Cosmic transparency and acceleration
aa r X i v : . [ a s t r o - ph . C O ] J a n Cosmic transparency and acceleration
R. F. L. Holanda , , , ∗ S. H. Pereira , † and Deepak Jain ‡ Departamento de F´ısica,Universidade Federal de Sergipe,49100-000 – S˜ao Cristov˜ao - SE, Brazil, Departamento de F´ısica,Universidade Federal de Campina Grande,58429-900 – Campina Grande - PB, Brazil, Departamento de F´ısica,Universidade Federal do Rio Grande do Norte,59078-970 – Natal - RN, Brazil, Universidade Estadual Paulista (Unesp)Faculdade de Engenharia, Guaratinguet´aDepartamento de F´ısica e Qu´ımica12516-410 – Guaratinguet´a - SP, Brazil Deen Dayal Upadhyaya College,University of Delhi, Sector 3,Dwarka, New Delhi 110078, India
In this paper, by considering an absorption probability independent of photon wavelength, weshow that current type Ia supernovae (SNe Ia) and gamma ray burst (GRBs) observations plushigh-redshift measurements of the cosmic microwave background (CMB) radiation temperaturesupport cosmic acceleration regardless of the transparent-universe assumption. Two flat scenariosare considered in our analyses: the ΛCDM model and a kinematic model. We consider τ ( z ) =2 ln(1 + z ) ε , where τ ( z ) denotes the opacity between an observer at z = 0 and a source at z . Thischoice is equivalent to deforming the cosmic distance duality relation as D L D − A = (1 + z ) ε and,if the absorption probability is independent of photon wavelength, the CMB temperature evolutionlaw is T CMB ( z ) = T (1 + z ) ε/ . By marginalizing on the ε parameter, our analyses rule out adecelerating universe at 99.99 % c.l. for all scenarios considered. Interestingly, by considering onlySNe Ia and GRBs observations, we obtain that a decelerated universe indicated by Ω Λ ≤ .
33 and q > σ c.l. and 2 σ c.l., respectively, regardless of the transparent-universeassumption. I. INTRODUCTION
The present stage of cosmic acceleration was proposed almost 20 years ago due to an unexpecteddimming in the observed light of type Ia supernovae (SNe Ia) [1, 2]. In general relativity witha homogeneous and isotropic space-time, such cosmic behavior requires the existence of an extracomponent called dark energy whose main characteristic is a negative pressure that overcomes theattractive character of matter. Nowadays, the existence of cosmic acceleration has been confirmed byseveral other complementary independent probes, such as the cosmic microwave background (CMB),the Hubble parameter and baryon acoustic oscillations (BAO) [3, 4, 5]. However, even after 20 yearswe still do not know if the energy density of this unknown component is constant or varies in time andspace (see the excellent reviews in Refs. [6, 7]). On the other hand, the cosmic acceleration scenarioalso originated the discussion about possible modifications of general relativity in order to explain the ∗ Electronic address: holandarfl@gmail.com † Electronic address: [email protected] ‡ Electronic address: [email protected] acceleration without dark energy [8, 9, 10]. Knowing whether dark energy exists and characterizingits equation of state is perhaps the greatest challenge in modern cosmology.Alternative scenarios that could contribute to the evidence for this acceleration or even mimic thebehavior of dark energy have been proposed over the years. Some examples include a possible intrinsicevolution in type Ia supernovae luminosity [11, 12], local Hubble bubble effects [13, 14], or a highredshift and replenishing dust mechanism [15, 16, 17]. However, these hypotheses were not supportedby data (see, for instance, Sec. 4.2 in Ref. [16] and Refs.