Using the large scale quasar clustering to constrain flat quintessential universes
aa r X i v : . [ a s t r o - ph . C O ] F e b Astronomy&Astrophysicsmanuscript no. ms c (cid:13)
ESO 2018October 24, 2018
Using the large scale quasar clustering to constrain flatquintessential universes
Ariel Zandivarez and H´ector J. Mart´ınez
Instituto de Astronom´ıa Te´orica y Experimental (IATE), CONICET-Observatorio Astron´omico, Universidad Nacional de C´ordoba.Laprida 854, C´ordoba X5000BGR. Argentina.e-mail: [email protected], [email protected]
Received XXX, 2008; accepted XXX , 2008
ABSTRACT
Context.Aims.
We search for the most suitable set of cosmological parameters that describes the observable universe. The search includes thepossibility of quintessential flat universes, i.e., the analysis is restricted to the determination of the dimensionless matter density andthe quintessential parameters, Ω M and w Q , respectively. Methods.
Our study is focused on comparing the position of features at large scales in the density fluctuation field at di ff erent redshiftsby analysing the evolution of the quasar two-point correlation function. We trace the density field fluctuations at large scales using alarge and homogeneous sample of quasars ( ∼ . z ≤ z = .
45) drawn from the Sloan DigitalSky Survey Data Release Six. The analysis relies on the assumption that, in the linear regime, the length scale of a particular featureshould remain fixed at di ff erent times of the universe for the proper cosmological model. Our study does not assume any particularcomoving length scale at which a feature should be found, but intends to perform a comparison for a wide range of scales instead.This is done by quantifying the amount of overlap among the quasar correlation functions at di ff erent times using a cross-correlationtechnique. Results.
The most likely cosmological model is Ω M = . ± .
02 and w Q = − . ± .
04, in agreement with previous studies. Theseconstraints are the result of a good overall agreement of the correlation function at di ff erent redshifts over scales ∼ − h − Mpc.
Conclusions.
Under the assumption of a flat cosmological model, our results indicate that we are living in a low density universe witha quintessential parameter greater than the one corresponding to a cosmological constant. This work also demonstrates that a largehomogeneous quasar sample can be used to tighten the constraints upon cosmological parameters.
Key words. cosmological parameters – cosmology: observations – large scale structure of the universe – quasars: general
1. Introduction
Observations of the cosmic microwave background (CMB, e.g.Spergel et al. 2007) and type Ia supernovae (e.g. Riess et al.1998; Perlmutter et al. 1999; Astier et al. 2006) support the ideathat the missing energy in the universe should possess negativepressure p and an equation of state p = w ρ . One possible can-didate for the missing energy is the vacuum energy density orcosmological constant Λ for which w = − Λ CDM, consists of a mix-ture of vacuum energy and cold dark matter. Another possibilityis the QCDM cosmology based on a mixture of quintessenceand cold dark matter (Ratra & Peebles 1988). The quintessence,which is defined as a fifth element di ff erent from baryons, neu-trinos, dark matter and radiation, is a slowly-varying spatially in-homogeneous component (Caldwell et al. 1998) whose w valueis less than 0. Many studies restrict the range of w to the inter-val ( − ≤ w ≤
0) since this range best fits current cosmologicalobservations.The distinction between Λ CDM and QCDM models isof key importance in cosmology. Even when both modelsmatch the CMB observations well, the QCDM models seemto be more suitable to describe the universe at high redshifts(Eisenstein et al. 2005). According to deep redshift surveys, the-ories should predict a strong large scale clustering structureand quasar formation at this stage. For this to happen, the cessation of growth of the structures should occur at earliertimes. Unlike Λ CDM, QCDM universes satisfy this requirementsince the larger the values of w , the earlier the growth ceases.Nevertheless, it is necessary to know how big the di ff erencesamong both cosmologies are in order to restrict the possible ini-tial conditions in the observed universe. To do so, a tuning pro-cess of the cosmological parameters is needed, combining thebest current observations with the most suitable statistical tools.Several attempts have been made to tighten the constraintson cosmological parameters. Among the tests suggested