Are Galaxy Clusters Suggesting an Accelerating Universe Independent of SNe Ia and Gravity Metric Theory?
aa r X i v : . [ a s t r o - ph . C O ] M a y Are Galaxy Clusters Suggesting an Accelerating Universe Independent of SNe Iaand Gravity Metric Theory?
J. A. S. Lima, ∗ R. F. L. Holanda, † and J. V. Cunha ‡ Departamento de Astronomia, Universidade de S˜ao PauloRua do Mat˜ao, 1226 - 05508-900, S˜ao Paulo, SP, Brazil
A kinematic method to access cosmic acceleration based exclusively on the Sunyaev-Zel’dovicheffect (SZE) and X-ray surface brightness data from galaxy clusters is proposed. By using theSZE/X-ray data from 38 galaxy clusters in the redshift range 0 . ≤ z ≤ .
89 [Bonament et al.,Astrop. J. , 25 (2006)], we find that the present Universe is accelerating and that the transitionfrom an earlier decelerating to a late time accelerating regime occurred relatively recent. Suchresults are fully independent on the validity of any metric gravity theory, the possible matter-energy contents filling the Universe, as well as on the SNe type Ia Hubble diagram from whichthe present acceleration was inferred. The ability of the ongoing Planck satellite mission to obtaintighter constraints on the expansion history through SZE/X-ray angular diameters is also discussed.Two simple simulations of future Planck data suggest that such technique will be competitive withsupernova data besides being complementary to it.
PACS numbers: 04.30.Db, 09.62 +v, 98.80.Hw
Introduction.
The dimming of distant type Ia super-novae observed by two different group of astronomers onedecade ago lead to unexpected and landmark conclusion:the universal expansion is speeding up and not slowingdown as believed since the early days of observationalcosmology[1, 2].Such phenomenon is normally interpreted as a dynamicinfluence of some sort of dark energy whose main effect isto change the sign of the decelerating parameter q ( z )[3].Another possibility is that the cosmic acceleration is amanifestation of new gravitational physics (rather thandark energy) that involves a modification of the left handside (geometric sector) of the Einstein field equations. Inthis sort of theory the Friedmann equation is modifiedand a late time accelerating stage is obtained even fora Universe filled only with cold dark matter (CDM)[4].At present, the space parameter associated with the cos-mic expansion is too degenerate, and, as such, it is notpossible to decode which mechanism or dark energy com-ponent is operating in the cosmic dynamics[3, 4].Currently, SNe type Ia are not only the powerful stan-dard candles available but still provides a unique directaccess to the late time accelerating stage of the Uni-verse. Naturally, this a rather uncomfortable situationfrom the observational and theoretical viewpoints evenconsidering that ten years later, the main observationalconcerns about errors in SNe type Ia measurements, likehost galaxy extinction, intrinsic evolution, possible selec-tion bias in the low redshift sample seem to be undercontrol[5].A promising estimator fully independent of SNe typeIa and other calibrators of the cosmic distance ladder is ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] the angular diameter distance ( D A ( z )) from a given set ofdistant objects. It has also been recognized that the com-bination of SZE [6] and X-ray surface brightness measure-ments may provide useful angular diameters from galaxyclusters [7, 8, 9, 10, 11].On the other hand, since the mechanism causing theacceleration is still unknown, it is interesting to inves-tigate the potentialities of SZE/X-ray technique from amore general viewpoint, that is, independent of the grav-ity theory and the matter-energy contents filling the Uni-verse. The better strategy available so far is to considerthe same kind of kinematic approach which has beensuccessfully applied for determining the transition decel-eration/acceleration in the past by using SNe type Iameasurements[12, 13, 14, 15, 16].In this letter, we employ a purely kinematic descrip-tion of the universal expansion based on angular diameterdistances of clusters for two different expansions of thedeceleration parameter. As we shall see, by using theBonamente et al. [11] sample we find that a kinematicanalysis based uniquely on cluster data suggests that theUniverse undergone a “dynamic phase transition” (de-celeration/acceleration) in a redshift z ≈ .
3. Further,it is also shown that the Planck satellite mission datamust provide very restrictive limits on the space param-eter, thereby opening an alternative route for accessingthe expansion history of the Universe.
Angular Diameter and Kinematic Approach.