[17] and [18]). On the other hand, there aresome cosmic opacity sources that could influence astronomical photometric measurements. For SNeIa observations the inferred opacities of the following sources are model dependent: the Milky Way,the host galaxy, intervening galaxies, and the intergalactic medium [12, 14, 19, 20, 21, 22]. Anotherpossible source is the oscillation of photons propagating in extragalactic magnetic fields, which couldconvert the photons into very light axions or chameleon fields [24, 25, 26, 27, 41]. Although such kindsof effects are very small, they may become significant at cosmological scales, imprinting signatures onelectromagnetic radiation. The contribution of these effects have been investigated in CMB, radio andoptical sources (see [28] and references therein). In this context, the authors of Ref. [29] consideredonly SNe Ia data (Union2) and two different scenarios with cosmic absorption (epoch-dependent and-independent absorption). The authors were able to validate the ΛCDM (flat) description with a highvalue for the cosmological constant Ω Λ in the extreme limit of perfect cosmic transparency (negligiblecosmic absorption).The question of whether some cosmic opacity has influence on photometric measurements of dis-tant SNe Ia remains open [22]. If some extra dimming is still present, the SNe Ia observations willgive us questionable values for main cosmological parameters and, consequently, for the accelerationrate. From a theoretical point of view, the opacity affects the fundamental relation between the lumi-nosity distance D L and angular diameter distance D A the so-called cosmic distance duality relation D L D − A = (1 + z ) [30]. In this way, several cosmic opacity tests have been performed recently. Theauthors of Refs. [31, 32] used angular diameter distances from BAO measurements and luminositydistances from SNe Ia data in a ΛCDM framework to constrain the optical depth of the Universe.The results indicated a transparent Universe, although not with significant precision. Several worksexploring cosmic opacity considered a cosmic distance duality relation deformed by a ε parameter,as D L D − A = (1 + z ) ε . For instance, the authors of Refs. [25, 26] used current measurements ofthe expansion rate H ( z ) and SNe Ia data in a flat ΛCDM model and showed that a transparentuniverse is in agreement with the data considered ( ε ≈ H ( z ) data and luminosity distances ofgamma ray bursts (GRBs) and SNe Ia in ΛCDM and ω CDM flat models were considered to explorethe possible existence of an opacity at higher redshifts ( z >
2) [33]. Again, the results indicated thetransparency of the universe, but with large error bars. Cosmological model independent analyseswere also performed: the authors of Refs. [34, 35] used current measurements of the expansion rate H ( z ) and SNe Ia data, those of Ref. [36] considered angular diameter distances from galaxy clustersand SNe Ia data, and, finally, the authors of Ref. [37] used 32 old passive galaxies and SNe Ia data.No significant opacity was found from these studies, although the results do not completely rule outthe presence of some dimming source.For the CMB temperature evolution law, if the cosmic expansion is adiabatic, the universe isisotropic and homogeneous, and the CMB spectrum at z ≈ T CMB ( z ) = T (1 + z ), where T is 2 . ± . T CMB ( z ) measurementsat intermediate and high redshifts are required to test the temperature law in the past. This T CMB ( z )prediction can be violated if some mechanism acts upon this radiative component [42, 43]. Adiabaticphoton production (or destruction) or deviations from isotropy and homogeneity could modify thisscaling [41]. Moreover, a source of dimming material (unless in perfect thermal equilibrium withthe CMB) would tend to change its blackbody spectrum. Infrared emission can also occur afterabsorption of visible photons by a diffuse component of intergalactic dust, besides discrete sources(dusty star-forming galaxies) [44, 45, 46].Other previously cited mechanisms, such as axion-photon conversion induced by intergalactic mag-netic fields [27, 28, 47], could cause excessive spectral distortion of the cosmic microwave back-ground, also, those with scalar fields with a nonminimal coupling to the electromagnetic Lagrangian[48, 49, 50, 51, 52], which could produce deviation from standard results. Thus, we may consider atemperature law deformed as T CMB ( z ) = T (1 + z ) − β , where the (constant) parameter β has beenlimited by different data sets: for z < T CMB ( z ) can be obtained via the Sunyaev-Zel’dovich effect(SZE) [53, 54], while for z > T CMB ( z ) has been measured from the analysis of quasar absorptionline spectra [55]. As a result, we may quote β = 0 . +0 . − . (1 σ ) from Ref. [56] and, more recently,the authors of Ref. [57] found β = 0 . ± .
