrain cosmological parameters, mainlythe matter density parameter, Ω M (see [27–32] for sev-eral analyses). The gas mass fraction as a cosmologicaltest is based on a basic hypothesis: the ratio betweenbaryons and total matter (baryons plus dark matter)in galaxy clusters is a fair sample for the Universe onlarge scales, being constant through the cosmic history.Over the years, this key hypothesis has been sup-ported by observational and hydrodynamical simula-tion results. For instance, the Ref.[33] investigated thebaryon distribution in groups and clusters. By con-sidering 123 systems (0 . < z <
1) they found thatthe gas mass fraction does not depend on the totalmass for systems more massive than 10 solar masses.Moreover, they obtained only a slight dependence ofgas mass fraction measurements with redshift for r (see their fig. 6). On the other hand, the hydrody-namical simulations of [34, 35] showed that hot, mas-sive galaxy clusters ( M > solar masses) anddynamically relaxed, do not show significant evolutionfor the depletion factor, γ = f gas / (Ω b / Ω M ), (see Ta-ble III in [34]). They considered γ = γ + γ z andfound − . < γ < .
07 considering the completesphere at r (this radii is that one within which themean cluster density is 2500 times the critical densityof the Universe at the cluster’s redshift). However, itis important to comment that thanks to new X-rayobservations, it has been possible detect the presenceof intrinsic scatter in the gas mass fraction measure-ments. In 40 measurements from the Ref. [32], forinstance, a 7.4% of intrinsic scatter was found. At themoment, it is not possible to distinguish observation-ally between the possible causes of this scattering. Inthis way, hydrodynamical simulations have also shownthat a similar level of dispersion may be due to pres-ence of a non-thermal pressure (see, for instance, [36]).Recently, some works have shown that the X-ray gasmass fraction measurement as a cosmological tool isstrongly dependent on the cosmic distance duality re-lation (CDDR) validity [37–40], D L D − A = (1 + z ) ,where D L and D A are the luminosity and angular di-ameter distances for a given redshift z . Particularly,the authors of Ref.[41] searched for systematics in SNeIa and galaxy cluster data using this relation, without advancing any hypothesis about the nature of dark en-ergy. This relation was proved in Ref. [42] and itonly requires sources and observers connected by nullgeodesics in a general Riemannian spacetime as well asconservation of photon number . Thus, even in a Rie-mannian spacetime, any departure from cosmic trans-parency could lead to dubious estimates of cosmologi-cal parameters if one uses cosmological tests dependenton flux, such as, SNe Ia distance module and X-raygas mass fraction (X-ray GMF). Therefore, althoughthe dark energy is supported by several other indepen-dent probes, if some extra opacity is still present, theobservations will give us unreal values to cosmologicalparameters, mainly to the Ω M if one considers thosefrom X-ray GMF of galaxy clusters.In this work we discuss how X-ray GMF observa-tions of galaxy clusters jointly with recent H ( z ) datacan be used to investigate a possible departure fromtransparency cosmic in a flat ΛCDM framework. TheX-ray GMF samples used separately in our analysesconsist of: 42 and 40 measurements obtained by theRefs.[30] and [32], respectively. The total redshiftrange is 0 . ≤ z ≤ . H ( z ) data consistof 38 points in the redshift range 0 . ≤ z ≤ . H , fromRef.[45] ( Planck collaboration ). In our analyses, thecosmic opacity is parameterized by τ ( z ) = 2 ǫz , whichcorresponds to a modification on the cosmic distanceduality relations such as D L D − A = (1 + z ) ǫ (if ǫ issmall and z ≤ σ c.l. ( ǫ ≈ M obtained from the X-ray GMFtest are strongly dependent on cosmic transparencyassumption even in the simple flat ΛCDM framework .This paper is organized as follows. In Section II wepresent the method. The cosmological data are de- For recent results of CDDR tests see Table I in [43]. It is worth to comment that the authors of Ref.[39] tested theCDDR with gas mass fraction and H ( z ) measurements in acosmological model independent approach. In this way, noinformation of how cosmological results from X-ray observa-tions are depend on the cosmic transparency hypothesis wasobtained. scribed in Section III. The analyses and results arepresented in Section IV and Section V shows our con-clusions. II. COSMIC OPACITY AND GAS MASSFRACTION OBSERVATIONS
In this section we first discuss how a cosmic opacitypresence affects the luminosity distance, and then wepresent the link between luminosity distance and X-rayGMF observations.