inthe literature we can mention measurements of the Hubbleparameter (e.g., Wei & Zhang 2008; Szydłowski et al. 2008),gravitational lensing (e.g., Zhang et al. 2007; Zhu & Sereno2008; Dor´e et al. 2007; La Vacca & Colombo 2008), angularsizes of distant objects (e.g., Daly et al. 2007; Santos & Lima2008), gas mass fraction in galaxy clusters (e.g., Chen & Ratra2004; Sen 2008) and baryon acoustic oscillation peaks atlarge scales (e.g., Lima et al. 2007; Sapone & Amendola 2007;S´anchez et al. 2008).In this work, we rely on the fact that if a particular spa-tial scale can be measured at di ff erent stages of the evolutionof the universe, that scale can then be used to determine thevalues of the cosmological parameters. At large enough scales( r > h − Mpc), density perturbations are in the linear regimeand above the present-day turnaround scale. Thus, structures atlarge scales, for instance features in the two-point correlation
Zandivarez & Mart´ınez: Constraining flat quintessential universes function, should be fixed in comoving length scales at di ff er-ent times in the evolution of the universe. Since quasars canbe observed at very large distances, studying the evolution ofquasar clustering, i.e., identifying similar features in the densityfield at di ff erent redshifts, could restrict the possible universesthat match the observations. This approach has been adopted byRoukema et al. (2002), hereafter R
02, using a relatively smallsample of quasars and finding a wide range of possibilities forthe quintessential parameter ( − . ≤ w < − . Ω M , and w Q ( w denoted withthe subscript Q to specify QCDM cosmologies). It is importantto note that QCDM cosmologies allow a wide spectrum of pos-sibilities for the equation of state w Q , which could be constant,uniformly evolving or oscillatory. In this work, we restrict ouranalysis to QCDM universes where w Q is constant since thereis evidence that this parameter does not evolve in time (e.g.,Wang & Mukherjee 2007; Gong et al. 2008). We follow the lineof thought of R
02 in which they did not assume the scale atwhich these features should occur, but only required consistencyof the scale between di ff erent times.This paper is organised as follows: in Sect. 2 we describe thesample of DR6 quasars; in Sect. 3 we constrain the values of thecosmological parameters Ω M and w Q , analysing features in thequasar correlation function at di ff erent redshifts; we discuss ourresults in Sect. 4. Throughout this paper we assume a Hubbleconstant H = h km s − Mpc − and all magnitudes are inthe AB system. As stated previously, we also assume that theuniverse is perfectly flat, i.e, Ω M + Ω Q =
2. The quasar sample
The SDSS DR6 provides the largest homogeneous sample ofquasars currently available. The identification of quasars isachieved with high e ffi ciency and completeness thanks to thewide-field five band photometric system (Gunn et al. 1998). Thehigh homogeneity of the SDSS spectroscopic sample is ac-complished by selecting the spectroscopic targets consistentlyon the basis of their photometric data (Richards et al. 2002;Blanton et al. 2003).To ensure further homogeneity in our quasar sample, we se-lect quasars that match the following criteria (Schneider et al.2005; Yahata et al. 2005): – The quasar primTarget flag is either ”QSO CAP”or ”QSO SKIRT” or ”QSO FIRST CAP” or”QSO FIRST SKIRT”; – i − band PSF magnitude corrected for Galactic reddening (us-ing the maps by Schlegel et al. 1998) satisfies 15 . ≤ i ≤ . – The redshift is z ≤ .
4, since the incompleteness becomesimportant for higher redshifts; – The absolute magnitude is brighter than M i = − . − h ), computed assuming Ω M = . w Q = − . Fig. 1.
Quasar redshift distribution shown as thin line histogramsin both panels. The upper panel shows in dotted lines the fiveGaussian functions whose sum (thick line) fits the redshift dis-tribution of DR6 quasars. The lower panel shows in thick linethe redshift distribution of our random quasar sample obtainedby using the fit shown in the upper panel. The vertical dashedline shows the 50th percentile (median value) corresponding toour two redshift bin analysis, while vertical dotted lines showthe 25th (first quartile) and 75th (third quartile) percentiles usedin the four redshift bins case. K − correction corresponding to a power-law quasar spectrumwith α = − . – We also exclude the southern Galactic region of SDSS dueto di ff erences in the selection compared to the main survey.Our final quasar sample comprises 38060 quasars with a medianredshift z med = .