018 (1 σ ). In Ref.[58] the authors showed that if thecosmic distance duality relation is violated, the black body spectrum changes to a greybody spectrum.By using the FIRAS/COBE data, they put a limit of the order of 0.01% on the possible deviationfrom the cosmic distance duality relation. However, this limit was obtained by using the radiationcoming from the surface of last scattering at z = 1100, so limits at low and intermediate redshifts alsohave to be considered.Very recently, the authors of Ref. [59] proposed an interesting and simple relation between β and ε in the presence of a dimming agent. By considering an absorption probability independent of photonwavelength (and preservation of the CMB blackbody spectrum), they found β = − ε . By using theresults on ε from Refs. [25, 26] together with direct constraints on β from Ref. [60], they found acompetitive constraint on β : β = 0 . ± .
016 (1 σ ). More recently, this method was performed withmore data in Ref. [61] and the value β = 0 . ± .
008 (1 σ ) was obtained.In this work, unlike in previous analyses, where the main aim was to constrain the cosmic opacity byusing SNe Ia (or GRBs) and other cosmic opacity free data sets, such as BAO and H ( z ) we use threecosmic-transparency-dependent data sets, namely, SNe Ia [62], GRBs [63] and T CMB ( z ), to constraincosmological parameters by adding ε as a free parameter in the analyses. In other words, we relaxthe cosmic transparency assumption in order to verify the cosmic acceleration. We consider two flatscenarios: ΛCDM and a kinematic model based on a parametrization of the deceleration parameter q ( z ). By considering departures from standard cosmology, such as D L D − A = (1 + z ) ε , we show thatcombinations of SNe Ia, GRB observations, and T CMB confirm the cosmic acceleration at 99.99% c.l.in both scenarios (marginalizing on ε parameter). Our results are obtained by assuming an absorptionprobability independent of photon wavelength; in such a framework, the T CMB ( z ) measurements canbe added in the analyses via a deformed temperature evolution law, such as T CMB ( z ) = T (1 + z ) − β (with β = − ε ) [59]. By considering only SNe Ia and GRB observations we obtain that Ω Λ ≤ . σ c.l. and q > σ c.l..This paper is organized as follows: In Section II we briefly describe the method. The cosmologicaldata are presented in Section III, and the analyses and results are described in Section IV. Finally,we conclude in Section V. II. BASIC EQUATIONSA. Cosmic opacity and luminosity distance
In this section we explain the method used in this paper. The methodology used in this analyses issimilar to the earlier work in Refs. [25, 26]. When cosmic opacity is taken into account the distancemoduli derived from SNe Ia are systematically affected, increasing their luminosity distances. If oneconsiders τ ( z ) as the opacity between an observer at z = 0 and a source at z , the flux received by theobserver is attenuated by a factor e − τ ( z ) . In this context, the observed luminosity distance ( D L,obs )is related to the true luminosity distance ( D L,true ) by D L,obs = D L,true e τ ( z ) . (1)Thus, the observed magnitude distance modulus is given by m obs ( z ) = m true ( z ) + 2 . e ) τ ( z ) . (2)In this work we perform the analysis based on luminosity distance from two different approaches: aflat ΛCDM cosmology and a kinematic model with a parametrization for the deceleration parameter q ( z ).
1. The flat Λ CDM model
In a flat ΛCDM model the luminosity distance is given by D L,true ( z ) = (1 + z ) c Z z dz ′ H ( z ′ ) , (3)where c is the speed of light and H ( z ) = H E ( z, p ) = H [Ω M (1 + z ) + (1 − Ω M )] / . (4)In the above expressions, Ω M = 1 − Ω Λ stands for the matter density parameter measured today, Ω Λ is the cosmological constant density parameter and H is the Hubble constant.
