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Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed16December 2015 (MNLATEXstylefilev2.2) High-z Supernova Type Ia Data: non-Gaussianity and Direction Dependence 5 Shashikant Gupta 1,2 and Meghendra Singh 3,4 1 1 Amity UniversityHaryana, Gurgaon, India 0 2 [email protected] 2 3 Mahamaya Tech. University,Noida, India c 4 meghendrasingh [email protected] e D 5 16December 2015 1 ] ABSTRACT O C We use the ∆χ2 statistic introduced in Gupta, Saini & Laskar (2008); Gupta, Saini (2010) to study directional dependence, in the high-z supernovae data. This depen- h. dence could arise due to departures from the cosmological principle or from direc- p tion dependent statistical systematics in the data. We apply our statistic to the gold - data set from Riess et al. (2004) and Riess et al. (2007), and Union2 catalogue from o Amanullah et al. (2010). Our results show that all the three data sets show a weak r t but consistent direction dependence. In 2007 data errorsare Gaussian, howeverother s two data sets show non-Gaussian features. a [ Key words: cosmology:cosmologicalparameters—cosmology:large-scalestructure of universe —supernovae: general 2 v 7 2 5 1 INTRODUCTION tionl=282◦,b=6◦.Theprobability offindingsuchaflow 4 in the ΛCDM cosmology is less than 1%. 1. A large and diverse variety of cosmological observations ◦ 0 (Perlmutter et al. 1999; Riess et al. 1998, 2002, 2004, 2007; • The planes normal to the quadrupole (l = 240 ,b = ◦ ◦ ◦ 4 Benoit et al. 2003; Page et al. 2007), during the last two 63 ) and octopole (l=308 ,b=63 ) are aligned with each ◦ 1 decades(notablytheNobelprizesin2006andin2011)have other and with the direction of the dipole (l = 264 ,b = ◦ : established that we live in a flat universe with an acceler- 48 ).Thisindicatesapreferredaxis;andisinconsistentwith v atedexpansion.ThisisconsistentwiththeEinstein’sgeneral the Gaussian random, statistically isotropic skies at about i X theory of relativity with a cosmological constant term. The 99%. r Cosmological Constant (Λ), also known as dark energy, is • Using HST key project data McClure & Dyer (2007) a treated as an ideal fluid with negative pressure; and with showed that a statistically significant variation of at least equation of state p=wρ, where w =−1. Dark energy den- 9 km/s/Mpc exists in the observed value of H0 with a di- sity dominates over that of baryonic and dark matter; and rectional uncertainty of about 10-20%. Gupta, Saini (2011) it constitute two third of the Universe. This model of the also found directional dependence in the HST key project Universe is known as the ΛCDM model or standard model data. of cosmology. • Recentworkprovidessomeevidenceforwhatisknown The foundation of the standard model of cosmology as the Hubble Bubble (Zehaviet al. 1998; Jha et al. 2007) (ΛCDMcosmology) istheCosmological principle(hereafter which suggests that we might be living inside a large void. CP), which states that the Universe is homogeneous and Value of the Hubble constant inside the bubble is different isotropic on the large scales (Peebles 1993). CP along with from what is outsidethe bubble.There is evidencefor such the fact that two third of the constituents of the Universe largescalevoidsintheCMBmapsaswell(Cruzet al.2005; is dark energy can explain many cosmological observations deOliveira-Costa & Tegmark2006;Cruz et al.2006,2007), and is the most successful model till date. Despite its suc- suggesting that such large voids are not implausible. If our cess, ΛCDM model has its shortcomings; few of which are positionisnearthecenterofsuchavoidonecanexplainthe summarized below: dimmingofSNewithoutinvokingdarkenergy.However,this • Observations indicate significantly larger amplitude challengestheCopernicanprinciplethatwedonotoccupya of flows than what ΛCDM predicts. (Watkinset al. specialplaceintheUniverse.Ifweassumethatourposition 2009) found the large scale peculiar velocity larger than is off-centered then one can explain the preferred axis (or 400km/Secatscalesupto100h−1 Mpc.Thisisinthedirec- tiny departuresfrom CP) also (Blomqvist et al. (2010)). (cid:13)c 0000RAS 2 Shashikant Gupta and Meghendra Singh In summary, some cosmological observations show depar- 2 METHODOLOGY: THE ∆χ2 STATISTIC ture from CP and hence are in contradiction to the stan- WeusetheGolddata(GD04andGD07)(Riess et al.2004, dard model of cosmology. Another threat to the model 2007)andtheUnion2dataAmanullah et al. (2010)forour comes from the fact that Observed value of Λ does analysis. The Gold data GD04 and GD07 contain 157 and not match its theoretical value and requires fine tuning. 