45. The redshift distribution of these objectscan be seen as a solid line histogram in Fig. 1.
3. Determining the parameters Ω M and w Q The key point of our work is to search for features in the redshift-space two-point correlation function of quasars that remain fixedin comoving scales as the universe evolves. We do not attempt tomodel theoretical quasar correlation functions to compare withobservations since this would require taking into account sev-eral not-well-known factors such as redshift space distortions,quasar bias, survey selection function, initial power spectrum,etc.. This might be a hard enterprise since it requires complexmodelling and exhaustive checks with numerical simulations asshown by S´anchez et al. (2008), all of them far beyond the scopeof this paper. Nor do we intend to relate the measured featureswith scales that resemble particular events in the history of theuniverse, such as the sound horizon at recombination as someauthors have already done (e.g. Guzik et al. 2007), since this canbe misleading as pointed out by S´anchez et al. (2008).Our analysis consists of comparing the positions of features(i.e. peaks and valleys) in the linear regime of the quasar two-point correlation function, ξ ( r ), at di ff erent redshifts. We are in-terested in a comparison among ξ ( r )s at di ff erent times over awide range of scales, and not just finding an agreement on a par-ticular comoving scale as has been done in previous works (e.g. andivarez & Mart´ınez: Constraining flat quintessential universes 3 Fig. 2. S / N analysis for the two redshift bin case. Upper panel:The percentage of ξ ( r ) at large scales with S / N ≥
1, aver-aged over two redshift bins, i.e., the percentage of scales in ξ ( r ) we use to derive cosmological parameters. Lower panel:Averaged S / N map for the ξ ( r ) in two redshift bins includingonly those scales where S / N ≥
1. Both panels corresponds toscales 100 h − Mpc ≤ r ≤ h − Mpc and show higher valueswith lighter colours. R ff ects of evolution in our results by dividing theminto four redshift bins. The first choice aims to construct sam-ples of quasars as large as possible to allow for a fair detectionof features in the ξ ( r ) with signal-to-noise ( S / N ) ratios as highas possible for a wide range of comoving scales. Nevertheless, itshould be noticed that the size of the redshift bins could allow for evolution of the quasar population within a single bin and thusbias the results. Therefore, we intend to understand possible evo-lution biases on the results that we found in the two redshift bincase by means of a four redshift bin analysis, even though thiscase will be a ff ected by larger errors bars in the ξ ( r ). In order to keep the noise as low as possible, we calculate the co-moving separations directly in three-dimensional, flat, comovingspace, assuming a perfectly homogeneous metric. That is, we donot attempt the commonly used procedure that consists of deriv-ing the spatial two-point correlation function from the projectedcorrelation function since that would require us to split pairs intoa two-dimensional array, which would increase shot noise.The cosmology is implicitly involved in the spatial two-point correlation function through the computation of the co-moving distances. The comoving distance-redshift relation in aquintessence flat universe is given by: r ( z ) = cH Z z dz p Ω M (1 + z ) + Ω Q (1 + z ) + w Q ) , (1)where Ω Q = − Ω M and w Q is constant. We explore di ff er-ent cosmologies, varying the parameters 0 . < Ω M ≤ .
55 and − . ≤ w Q ≤ − . ∆Ω M = .