2. The kinematic approach
For the kinematic approach, we consider the deceleration parameter, given by q = − ¨ aaH , where H = ˙ a/a is the Hubble parameter and a ( t ) = (1 + z ) − , from which one may write q ( z ) = 1 + zH dHdz − d ln H d ln(1 + z ) − . (5)In this approach, we use one of the most frequently used parametrization of q ( z ) [64, 65, 66], q ( z ) = q + q z z , (6)where q is the current deceleration parameter value and the second term has the property that in theinfinite past q → q + q . Such a parametrization mimics the behavior of a wide class of acceleratingdark energy models. Since we expect the Universe to be matter-dominated at early times, whichimplies q ( z ≫
1) = 1 /
2, we have q + q = 1 /
2, from which Eq. (6) becomes q ( z ) = (cid:18) q + z (cid:19)
11 + z . (7)Then, the luminosity distance in terms of q can be written as D L,true ( z ) = (1 + z ) cH Z z exp (cid:20) − Z u [1 + q ( u )] d ln(1 + u ) (cid:21) du. (8)At this point it is worth stressing that in previous work the cosmic opacity was taken as τ ( z ) = 2 εz .The authors argued that for small ε and z ≤ D L D − A = (1 + z ) ε . However, in order to obtain more robust results, wedo not exactly follow this approach, as we are using data sets that are also present at higher redshift.So, a complete expression for the observed distance modulus is used, which can be obtained from thedeformed CDDR, D L,obs = D L,true (1 + z ) ε , such as m obs ( z ) = m true ( z ) + 5 log(1 + z ) ε . (9)Then, the τ ( z ) function in our case is τ ( z ) = 2 ln(1 + z ) ε . D L ( G p c ) z a) T C M B ( K ) z b) D L ( G p c ) z c) FIG. 1: In Fig.(a) we plot the 740 luminosity distances of SNe Ia [62]. In Fig.(b) we plot the 42 T cmb measurements. In Fig.(c) we plot the 162 luminosity distances of GRBs [63]. B. Cosmic opacity and CMB temperature law
We have seen that luminosity distance is affected by a departure from cosmic transparency, sinceflux goes inversely with the square of luminosity distance. Changes also occur in the CMB temperaturelaw. Deviations of the standard CMB temperature law have been written using the parametrization T CMB ( z ) = T (1 + z ) − β , where β is a constant. This scaling is presumed to be a consequenceof photon number and radiation entropy nonconservation [43, 67]. Actually, β may be a functionof redshift, but since a possible deviation is expected to be small, a constant β can be justified bycurrent error bars. In this context, the authors of Ref. [59] considered the CMB spectrum in presenceof a dimming source (as dust or axion-photon conversion) and by assuming that the photon survivalprobability is independent of wavelength, they found a direct relation between ε and β : β = − ε . Inthis way, we have T CMB ( z, ε ) = T (1 + z ) ε/ . (10)This equation and Eq. (9) are crucial for our analyses. However, as it is well known, a photon dimmingis expected to be stronger at high photon energies, so one must be careful when using indirect boundson SNe Ia or GRB brightness coming from T CMB ( z ) measurements. In this way, in Table I we showrecent constraints on ε parameter from observations at different wavelengths: optical from SNe Iaobservations, microwave obtained from T CMB ( z ) measurements considered in this paper (see nextsection) by using Eq. (10), X-ray from gas mass fraction (GMF) measurements and gamma-ray fromGRBs observations. As one may see, the obtained values for ε from different wavelengths are in fullagreement with each other within 1 σ , so, the relation β = − ε is verified with observational data andit will be used in our analyses.At this point, it is worth commenting that the authors of Ref.[68] considered a source with ablackbody spectrum (BBS) at temperature T BBS and they obtained that if T BBS ( z ) = T BBS (1 + z ) ε , the CDDR for blackbody sources should change for D L = D A (1 + z ) ε ) . The authorsintroduced a species of dark radiation particles to which photon energy density is transferred andobtained ε < . − at 2 σ c.l. by using Planck data [5]. However, Eq. (10) is more generalsince it relates possible departures from the standard CDDR (at any band in the electromagneticspectrum used to obtain D L ) with possible departures from the CMB temperature evolution law.The basic assumption is that the probability that photons are created or destroyed is independent oftheir wavelength. III. DATA
In this paper, we consider the following data sets: •