A. Luminosity distance and cosmic opacity
The methodology used in our analyses was proposedby [15]. It was initially applied for SNe Ia data, how-ever, it also can be applied for X-ray GMF observa-tions. As it is well known, the distance modulus de-rived from SNe Ia or gamma-ray bursts and the X-raysurface brightness observations may be systematicallyaffected if there are cosmic dimming sources. In fewwords, a direct consequence of the photon number re-duction is an increasing of D L . Hence, if τ ( z ) denotesthe opacity between an observer at z = 0 and a sourceat z , the flux received by the observer in z = 0 is atten-uated by a factor e − τ ( z ) and, therefore, the observedluminosity distance ( D L,obs ) is related to the true lu-minosity distance ( D L,true ) by D L,obs = D L,true e τ ( z ) . (1)Then, the observed distance modulus is [9, 10] m obs ( z ) = m true ( z ) + 2 . e ) τ ( z ) . (2)In this paper, D L,true ( z ) comes from a flat ΛCDMmodel, such as D L,true ( z, Ω M , H ) = (1 + z ) c Z zo dz ′ H ( z ) , (3)where c is the speed of light and H ( z ) = H E ( z, p ) ,E ( z, p ) = [Ω M (1 + z ) + (1 − Ω M )] / . (4)In the above expression, Ω M stands for the matterdensity parameter measured today. In order to usethe full redshift range of the available data, we fol-low the Refs.[15, 16] and considered the parameteri-zation D L = D A (1 + z ) (2+ ǫ ) , with ǫ parameterizing departures from transparency cosmic. These authorsargued that for small ǫ and z ≤ τ ( z ) = 2 ǫz or τ = (1 + z ) α − α = 2 ǫ .In this way, in our analyses, we consider a simple lin-ear parameterization for τ ( z ), such as: τ ( z ) = 2 ǫz .The measurements of m obs (or D L,obs ) are obtainedfrom the X-ray GMF data. The unknown parametersΩ M and ǫ are constrained by fitting the X-ray GMFdata separately and jointly with H ( z ) measurementson a flat ΛCDM model. As comment earlier, the basicidea behind this test is that while the X-ray surfacebrightness of galaxy clusters can be affected by cos-mic opacity, the H ( z ) measurements are obtained viatwo ways: from measurements of radial BAO and fromthe differential ages of old passively evolving galaxies,which relies only on the detailed shape of the galaxyspectra but not on the galaxy luminosity. Both meth-ods are cosmic opacity independent. Therefore, H ( z )values are not affected by a non-zero τ ( z ) since τ isassumed not to be strongly wavelength dependent inthe optical band. B. Luminosity distance from galaxy clusters
In galaxy clusters, the GMF is defined by [30] f gas = M gas M tot , (5)where M tot is the total mass (dominated by dark mat-ter) and M gas is the gas mass. The total mass withina given radius R can be obtained by assuming that theintracluster gas is in hydrostatic equilibrium. On theother hand, the intracluster gas emits X-ray predom-inantly via thermal bremsstrahlung and its mass canbe estimated by integrating a gas density model. The f gas is expected to be same at all z since these struc-tures are the largest virialized objects in the Universe,consequently, a faithful representation of the cosmolog-ical average baryon fraction can be found in clusters.Thus, in order to constrain cosmological parameters,the X-ray GMF of galaxy clusters can be used via thefollowing expression [30] f obsX − ray ( z ) = N " D ∗ L D ∗ / A D L D / A , (6)where the symbol * denotes quantities from a fiducialcosmological model used in the observations (usually f ob s z a) H ( z ) k m / s / M p c z b) FIG. 1: Fig.(a) shows the X-ray GFM data. The open circles and filled squares correspond to samples from Ref.[32] andRef.[30], respectively. Fig.(b) shows the H ( z ) data (in units of km/s/Mpc ). The open circles and filled squares correspondto measurements from cosmic chronometers and radial BAO, respectively. a flat ΛCDM model with Ω m = 0 . H = 70km/s/Mpc). The parameter N defines an arbitrarynormalization on which we marginalize. The ratio mul-tiplying in brackets computes the expected measuredgas fraction f obsX − ray when the cosmology is varied. Onthe other hand, in Ref.[37], the authors showed thatthe gas mass fraction measurements extracted from X-ray data are affected if there are cosmic opacity sources(consequently, departure from the CDDR validity). Ina such framework, if one considers our parametriza-tion τ ( z ) = 2 ǫz (or D L = D A (1 + z ) (2+ ǫ ) ), the Eq.(6)is rewritten as f obsX − ray ( z ) = N " (1 + z ) ǫ/ D ∗ / L D / L . (7)Finally, we define the distance modulus of a galaxycluster as m obs ( z, N, ǫ ) = 5 log[(1+ z ) ǫ/ D ∗ L [ N/f obsX − ray ( z )] / ]+25 , (8)which depends on cosmic opacity ( D ∗ L is in Mpc). III. DATA