025 and ∆ w Q = . Ω M values since there is com-pelling evidence that we are living in a low matter density uni-verse (e.g. S´anchez et al. 2006; Spergel et al. 2007).For each pair of parameters, Ω M and w Q , we assign comov-ing positions of quasars by using their redshifts in Eq. 1 andcompute the correlation functions according to Landy & Szalay(1993): ξ ( r ) = DD ( r ) − DR ( r ) + RR ( r ) RR ( r ) , (2)where DD ( r ), DR ( r ) and RR ( r ) are the normalised number ofquasar-quasar, quasar-random and random-random pairs withcomoving separations within r ± ∆ r /
2. For the purposes ofthis work we use linear bins in comoving distances with ∆ r = h − Mpc.The random sample was constructed to have the same angu-lar coverage of the survey and a smooth redshift distribution thatmatches that of the quasars. The angular mask was constructedusing routines included in the HEALPix package to pixelizethe sky coverage of the SDSS DR6. As regards the redshift dis-tribution of the random points, we have found that a remarkablygood fit to the redshift distribution of quasars is the sum of fiveGaussian functions as shown in the upper panel of Fig. 1. Byfitting this function we have a smooth redshift distribution thatclosely follows that of the quasars. We then draw the redshiftsof the individual random points from this distribution and obtainthe distribution shown in the lower panel of Fig. 1. The randomsample is 20 times denser than the sample of quasars in order toreduce shot noise.For the comoving scales that we are dealing with ( r > h − Mpc), we compute ξ ( r ) error bars using the bootstraptechnique. This method becomes an unbiased estimate of theensemble error when the number of pair counts is large com-pared to the numbers of objects in the sample (Mo et al. 1992; Hierarchical Equal Area iso-Latitude Pixelization, G´orski et al.(2005) Zandivarez & Mart´ınez: Constraining flat quintessential universes
Fig. 3. ( Ω M − w Q ) maps for the ” cross-correlation ” ( CC (2 z ) ) of the ξ ( r ) in two redshift bins. The upper left panel shows CC (2 z )[1 , taking into account peaks and valleys in the ξ ( r ), while the lower panels show the CC (2 z )[1 , only for peaks (left) and valleys (right).The cross-correlation in these panels were computed only for S / N ≥
1, except for the upper right panel, where CC (2 z )[1 , is showntaking into account peaks and valleys without any S / N restriction. Higher values of the cross-correlation correspond to lightercolours. Labels in contour levels show the percentage of the map outside the contours.Shanks & Boyle 1994). Hence, for each correlation function, wealso have 20 bootstrap estimates.In order to avoid spurious features, we smooth the correla-tion function and each bootstrap estimation using a Gaussian fil-ter with σ smooth = h − Mpc. It is worth mentioning that smooth-ing the pair counts instead of the correlation function producescomparable results. We use these smoothed correlation functionsto extract cosmological information and the smoothed bootstrapestimates to compute error bars. Ω M and w Q As it has been stated in previous works (e.g. Roukema2001, R ξ ( r )s at di ff erent redshifts. To achieve this, we do afull scale comparison among ξ ( r )s. This approach should intro-duce tighter constraints on the cosmological models. To quantify the goodness of each cosmological model pair( Ω M , w Q ), we use a function that allows us to check for sim-ilarities among ξ ( r )s at di ff erent redshifts. This function is the cross correlation and it is an integral that expresses the amountof overlap between two functions. Formally, the cross correla-tion of two real functions f and g is defined as( f ⋆ g )( τ ) = Z ∞−∞ f ( t ) g ( τ + t ) dt (3) andivarez & Mart´ınez: Constraining flat quintessential universes 5 Fig. 4. ( Ω M − w Q ) map shifted to lower values of w Q . The mapshows CC (2 z ) of the ξ ( r ) for two redshift bins. The minimumcomoving distance of the map is r min = h − Mpc. Each modelis computed only for S / N ≥
1. Labels in contour levels show thepercentage of the map outside the contours.where τ is a lag applied to g . Particularly, we are interested inthe case of a lag τ =
0. Hence, the discrete formula for the crosscorrelation in our study is CC [ i , j ] = ( ξ i ⋆ ξ j ) = r max X r = r min ξ i ( r ) ξ j ( r ) (4)where ξ is the two-point correlation function and the subscripts i and j denote particular redshift bins. It should be taken intoaccount that we have restricted the sum over a finite range ofcomoving distances ( r min , r max ) at which we perform our analy-sis of the ξ ( r )s. Particularly, we choose as a maximum distance r max = h − Mpc since larger scales have very low S / N , while r min is used as a variable taking values ≥ h − Mpc.Since we are interested in considering only the features thatshow a true signal, from now on (unless specified otherwise) weonly use those points in ξ ( r ) whose S / N ratio is greater thanunity. Points that do not exceed this threshold are set to zero inthe computation of the quantity CC . Our first approach is to split the quasar redshift distribution intotwo equal number subsamples. These subsamples are called thelow redshift subsample, 0 < z ≤ .
45, and the high redshift one,1 . < z ≤ . ξ ( r ) ? In order to answer this question we con-struct two significance maps which are shown in Fig. 2. Thefirst map (upper panel) shows the percentage of scales in the ξ ( r ) that shows S / N ≥ S / N for the ξ ( r ) for two redshift bins, also using Fig. 5.