740 SNe Ia distance moduli from the JLA compilation [62] obtained through the SALT II fitter[72]. The distance modulus of a SNe Ia is obtained by a linear relation from its light curve as
TABLE I: Constraints on ε from several wavelength observations. The values of works with the symbol ∗ wereobtained considering an absorption probability independent of photon wavelength.Reference Data set Model τ ( z ) ε (1 σ )[25] 307 SNe Ia + 10 H ( z ) flat ΛCDM 2 εz − . +0 . − . [26] 307 SNe Ia + 12 H ( z ) flat ΛCDM 2 εz − . +0 . − . [34] 581 SNe Ia + 28 H ( z ) model independent 2 εz . ± . H ( z ) flat ΛCDM εz . ± . H ( z ) flat ΛCDM εz . ± . H ( z ) flat XCDM εz . ± . H ( z ) flat XCDM εz . ± . H ( z ) model independent 2 εz . +0 . − . [71] 40 GMF + 38 H ( z ) flat ΛCDM 2 εz . ± . GMF + 38 H ( z ) flat ΛCDM 2 εz . ± . εz . ± . ∗ T CMB ( z ) model independent 2 ln(1 + z ) ε − . +0 . − . [57] ∗ T CMB ( z ) model independent 2 ln(1 + z ) ε − . ± . ∗ T CMB ( z ) model independent 2 ln(1 + z ) ε − . ± . µ = m b − ( M − αx + βC ), where m b is the observed rest-frame B-band peak magnitude ofthe SNe Ia, x is the time stretching of the light curve, C is the supernova color at maximumbrightness, M is the absolute magnitude, and α and β are the shape and color corrections of thelight curve. One can fix the values of M , α and β for different models. Here we use the boundson these parameters given by [62] for ΛCDM model: M = − . ± . α = 0 . ± . β = 3 . ± . α and β act like global parameters, regardlessof the prior cosmological model one chooses to determine the distance modulus of each SNe Ia(see Fig. 1a). The data points include statistical plus systematic errors. •
162 GRB distance moduli from Ref. [63]. This sample is in redshift range 0 . ≤ z ≤ . • T CMB ( z ) data. The current CMB temperature, T = 2 . ± . . ≤ z ≤ .
55) and 18from Ref. [74] (0 . ≤ z ≤ . T CMB ( z )obtained from observations of spectral lines [75, 76, 77, 78, 79, 80]. In total, this represents 42observations of the CMB temperature at redshifts between 0 and 3.025 (see Fig. 1b). IV. ANALYSES AND RESULTS
We obtain the constraints on the set of parameters p , where p = (Ω Λ ) and p = ( q ) when theflat ΛCDM and the kinematic approach are considered, respectively, based on the evaluation of thelikelihood distribution function, L ∝ e − χ / , with χ = X i =1 [ m obs ( z i ) − D L,true ( z i , p ) − µ − z i ) ε ] σ m SNobsi + X i =1 [ T obs ( z i ) − T CMB ( z i , ε )] σ T obsi + X i =1 [ m obs ( z i ) − D L,true ( z i , p ) − µ − z i ) ε ] σ m GRBobsi (11)where σ m SNobsi , σ m GRBobsi and σ T obsi are the error associated to SNe Ia and GRB distance moduliand the error of the T CMB ( z ) measurements, respectively. D L,true is given by equations (3) and (8)for ΛCDM and kinematic frameworks, respectively, T CMB ( z, ε ) is given by Eq. (10). The quantity µ is the so-called nuisance parameter: µ = 25 − H .As is commonly done in the literature, all of the results in our analyses from SNe Ia and GRB dataare derived by marginalizing the likelihood function over the pertinent nuisance parameter [16, 82](see also the section III in Ref.[81]). Then, one may obtain a new likelihood distribution function, e L ∝ e − e χ / , where e χ is given by e χ = a SNe − b SNe c SNe + ln( c SNe π ) + a GRB − b GRB c GRB + ln( c GRB π ) + X i =1 [ T obs ( z i ) − T CMB ( z i , ε )] σ T obsi , (12)with a SNe/GRB = X SNe/GRB [ m ( SNe/GRB ) obs ( z i ) − m ∗ ( z i , p )] σ m SNe/GRBobsi b SNe/GRB = X SNe/GRB [ m ( SNe/GRB ) obs ( z i ) − m ∗ ( z i , p )] σ m SNe/GRBobsi c SNe/GRB = X SNe/GRB σ m ( SNe/GRB ) obsi . (13)In this equation m ∗ ( z i , p ) = 5 log D L,true ( z i , p ) + 5 log(1 + z i ) ε .Our results for the flat ΛCDM model from the SNe Ia and GRB observations are plotted in Fig.(2a).All contours are for 68.3%, 95.4% and 99.73% c.l.. By using exclusively the SNe Ia data (black solidline), we obtain Ω Λ = 0 . +0 . . − . − . and the absorption parameter ε = 0 . +0 . . − . − . for 68.3%and 95.4% c.l.. As one may see, although distance moduli of SNe Ia have been obtained in a ΛCDMframework, the analysis by using only SNe Ia data supports a decelerated expansion even within 68.3%c.l. if some cosmic absorption mechanism is present. Moreover, the Einstein-de Sitter model (Ω M = 1)and de Sitter model (Ω M = 0) are allowed almost within 95.4% c.l.. Thus, as stressed in Ref. [29], anaccelerating dark energy component must be invoked via SNe Ia data only in a transparent universe.By using exclusively the GRB data (red dash-dotted line), we obtain Ω Λ ≤ .