Let usnow assume that the Universe is spatially flat as moti-vated by inflation and WMAP measurements [17]. In thiscase, the angular diameter distance in the FRW metricis defined by (in our units c = 1), D A = (1 + z ) − H − Z z duH ( u ) = (1 + z ) − H Z z exp (cid:20) − Z u [1 + q ( u )] d ln(1 + u ) (cid:21) du, (1) -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0-6-4-20246810121416182022 a) No Transition Region z T = 0.15z T = 0.8 Best Fit: q = - 1.35q = 4.2 q(z)= q + q z q q b) (SZE/X-Ray from Galaxy Clusters)q(z)= q + q z z T = 0.32 P r ob a b ili t y z T FIG. 1: a) Contours in the q o − q plane for 38 galaxy clusters data [11] considering q ( z ) = q + q z . The best fit to the pairof free parameters is ( q , q ) ≡ ( − . , . z t = 0 . Λ = 0 . z t = 0 .
15 for Ω Λ = 0 . b) Likelihood function for the transitionredshift. The best fit is z t = 0 . where H ( z ) = ˙ a/a is the Hubble parameter, and, q ( z ),the deceleration parameter, is defined by q ( z ) ≡ − a ¨ a ˙ a = dH − ( z ) dt − . (2)In the framework of a flat FRW metric, Eq. (1) isan exact expression for the angular diameter distance.As one may check, in the case of a linear two-parameterexpansion for q ( z ) = q + zq , the above integral cananalytically be represented as D A ( z ) = (1 + z ) − H e q q q − q [ γ ( q − q , ( z + 1) q ) − γ ( q − q , q )] , (3)where q and q are the values of q ( z ) and its redshiftderivative, dq/dz evaluated at z = 0 while γ is an incom-plete gamma [18]. By using the above expressions we mayget information about q , q and, therefore, about theglobal behavior of q ( z ). In principle, a dynamic “phasetransition” (from decelerating to accelerating) happensat q ( z t ) = 0, or equivalently, z t = − q /q . Another inter-esting parametrization is q ( z ) = q o + q z/ (1 + z ) [15, 16].It has the advantage to be well behaved at high redshiftwhile the linear approach diverges at the distant past.Now, the integral (1) assumes the form: D A ( z ) = (1 + z ) − H e q q − ( q + q )1 [ γ ( q + q , q ) − γ ( q + q , q / (1 + z ))] , (4)where q now is the parameter yielding the total cor-rection in the distant past ( z ≫ , q ( z ) = q + q )and γ is again the incomplete gamma function. Notethat in this case the transition redshift is defined by z t = − q / ( q + q ). Constraints from Galaxy Clusters.
The SZE is a smalldistortion on the Cosmic Microwave Background (CMB)spectrum provoked by the inverse Compton scattering ofthe CMB photons passing through a population of hotelectrons. Observing the temperature decrement of theCMB spectrum towards galaxy clusters together the X-rays observations, it is possible to break the degeneracybetween concentration and temperature thereby obtain-ing D A ( z ). Therefore, such distances are fully indepen-dent of the one given by the luminosity distance, D L ( z ).Let us now consider the 38 measurements of angu-lar diameter distances from galaxy clusters as obtainedthrough SZE/X-ray method by Bonamente and cowork-ers [11]. In our analysis we use a maximum likelihooddetermined by a χ statistics χ ( z | p ) = X i ( D A ( z i ; p ) − D Ao,i ) σ D Ao,i + σ stat , (5)where D Ao,i is the observational angular diameter dis-tance, σ D Ao,i is the uncertainty in the individual distance, σ stat is the contribution of the statistical errors added inquadrature ( ≈ p ≡ ( H , q , q ). For the sake consistency, theHubble parameter H has been fixed by its best fit value H ∗ = 80 km/s/M pc . Linear Parameterization: q = qo + q z . In Figs. 1(a)and 1(b) we show, respectively, the contour in the plane q − q (68 . .
4% c.l.) and likelihood ofthe transition redshift from the Bonamente et al. sam-ple. The confidence region (1 σ ) are − . ≤ q ≤ − . ≤ q ≤ −
3. Such results favor a Universe withrecent acceleration ( q <
0) and a previous decelerat-ing stage ( dq/dz > q = − . , q = 4 . z t = 0 .
32 (see Fig.1(b)). Note the presence of a forbidden region forming a -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0-10-5051015202530 a) No Transition Region z T = 0.9 z T = 0.15 Best Fit: q = - 1.43q = 6.18 q(z)=q +q z/(1+z) q q b) q(z)= q + q z/(1+z) z T = 0.30 P r ob a b ili t y z T (SZE/X-Ray from Galaxy Clusters) qcpfFIG. 2: a) Contours in the q − q plane for 38 galaxy clusters data [11] considering q ( z ) = q + q z/ (1 + z ). The best fit tothe pair of free parameters is ( q , q ) = ( − . , . b) Likelihood function for the transition redshift and the associated bestfit at z t = 0 .