The two point correlation functions for the preferred (leftpanels) and standard (right panels) cosmological models. Upperand lower panels display the ξ ( r )s for the highest and lowestredshift bins respectively. Error bars were computed using thebootstrap re-sampling technique and are shown as a grey region.Up and down arrows point out features with S / N ≥ ξ ( r )s are restricted to the range100 h − Mpc ≤ r ≤ h − Mpc. In both panels of Fig. 2 lightercolours indicate higher values. From the first map we observethat our restriction on S / N implies that the useful range of scaleson the ξ ( r ), on average, runs from 23% to 33%. On the otherhand, the second map shows that the averaged S / N lies in therange 1.39 to 1.51. From the contour levels and colours in thisFig. we can observe that there are many regions in both mapsthat are prone to a significant range of scales that are useful anda relatively high average S / N . Beyond that, an important issuethat should be noticed is that all cosmological pairs on the maphave useful scales in the ξ ( r ) and therefore all models are plau-sible in showing coincident features at di ff erent times.To quantify the level of agreement among ξ ( r )s, we use thecross correlation denoted by CC (2 z )[1 , , where we have added anupper-script, (2 z ), denoting the current case. This statistic hasthe capability of enhancing features that occur at similar scaleswhile erasing those that do not appear at the same distance.Moreover, since there is a sum over a wide range of comovingscales, several coincident signals for a particular cosmologicalpair produce higher values of CC implying a better match be-tween the ξ ( r )s at di ff erent redshifts.The CC (2 z )[1 , maps with r min = h − Mpc are shown inFig. 3. All panels show contour levels around the CC (2 z )[1 , max-ima that indicate the percentage of the map outside their bound-aries. The upper left panel shows the CC (2 z )[1 , computed takinginto account all features (peaks and valleys) in the ξ ( r )s. Fromthis panel, we can see that the most likely cosmological modelsare in a small region in the lower left corner of the map. The Zandivarez & Mart´ınez: Constraining flat quintessential universes
Fig. 6. S / N analysis for the four redshift bin case. Left panels: The percentage of the ξ ( r ) at large scales with S / N ≥
1, averagedover the redshift bins 1-3 (upper panel) and 2-4 (lower panel). Right panels: Averaged S / N map for the ξ ( r ) for redshift bins1 and 3 (upper panel) and 2 and 4 (lower panel) including only those scales where S / N ≥
1. All panels correspond to scales100 h − Mpc ≤ r ≤ h − Mpc and show higher values with lighter colours.black cross shows the maximum CC (2 z )[1 , value that correspondsto Ω M = .
175 and w Q = − . ξ ( r )s,an interesting question is whether a particular feature (peaks orvalleys) contributes to a particular region in the map. This issuecan be answered by computing the cross correlation statisticsusing eq. 4, but taking into account positive (peaks) or negative(valleys) significant ( S / N ≥
1) values of the ξ ( r ) only. The re-sulting maps can be seen on the lower left (peaks only) and lowerright (valleys only) panels in Fig. 3.From the left panel, we can see that peaks restrict the areadisplayed by the contour levels compared with the ones obtainedwith all features. The contours are reduced, ruling out some cos-mological pairs with w Q ≥ − . Ω M ). Even so, the preferred cos-mological pair remains unchanged. On the other hand, valleysalone give lower Ω M values ( ∼ . w Q . Nevertheless, their contribution to the combined feature analysis is not strongenough to prevail over the signal preferred by peaks.Finally, it is interesting to know what would be the e ff ect ofrelaxing the restriction imposed on the S / N of the features. In theupper right panel of Fig. 3 we construct the CC (2 z )[1 , map withoutimposing any value on the S / N of the ξ ( r )s. From the colourintensity of the map it is clear that the main e ff ect (comparedto the upper left panel) is the widening of the area of preferredcosmologies. Even when this map enhances the importance ofsome cosmological models that are not statistically significant,the resulting preferred cosmology turns out to be very similar tothe one obtained from significant features only.Since our result on w Q is close to the limit of the w Q rangeused so far, it would be interesting to test whether the exclusionof w Q < − . − . ≤ w Q ≤− .