79 for 68.3% and nolimits are obtained for 95.4% and 99.73% c.l. on the parameter space. The absorption parameter inthis case is: ε = 0 . +0 . . − . − . for 68.3% and 95.4% c.l.. From the joint analysis by using SNe Ia andGRB observations (filled region) one may see that a decelerated universe is allowed only within 95.4%c.l.. We obtain Ω Λ = 0 . +0 . . . − . − . − . for 68.3%, 95.4% and 99.73% c.l. (two free parameters).Now, if one assumes an absorption probability independent of photon wavelength, T CMB ( z ) mea-surements can be added to analyses via a deformed temperature evolution law, such as T CMB ( z ) = T (1 + z ) ε [59]. In Fig.(2b), from T CMB ( z ) data (blue dashed line), the absorption parameter is ε = − . ± . ± .
07 for 68.3% and 95.4% c.l. and no limit on Ω Λ is obtained. On the other hand,in a joint analysis involving SNe Ia, GRB data, as well as T CMB ( z ), tighter limits on (Ω Λ , ε ) plane areobtained and a cosmic acceleration is allowed with high confidence level (filled region) even if thereis some cosmic achromatic absorption mechanism. In this case we obtain: Ω Λ = 0 . +0 . . . − . − . − . for 68.3%, 95.4% and 99.99% (two free parameters). In Fig.(4a) we plot the likelihood of Ω Λ by DeceleratedExpansion AcceleratedExpansion a) DeceleratedExpansion AcceleratedExpansion b) FIG. 2: In Fig.(a) the black solid and red dash-dotted lines correspond the analyses using SNe Ia andGRBs, respectively, within the flat ΛCDM framework. In Fig.(b) the black solid, red dash-dotted and bluedashed lines correspond to the analyses using SNe Ia, GRBs and T CMB ( z ), respectively, within the flat ΛCDMframework. The filled contours are from the joint analysis. All contours are for 68.3%, 95.4% and 99.73% c.l.(except for the filled region in Fig.(b), for this case the contours are for 68.3%, 95.4% and 99.99%). -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0-0.6-0.4-0.20.00.20.40.6 DeceleratedExpansion a) q AcceleratedExpansion -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8-0.6-0.4-0.20.00.20.40.6
DeceleratedExpansion b) q AcceleratedExpansion
FIG. 3: In Fig.(a) the black solid and red dash-dotted lines correspond the analyses using, separately, SNeIa and GRBs, respectively, within the kinematic model. The filled contours are from the joint analysis. InFig.(b) the black solid, red dash-dotted and blue dashed lines correspond the analyses by using, separately,SNe Ia, GRBs and T CMB ( z ), respectively within the kinematic framework. The filled contours are from thejoint analysis. All contours are for 68.3%, 95.4% and 99.73% c.l. (except for the filled region in Fig.(b), forthis case: 68.3%, 95.4% and 99.99%). marginalizing on the ε parameter (by using a flat prior − ≤ ε ≤ Λ ≤ . σ c.l. .Our results for kinematic model are plotted in Figs.(3a) and (3b). In Fig.(3a) the nonfilled contoursare for 68.3%, 95.4% and 99.73% c.l.. By using exclusively the SNe Ia data (black solid line), we obtain q ≤ q = 1 / q ≥ − .
75 for 68.3% and q ≥ − .