30. Comparing with Figs. 1(a) and 1(b) we see that the results are weekly dependent on the parameterizations. trapezium. The horizontal line at the top is defined by q = 0, which leads to an infinite (positive or negative)transition redshift. Note also that the segment at 45%defines the infinite future ( z t = − z t ≤ −
1, thereby demonstrating that the hatched trapez-ium is actually a physically forbidden region (for a sim-ilar analysis involving luminosity distance see [15]). Forcomparison we have also indicated in Fig. 1(a) the tran-sition redshifts z t = 0 .
15 corresponding to a flat ΛCDMwith Ω Λ = 0 .
43, as well as, z t = 0 . Λ ≃ . q = qo + q z/ z . In Figures2(a) and 2(b) we display the corresponding plots for thesecond parameterization. The confidence region (1 σ ) isnow defined by: − . ≤ q ≤ − . . ≤ q ≤ q <
0) and a previous decelerating stage ( dq/dz > q = − .
43 and q = 6 .
18 while for thetransition redshift is a little smaller z t = 0 . q >
0) is onlymarginally compatible at 2 σ of statistical confidence.The results in the q − q planes for both cases suggestthat: (i) the Universe had an earlier decelerating stage( q = dq/dz > q <
0) since z ∼ .
3. A similar result hasbeen previously obtained using SNe type Ia as standardcandles by Shapiro and Turner[12].
Prospects for Planck Satellite Mission.
Let us discussthe potentiality of the SZE/X-ray technique when futuredata from Planck satellite mission become available[19]. The mission is a project from European Space Agencywhose frequency channels were carefully chosen for mea-surements of thermal Sunyaev-Zeldovich effect. In prin-ciple, the Planck satellite will see (through SZE) about30,000 galaxy clusters over the whole sky with a signif-icant fraction of clusters near or beyond redshift unity.However, since accurate angular diameter measurementsrequire long SZE/X-ray integrations nobody expects thatall observed clusters might have useful distance measure-ments to constrain cosmological parameters. Therefore,it is interesting to simulate two realistic samples of angu-lar diameter distances (ADD) by using a fiducial modelto D trueAo,i = D A ( z i , q ∗ , q ∗ , H ∗ ), where H ∗ , q ∗ and q ∗ arethe best fit values to the linear case obtained from Bona-mente et al. sample[11]. Table 1 z range Clusters bins Clusters/bin ADD i Error(P, O) (P,O) (P,O)[0 . , .
5] 100, 500 10 10, 50 15%, 10%[0 . , .
0] 70, 350 10 7, 35 17%, 12%[1 . , .
5] 40, 200 10 4, 20 20%, 15%The first simulation (termed pessimistic - P), assumesthat only 210 clusters are distributed in the redshiftranges in the following form: 0 ≤ z ≤ . . ≤ z ≤ ≤ z ≤ . ≤ z ≤ . . ≤ z ≤ ≤ z ≤ . z = 0 . H parameter in D A ( z i , p ) in Eqs. (1) and (5). -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0123456789 Optimist Projection q(z) = q + q z Pessimist Projection q q FIG. 3: Contours on the plane ( q o , q ) from synthetic Planckdata as defined in Table 1. In the pessimistic case, 210 P(pessimistic projection) and 1050 O (optimistic projection)clusters were considered by assuming a random error between10% and 20% in the resulting angular diameter distances. In Fig. 3 we display the results of our simulations forthe linear parameterization. The contours correspond68%, 95% and 99 .
7% c.l. for the optimistic (colored in-ner contours) and pessimistic case (outer contours), re-spectively. Comparing with Fig. 1(a) we see that theallowed region was remarkably reduced even in the pes-simistic case. This means that angular diameters fromSZE/X-ray data will become a potent tool for measuringcosmological parameters fully independent and competi- tive with SNe type Ia luminosity distances.
Conclusions.
We have shown that the combination ofSunyaev-Zeldovich/X-ray data from galaxy clusters is aninteresting technique for accessing the present accelerat-ing stage of the Universe. This result follows from a newkinematic approach based on the angular diameter dis-tance of galaxy clusters obtained from SZE/X-ray mea-surements. By using two different parameterizations, itwas found that the existence of a transition from a decel-erating to an accelerating expansion occurred relativelyrecent ( z t ≃ . Acknowledgments
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