50. As can be seen from Fig. 4 our results are stable, henceno bias arises from our original choice for the w Q range. andivarez & Mart´ınez: Constraining flat quintessential universes 7 To illustrate the ability of our method to compare ξ ( r )s at dif-ferent times, in Fig. 5 we show a comparison between the bestcosmological pair obtained using the CC (2 z )[1 , map ( Ω M = . w Q = − .
95, left panels) and the standard model ( Λ CDM , Ω M = .
250 and w Q = − .
00, right panels). The grey region ineach panel shows the 1 σ bootstrap errors bars. The best modelis clearly better than the standard one given the number of coin-cident significant features at di ff erent scales. The arrows in thisFig. show features observed at the same comoving scale that sat-isfy S / N ≥
1. While the preferred cosmological model obtainedfrom our maps has four coincident features, we can only observeone matching feature for the standard cosmological model.We have also checked whether the resulting best model de-pends on the choice of r min by computing CC (2 z )[1 , maps varying r min from 120 to 190 h − Mpc. In all cases we found that the pre-ferred pair of cosmological parameters is the same obtained for r min = h − Mpc, indicating the stability of our results.
Given the size of the redshift bins used in the previous case,the evolution of the quasar population within each bin might bebiasing the results. In this subsection, we use four redshift binsto avoid too much evolution within a single bin. This analysistests the stability of the results obtained with two redshift binseven when narrower redshift bins imply larger errors in the ξ ( r ).We divide the sample into four equal number redshift bins:1. 0 . < z ≤ . . < z ≤ . . < z ≤ . . < z ≤ . ξ ( r ), we show in Fig. 6 thesignificance maps for the redshift bins 1-3 (upper panels) and 2-4 (lower panels). The percentage maps (left column) are shownfor significant features averaged over two redshift bins. Whencombining redshift bins 1-3 (upper panel) we observe that 18%-26% is the range of scales for the ξ ( r )’s that are useful for ourstudy while 19%-31% is the range of scales obtained for redshiftbins 2-4 (lower panel). On the other hand, when analysing theaveraged S / N maps (right column) for significant features only,the redshift bins 1-3 (upper panel) show a variation from 1.25to 1.59 while the average over redshifts 2-4 (lower panel) lies inthe range 1.46 to 1.56. It is clear that all cosmological pairs havesome portion of the ξ ( r ) useful for our study.These S / N maps span similar value ranges to those corre-sponding to the two redshift bin case (see Fig. 2). Even whencomputing the ξ ( r )s within narrower redshift bins produceslarger error bars, on average, the S / N show no degradation com-pared to the previous case since some peaks and valleys in ξ ( r )are larger now. This could be a consequence of less evolutionwithin a single redshift bin. Under this assumption, some fea-tures in the ξ ( r ) in the two redshift bin case could have been Fig. 7.
Comparison of the ξ ( r )s between the two (blue and redcurves) and four (grey areas) redshift bin cases for the preferredmodel obtained in the two redshift bin case. Error bars werecomputed using the bootstrap re-sampling technique. Each panelquotes the corresponding redshift ranges.eroded away by evolution. In Fig. 7 we observe a comparisonbetween the ξ ( r )s obtained in the two (blue and red curves) andfour (grey areas) redshift bins cases for the preferred cosmolog-ical model found in the previous subsection. Examples of ourstatement are the regions with 200 h − Mpc ≤ r ≤ h − Mpcin the two upper panels, and the peak at ∼ h − Mpc in thelower left panel of this Fig.Figure 8 shows the cross correlation maps corresponding tothe calculation of equation 4 for redshift bins 1-3 ( CC (4 z )[1 , , leftpanel) and 2-4 ( CC (4 z )[2 , , right panel). Each map is computed us-ing r min = h − Mpc. The preferred model in the CC (4 z )[1 , map is Ω M = .
225 and w Q = − .
90. On the other hand, the CC (4 z )[2 , mapbetter fits with Ω M = .
200 and w Q = − .
95. We have checkedthat these results do not change when varying r min from 120 to190 h − Mpc. From these results the mean cosmological valuesand their corresponding standard errors are Ω M = . ± . w Q = − . ± .
04. We observe that this result is consistentwithin a 1 σ interval with the one obtained for the two redshiftbin case.In the light of our results, we consider as the most likelycosmological model to be the one obtained for the four redshiftbin case since it is less a ff ected by evolution e ff ects while havingsimilar S / N values to those obtained when using larger redshiftbins.