23 for 95.4%. The best fit for In figs. 4a and 4b the blue curves are non-Gaussian, so the horizontal black lines provide only approximate valuesfor the intervals of 1 and 2 σ c.l.. Accelerated Expansion SNe + GRB + T
CMB
SNe + GRB L i ke li hood Decelerated Expansion -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.00.10.20.30.40.50.60.70.80.91.0
Decelerated ExpansionAccelerated Expansion
SNe Ia + GRB + T
CMB
SNe Ia + GRB L i ke li hood q FIG. 4: In Fig.(a) we plot the likelihoods for Ω Λ by marginalizing on ε parameter by using only SNe Ia plusGRB observations (blue line) and SNe I, GRBs plus T CMB ( z ) (black line). In Fig.(b) we plot the likelihoodsfor q by marginalizing on ε parameter by using only SNe Ia plus GRBs observations (blue line) and SNe I,GRBs plus T CMB ( z ) (black line). The horizontal lines correspond to 1 σ c.l. (68.3%) and 2 σ c.l. (95.4%). the absorption parameter in this case is: ε = 0 .
18. The joint analysis is plotted in the filled regions.Then, by using SNe Ia and GRB observations (filled region), one may see that a decelerated universeis ruled out by the data around 95.4% c.l.. We obtain q = − . +0 . . − . − . (two free parameters).Again, from T CMB ( z ) data, the absorption parameter is ε = − . ± . ± .
07 for 68.3% and95.4% c.l. and no limit on q is obtained. On the other hand, in a joint analysis involving SNe Ia, GRBdata as well as T CMB ( z ), tighter limits on the ( q , ε ) plane are obtained and a cosmic acceleration isallowed with a high confidence level (filled region) even if there is some cosmic achromatic absorptionmechanism. In this case we obtain: q = − . ± . ± . ± .
27 for 68.3%, 95.4% and 99.99%c.l.. In Fig.(4b) we plot the likelihood of q by marginalizing on the ε parameter. The horizontallines correspond to 1 σ c.l. (68.3%) and 2 σ c.l. (95.4%). Now, as one may see, the intervals of q by using SNe Ia, GRBs and T CMB ( z ) data (black line) are fully within those supporting the cosmicacceleration: q <
0. By considering only SNe Ia and GRB observations we obtain that q > σ c.l.. V. CONCLUSIONS
As it is well known, the dimming of the distant SNe Ia has been interpreted as a consequence ofthe present accelerated stage of evolution of the Universe. However, such an interpretation dependson a assumption of a transparent universe . In fact, an absorption source leads to a reduction of thephoton number from a luminosity source by a factor of e − τ ( z ) and, consequently, an increase of itsinferred luminosity distance. We have considered τ ( z ) = 2 ln(1 + z ) ε , where τ ( z ) denotes the opacitybetween an observer at z = 0 and a source at z .In this work, we have reviewed the well-known result in which only SNe Ia data in a flat ΛCDMmodel allows the Einstein-de Sitter model (without acceleration) if some opacity source exists (seeFig.2a, black solid line). Moreover, by considering a flat kinematic approach, we have also verifiedthat a positive deceleration parameter ( q >
0) is allowed within 95 .
4% (see Fig. 3a, black solidline). However, by considering a joint analysis of SNe Ia along with gamma-ray burst data, whichis also affected if some cosmic opacity source is present, we have shown that observations rule out adecelerated universe ( q >
0) at 95 .
4% c.l. even in the presence of cosmic opacity (see Fig.4b). Wehave also obtained that Ω Λ ≥ .
33 is favored by these data in flat ΛCDM model (see Fig.4a). In thesefigures we have marginalized on the ε parameter.On the other hand, under the assumption of an absorption probability that is independent of photon0wavelength, we have added T CMB ( z ) measurements to the analyses via a deformed temperatureevolution law, and the cosmic acceleration evidence is reinforced (see filled contours in Figs. 2band 3b). These joint analyses were performed by using the following deformed equations relatedto the cosmic distance duality relation and the evolution law of CMB: D L D − A = (1 + z ) ε and T CMB ( z ) = T (1+ z ) − β , where β = − ε if one considers an absorption probability that is independentof photon wavelength. It is important to stress that such an assumption is well verified for currentanalyses by using observational data at different wavelengths (see Table I). Finally, it is notable thatthree different types of observations that are affected by cosmic opacity provide an accelerated universewhen analyzed together in a framework without the assumption of cosmic transparency. Acknowledgments
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