In this paper, we consider two types of data sets:
A. Cosmic opacity dependent data
Here we have two GMF samples, namely: •
42 X-ray GMF measurements obtained by theChandra telescope for hot ( kT > keV ), massive, X-ray luminous and dynamically relaxed galaxyclusters spanning the redshift range 0 . ≤ z ≤ . r ra-dius in the reference ΛCDM cosmology. This ra-dius corresponds that one for which the meanenclosed mass density is 2500 times the criticaldensity of the Universe at the redshift of the clus-ter (see Fig.1a). •
40 X-ray GMF measurements from massive, dy-namically relaxed galaxy clusters compiled bythe Ref.[32]. These authors significantly reducesystematic uncertainties compared to previouspapers by incorporating a robust gravitationallensing calibration of the X-ray mass estimates.Moreover, as an unprecedented approach, theGMF measurements were obtained in sphericalshells at radii near r , rather than X-ray GMFintegrated at all radii < r . This procedureexcludes cluster centers and reduces the theoret-ical uncertainty in gas depletion from hydrody-namic simulations. As a result, the error barsof this sample are smaller than those in Ref.[30](see Fig.1a). Decelerated expansion M =1- a) Accelerated expansion
Accelerated expansion M =1- b) Decelerated expansion -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.58090100110120 c) -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.560708090100110 d) FIG. 2: Fig.(a) shows the ( ǫ, Ω M ) plane by the using X-ray GMF sample from Ref.[32] and the H ( z ) data. Fig.(b)shows the X-ray GMF sample from Ref.[30] and the H ( z ) data. In all these panels the black and blue lines correspondto analyses by using X-ray GMF samples and H ( z ) data separately. In each case, the filled contours are the results fromjoint analysis. Panels (c) and (d) show the χ values for ǫ by using the X-ray GMF samples from Ref.[32] and Ref.[30],respectively, jointly with H ( z ) data (marginalizing on the Ω M ). B. Cosmic opacity independent data
Here, we consider 38 H ( z ) measurements, namely,30 from cosmic chronometers (see table 4 in Ref.[47])plus 8 H ( z ) measurements from radial baryon acousticoscillations (see Fig.1b). Briefly, the cosmic chronome-ters approach uses relative ages of the most massiveand passively evolving galaxies to measure dz/dt , fromwhich H ( z ) is inferred. The method of getting agesof old passively evolving galaxies depends only on thedetailed shape of the galaxy spectra but not on thegalaxy luminosity, which turns this quantity indepen-dent on cosmic opacity . The 8 H ( z ) measurementsfrom radial baryon acoustic oscillations can be found We consider τ not to be strongly wavelength dependent inthe optical band (see Refs.[48, 49]). in Refs.[50–54]. A complete data table also can befound in Ref.[55]. Recently these measurements of theHubble parameter have been used to constrain severalcosmological parameters [47, 56–62]. We considered inour analyses H = 67 . ± .
9, (in km/s/M pc ), obtainedby the
Planck collaboration for a flat ΛCDM universefrom a combination of temperature and lensing data ofthe cosmic microwave background [45].
IV. ANALYSES AND RESULTS
We obtain the constraints to the set of parameters( ǫ, N, Ω M , H ), by evaluating the likelihood distribu-tion function, L ∝ e − χ / , with TABLE I: Constraints on ǫ from different analyses. The symbols * and ** denote the samples from Refs.[32] and [30],respectively. Reference Data set Model τ ( z ) ǫ (1 σ )[15] 307 SNe Ia + 10 H ( z ) flat ΛCDM τ ( z ) = 2 ǫz − . +0 . − . [16] 307 SNe Ia + 12 H ( z ) flat ΛCDM τ ( z ) = 2 ǫz − . +0 . − . [19] 581 SNe Ia + 28 H ( z ) model independent τ ( z ) = 2 ǫz . ± . H ( z ) flat ΛCDM τ ( z ) = ǫz . ± . H ( z ) flat ΛCDM τ ( z ) = ǫz . ± . H ( z ) flat XCDM τ ( z ) = ǫz . ± . H ( z ) flat XCDM τ ( z ) = ǫz . ± . H ( z ) model independent τ ( z ) = 2 ǫz . . . This paper 40
GMF ∗† + 38 H ( z ) flat ΛCDM τ ( z ) = 2 ǫz . ± . GMF ∗∗† + 38 H ( z ) flat ΛCDM τ ( z ) = 2 ǫz . ± . χ = P z [ m obs ( z,N,ǫ ) − m true ( z, Ω M ,H ) − . ǫz ] σ mobs (9)+ P z [ H ( z, Ω M ,H ) − H obs ( z )] σ Hobs + ( H − H ∗ ) σ H ∗ where σ m obs , σ H obs and σ H ∗ are the er-rors associated to m obs ( z, N, ǫ ) of the galaxycluster data, H ( z ) obs measurements and H prior ( H = 67 . ± .