4. Summary and discussion
In this work we have studied the evolution of the quasar two-point correlation function, ξ ( r ), to constrain cosmological pa-rameters for flat quintessential universes, namely the matter adi-mensional density parameter Ω M and the quintessence equationof state, w Q . For this purpose, we have drawn a large homo- Zandivarez & Mart´ınez: Constraining flat quintessential universes
Fig. 8. ( Ω M − w Q ) maps for the CC (4 z ) of the ξ ( r ) for four redshift bins. The left panel show the CC (4 z )[1 , , while the right panels show CC (4 z )[2 , . The minimum comoving distance for each map is r min = h − Mpc. Each model is computed only for S / N ≥
1. Highervalues of the cross-correlation correspond to lighter colours. Labels in contour level show the percentage of the map outside thecontours.geneous quasar sample from the SDSS DR6, the largest quasarcatalogue available at present. The size and homogeneity of thesample allow for highly reliable statistical studies of quasars.As in previous studies of this aspect (e.g. Vallinotto et al. 2007;S´anchez et al. 2008; Crocce & Scoccimarro 2008), we have fo-cused on analysing the position of features in ξ ( r ), at large co-moving scales ( r > h − Mpc) well into the linear regime ofstructure formation. The key point of the analysis lies in the as-sumption that, for the proper cosmological model, the comovinglength scale of such features in ξ ( r ) should remain fixed at dif-ferent stages in the evolution of the universe. In this work, we donot assume any particular scale at which the features (peaks andvalleys) should be found, but we only require consistency amongdi ff erent redshifts ranges, imposing purely geometric constraintson the cosmological parameters.The study is focused on two di ff erent complementary ap-proaches: splitting the quasar redshift distribution into two andfour redshifts bins. The first analysis is performed in order tohave reliable estimations of ξ ( r ) at large scales (where the signalis expected to be low) while the second one is performed aimingto rule out a possible bias in the previous results due to an evolu-tion of the quasar population within a particular redshift bin. Thecomparison among ξ ( r )s of di ff erent redshift bins is carried outby means of the cross-correlation function ( CC ). This particularstatistic has the advantage of quantifying the amount of over-lap between two functions. In our case, the CC s are computedfor modified ξ ( r )s, i.e, we have imposed a significance criterionwhich keeps the ξ ( r )s values when they satisfy the constraint | ξ ( r ) /σ ( r ) | ≥
1, otherwise are set to zero.With regard to the two redshift bin case, the most likelymodel is Ω M = . ± .
02 and w Q = − . ± .
04. The co-incidence among the results for the di ff erent reference scales( r min ) analysed is remarkable, implying that ξ ( r )s for both red-shift bins agree over a wide range of comoving scales, from ∼
100 to ∼ h − Mpc. From the four redshift bin case we ob-tain that the preferred cosmological model is Ω M = . ± . w Q = − . ± .
04. When analysing the four redshift bincase, it becomes clear that the two redshift bin analysis is af- fected by evolution. The main observable e ff ect of evolution isthat, on average, the amplitude of some features in the ξ ( r ) hasbeen diminished, and as a consequence, the best model has beenslightly shifted towards lower values of Ω M .The closest reference in the literature of an analysis similar tothat performed in this work is the study of R
02. Some of the sim-ilarities with this work are: we both only use a sample of quasarsto constrain cosmological parameters, and neither of us searchfor a particular comoving scale. Beyond quasar sample sizes,the methodology di ff erences can be enumerated as follows: – We extract information from the position of peaks and val-leys in the ξ ( r )s while R
02 used peak information only. – We restrict the ξ ( r )s with a significance criterion in order toavoid spurious detections while R
02 used all the informationwithout any restriction on S / N ratio. – The statistical tool proposed in this work intends to performa full scale comparison among ξ ( r )s, i.e, to take into accountmore than one feature agreement among di ff erent redshiftbins. On the other hand, R
02 developed a method based onsearching for the largest probability that the first local maxi-mum in ξ ( r ) above a given reference scale occurs at the samecomoving distance for consecutive redshift bins. – By construction, our method gives larger weights to strongerfeatures while the probability method of R
02 only considersthe comoving length scale at which a peak is present. – We performed an analysis involving two and four redshiftbins and found that evolution cannot be neglected. R
02 per-formed their analysis using three redshift bins, extractinginformation from consecutive redshift bins. Their approachcan still be a ff ected by evolution e ff ects.In their work, R
02 found a matter density parameter in goodagreement with our findings (see Table 1). Nevertheless, theycannot impose strong constraints on the quintessence parameter,finding only an upper limit ( w Q ≤ − . ξ ( r )s providestighter constraints on cosmological parameters rather than bas- andivarez & Mart´ınez: Constraining flat quintessential universes 9 Table 1.
Constraints on cosmological parameters Ω M and w Q found in the literature under the assumption of perfectly flat universes.The last column quotes the data sets used in each work. Author Ω M w Q DataRoukema et al. (2002) 0 . ± .
10 [ − . , − .
5] 10K 2dFQSO Tegmark et al. (2004) 0 . ± . − . ± .
30 WMAP + SDSS MGS Eisenstein et al. (2005) 0 . ± . − . ± .
18 WMAP + SDSS MGS + LRG Wang & Mukherjee (2006) ... − . + . + . − . − . R04 WMAP3 + SNIa + LRGWang & Mukherjee (2006) ... − . + . + . − . − . A06 WMAP3 + SNIa + LRGS´anchez et al. (2006) 0 . ± . − . + . − . WMAP + Spergel et al. (2007) 0 . ± . − . ± .
073 WMAP3 + SNLS This work 0 . ± . − . ± .
04 DR6 SDSS QSO
1: Quasars from the 10K release of the 2 degree Field Quasar Redshift Survey, Croom et al. (2001).2: First year Wilkinson Microwave Anisotropy Probe, Spergel et al. (2003).3: Main Galaxy Sample, Strauss et al. (2002).4: Luminous Red Galaxies, Eisenstein et al. (2001).5: Riess et al. (2004).6: Third year Wilkinson Microwave Anisotropy Probe, Spergel et al. (2007).7: Astier et al. (2006).8: 2 degree Field Galaxy Redshift Survey, Colless et al. (2001).9: Supernovae Legacy Survey, Astier et al. (2006). ing the analysis on the coincidence of a single particular scaleonly.From the comparison with results from a variety of dataand techniques (see Table 1), we observe that our estimation of Ω M is in good agreement with R
02, S´anchez et al. (2006) andSpergel et al. (2007). In addition, our result for w Q agrees withall sources quoted in Table 1, and has one of the smallest uncer-tainty intervals.What are the possible implications of our results in termsof structure formation compared to the initial conditions estab-lished by the Λ CDM concordance model? We observe a smalldiscrepancy in the mean values obtained for the Ω M and w Q compared with a standard universe with cosmological constant( Ω M = .
25 and w Q = − . Ω M and higher valuesof w Q than those of the concordance model. Both tendencies ob-served in the parameters have similar consequences for structureformation. If all cosmological models were normalised to pre-dict the observed amplitude of density fluctuations at present,lower values of density would predict higher levels of densityfluctuations in the past (e.g. Spergel 1998). Under the same as-sumptions of density perturbation normalisation, earlier cloudformation and higher core densities are observed if the dynam-ics of the dark energy is enhanced, i.e. for models representedby an equation of state parameter w Q > − Ω M and w Q . Acknowledgements
We thank to the anonymous referee for important sugges-tions that have greatly improved the original manuscript.We also thank E.D. and Diego G. Lambas for carefullyreading the manuscript and suggestions. HJM acknowledgesthe support of a Young Researchers’ grant from AgenciaNacional de Promoci´on Cient´ıfica y Tecnol´ogica Argentina,PICT 2005 / // / . The SDSS is managed by theAstrophysical Research Consortium (ARC) for the ParticipatingInstitutions. The Participating Institutions are The Universityof Chicago, Fermilab, the Institute for Advanced Study, theJapan Participation Group, The Johns Hopkins University, theKorean Scientist Group, Los Alamos National Laboratory,the Max Planck Institut f¨ur Astronomie (MPIA), the MaxPlanck Institut f¨ur Astrophysik (MPA), New Mexico StateUniversity, University of Pittsburgh, University of Portsmouth,Princeton University, the United States Naval Observatory, andthe University of Washington. References
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