9, in km/s/M pc ), re-spectively. m true ( z, Ω M , H ) is obtained via m true ( z, Ω M , H ) = 5 log D L,true ( z, Ω M , H ) + 25,while D L,true ( z, Ω M , H ) is given by equation Eq.(3)and H ( z, Ω M , H ) from Eq.(4). We marginalize onthe N parameter.The Fig.(2) shows all the results from our analyses.The black solid contours in the Figs. (2a) and (2b) arethe confidence intervals of ∆ χ = 2 .
30 (1 σ ), 6.17 (2 σ )and 11.82 (3 σ ) on the (Ω M − ǫ ) plane from analyseswith the X-ray GMF samples present in the Refs. [32]and [30], respectively. From these results, it is very im-portant to point out that, even considering the simpleflat ΛCDM model, the constraints on the Ω M parame-ter exclusively from X-ray GMF data depend stronglyon the transparency cosmic assumption. This meansthat using only this kind of observation we can notconstrain simultaneously the energy content of the flatΛCDM model and the ǫ parameter. In other words,there is a degeneracy between the Ω M and ǫ param-eters. Moreover, a decelerated universe is allowedwithin ≈ σ in Fig.(a) and within 1 σ in Fib.(b) (seethe vertical black dashed-dot line). In both figures, thevertical blue lines correspond to results by using exclu-sively the H ( z ) data (the confidence intervals are for 1 σ , 2 σ and 3 σ ). As one may see, the Ω M parameteris well constrained when the H ( z ) data are added inthe analyses and, therefore, limits on ǫ can be found.In each figure, the results from the joint analysis byusing X-ray GMF + H ( z ) are displayed by the filledcontours.On the other hand, Figs. (2c) and (2d) show the χ values for ǫ by using the X-ray GMF samples fromRef.[32] and Ref.[30], respectively, jointly with H ( z )data (marginalizing on Ω M ). The intervals found are(at 1 σ ): • Fig.(2a): Ω M = 0 . ± .
02 and ǫ = 0 . ± . • Fig.(2b): Ω M = 0 . ± .
02 and ǫ = 0 . ± . • Fig.(2c): ǫ = 0 . ± .
080 (by marginalizing onΩ M ). • Fig.(2d): ǫ = 0 . ± .
13 (by marginalizing onΩ M ).As one may see, although in this case we have ǫ > ǫ = 0). Our Ω M value is in full agreementwith that one from Planck results [45].Table I shows some recent constraints on the cosmicopacity by using approaches involving SNe Ia, gamma-ray bursts and H ( z ) observations as well as the resultsof the present paper. As commented earlier, the ap-proach used in Refs.[15, 16, 21] is similar to this pa-per, but the bands of the electromagnetic spectrumexplored were other, namely, optical and gamma-ray.On the other hand, the Refs.[19, 63] considered cosmo-logical model independent approaches by using SNe Iaand H ( z ) data. As one may see, the results from differ-ent bands of the electromagnetic spectrum are in fullagreement each other and no significant deviation fromtransparent universe is verified. However, these resultsdo not rule out ǫ = 0 with high statistical significanceyet. V. CONCLUSIONS
As it is largely known, cosmic opacity can mimica dark energy behavior and its presence has been in-vestigated along the years by different methods. Re-cently, type Ia supernovae and gamma ray bursts ob-servations have been used along with cosmic expan-sion rate measurements, H ( z ), to constrain possibledepartures from cosmic transparency. In this context,dependent and independent cosmological model anal-yses were performed. In this paper, by considering aflat ΛCDM framework we showed how is possible usegalaxy cluster X-ray gas mass fraction samples jointlywith the most recent H ( z ) data to impose limits oncosmic opacity. We considered the H prior in ouranalyses, namely: H = 67 . ± . τ ( z ) quantifies the cosmic opacity and was pa-rameterized by τ ( z ) = 2 ǫz in our case. This kindof τ ( z ) function is directly linked to a violation of the cosmic distance duality relation validity such as D L D − A = (1 + z ) ǫ if ǫ is small and z ≤
1. Asone may see from Table I, we did not find any signifi-cant departure from cosmic transparency ( ǫ ≈
0) andour results are in full agreement with previous studieswhere type Ia supernovae and gamma-ray burst obser-vations were used in similar approaches. However, itis very important to stress that these analyses did notrule out ǫ = 0 with high statistical confidence leveland additional tests are still required with forthcom-ing data. Moreover, from panels (a) and (b) in Fig.(2)it is possible to conclude that constraints on Ω M ob-tained from X-ray gas mass fraction test (black solidcontours) depend strongly on the cosmic transparencyhypothesis even if the simple flat ΛCDM model is con-sidered. Acknowledgments
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