182 SNe respectively; while the most recent and largest set To avoid this several alternative explanations have been Union2contains557Supernovae.Foragivensupernovathe suggested (Silvestri & Trodden 2009; Frieman et al. 2008; measuredquantity,thedistancemodulusµ,isthedifference Sahni& Starobinsky 2006). In many of these alternative between the apparent and the absolute magnitude models,wisallowedtobedifferentfrom−1inthepast;and approachesw≃−1atlowred-shifts.Itisdifferentfromcos- µ(z)=m(z)−M, (1) mological constant, sincetherethevalueof w isalways −1. Cosmological observations suggest w ≃ −1 at the present wheretheapparent magnitude m(z)dependson theintrin- epoch, which is consistent with ΛCDM as well as the alter- sic luminosity of a supernova, the redshift z and the cos- nativecosmologies. Tobeabletodistinguishamongvarious mological parameters; and M is the absolute magnitude of models,werequiredatathatispreciseenoughtodiscerntiny a type Ia supernova. It can be expressed in terms of the variations in the dark energy. It is also required that data luminosity distance DL as be available at a large number of redshifts to constrain the µ(z)=5log(DL(z)/Mpc)+25, (2) detailed behavior of dark energy with time. At the present the only data that comes reasonably close to these require- where theluminosity distance is given by ments is provided by the observations of the high-redshift c(1+z) z dx supernovae, which are believed to be standard candles. Be- DL(z)= , (3) sides,thequalityoftheSNeIadatamaybedoubtfuldueto H0 Z0 h(x) thefollowing reasons: where h(z;ΩM,ΩX)=H(z;ΩM,ΩX)/H0, and thus depends onlyonthecosmologicalparametersmatterdensityΩM and the dark energy density ΩX. We assume that the prescrip- • Thephysics of SNeIa is relatively poorly understood. tion for the variation of dark energy with redshift is sepa- • Thepossibility ofphysicalmechanisms,suchasdustin ratelyspecified,forexampleintheΛCDMmodeltheenergy theinter-galactic medium that systematically dims them. density in the dark energy remains a constant. In Eq 1 the • The supernova data are usually collated from several dependenceofthemeasuredquantityµonM islinear.Since different sources that might have slightly different system- µ depends on the logarithm of the luminosity distance it is atics, due either to instrumental effects or to the fact that clearthatitdependslinearlyonthelogarithmoftheHubble theyoccurindifferentdirectionsinthesky.Sincewehaveto parameter H0. Usually the data is given in terms of Eq 2, correctforthegalacticdustextinction,whichmightnotget where the constant M has already been marginalized over. completely removed from the samples, this might produce Thus, instead of two nuisance parameters we are left with anisotropy in data. only one parameter, theHubbleconstant H0. We now give an introduction of the ∆ statistic pre- viously introduced and used in GSL08 and GS10. A gen- These considerations imply that to have precise informa- eralization was given in GS10, where we had numerically tion about the behavior of dark energy we should have marginalized overtheHubbleconstant H0.Someof thede- a good knowledge of the statistical properties of super- tailsarerepeatedheretomakethisworkself-contained.For novae,both random as well as systematic. There havebeen ouranalysiswehaveassumedaflatΛCDMuniverse.Asim- several attempts to search for the direction dependence ilar analysis could be carried out for a more general model in the SNe Ia data. Gupta, Saini & Laskar (2008) (here- of dark energy. afterGSL08)and(Gupta, Saini2010)(hereafterGS10)used Weconsidersubsetsofthefulldatasettoconstructour the extreme value statistics to show that the two super- statistic consisting of Nsubset data points. Since theΛCDM nova data sets, Riess et al. (2004) (GD04) and Riess et al. modelfitsthegolddatasetsGD04,GD07andUnion2well, (2007) (GD07), do show some evidence for direction de- wefirstobtainthebestfittothefulldatasets byminimizing pendence. Antoniou & Perivolaropoulos (2010) have also theχ2, which we defineas: sAdhimcoawatnnedulaleaihpthreeetfrearslry.esdt(e2am0x1ai0st)ic.uSspienrvogebrltaehlmeostUhwenriitohwno2trhkcesathhaialgovhge-uraeeldsofsrhoiinmft- χ2=ΣNi=1(cid:2)µi−µσΛiCDM(cid:3)2, (4) supernova data or directional dependence in the supernova whereσi isobservedstandarderrorinµi.Bythisweobtain data and other probes (Nesseris & Perivolaropoulos 2004, the best fit values of the parameters ΩM and H0. Then for 2005, 2007; Jain & Ralston 2006, 2007; Amendola et al. eachsupernovawecalculate χi =[µi−µΛCDM(zi;ΩM)]/σi, 2013). where µΛCDM(zi;ΩM) is calculated using thebest fit values Inthispaperourmaintaskis1)tolookfordirectionde- of ΩM and H0. We assume that all the supernovae are sta- pendentsystematic effects in thelatest SNeIadata (Union tistically uncorrelated. 2catalogue)and2)tocomparethequalityofthisdatawith We define χ2M = Σiχ2i and the normalized quantity thepreviousdatasets(Golddata2004and2007).Theplan χ2R =χ2M/Nsubset,χ2R indicatesthestatistical scatterofthe of the paper is as follows. In § 2 we introduce the statistic subset from the best fit ΛCDM model and its expectation we have used, in § 3 we provide our results and end with value is unity that is hχ2i = 1. If CP holds then the ap- R conclusions in § 4. parent magnitudeof a supernovashould not depend on the (cid:13)c 0000RAS,MNRAS000,000–000 High-z Supernova Type Ia Data: non-Gaussianity and Direction Dependence 3 Figure1.Acomparisonoftheoreticalandbootstrapprobability Figure 3. The theoretical and the bootstrap probability distri- distributions for simulated data. The data comprises 157 supernovae, whosepositionsontheskyweregeneratedrandomly. butions for GD04 for the ∆χ2 statistic. Theoretical distribution isshiftedtotheleftcomparedtowhatwefindforoursimulated data in Figure 1, which uses Gaussian deviates suggesting evi- dencefornon-Gaussianity. thisquantityanditisobviousthatforeveryvalueof∆χ2nî, theantipodalpoint hasthenegativeof thatvalue.Wethen vary thedirection nˆ across thesky to obtain themaximum absolute difference 2 ∆χ2 =max{|∆χnˆ|} . (5) As shown in GSL08, the distribution of ∆χ2 follows a sim- ple, two parameter Gumbel distribution, characteristic of extremevalue distribution typeI (Kendall & Stuart 1977), P(∆χ2)= 1sexp(cid:20)−∆χ2s−m(cid:21) exp(cid:20)−exp(cid:18)−∆χ2s−m(cid:19)(cid:21) , (6) where the position parameter m and the scale parameter scompletelydeterminethedistribution.Toquantifydepar- turesfromisotropyweneedtoknowthetheoreticaldistribu- tion, which is calculated numerically by simulating several sets of Gaussian distributed χi on the gold set supernova positions and obtaining ∆χ2 from each realization. And as inGSL08,wecomputeabootstrapdistributionbyshuffling Figure2.Acomparisonoftheoreticalandbootstrapprobability thedatavalueszi,µ(zi)andσµ(zi)overthesupernovaepo- distributions for simulated data. The data comprises 557 super- sitions (for further details see GSL08). A modified version novae, whosepositionsontheskyweregeneratedrandomly. ofthisstatisticwasintroducedinGS10,whichmarginalizes overtheHubbleConstant.However,bythiswelooseallthe information about H0 and hence, here we do not marginal- direction in which it is observed but only on the cosmol- ize. ogy and thus supernovae in different directions should be scattered similarly with respect to thebest fit model. We divide the data into two hemispheres labeled by 3 RESULTS the direction vector nˆ, and take the difference of the χ2 R computed for the two hemispheres separately to obtain InGSL08andGS10wediscussedaspecificbiasintheboot- ∆χn2ˆ =χ2R1−χ2R2,wherelabel’1’correspondstothathemi- strap distribution, showing that it is shifted slightly to the sphere towards which the direction vector nˆ points and la- leftofthetheoreticaldistributionduetothefactthattheo- bel’2’ refers to theotherhemisphere. Wetaketheabsolute reticaldistributionisobtainedbyassumingχistobeGaus- valueof ∆χ2nî sinceweareinterested inthelargest valueof sian random variates with a zero mean and unit variance. (cid:13)c 0000RAS,MNRAS000,000–000 4 Shashikant Gupta and Meghendra Singh Table1. Themodelparameters(ΛCDM)forthethreedatasets aretabulated here. Set #SNe ΩM H0 χ2/dof GD04 157 0.32 64.5 1.14 GD07 182 0.33 63.0 0.88 Union2 557 0.27 70 0.97 Table 2. Direction for maximum ∆ in the three data sets are tabulatedhere. Model Set ∆χ2 longt lat ΛCDM GD04 0.83 96.5 44.5 ΛCDM GD07 0.53 347.1 27. ΛCDM Union2 0.22 65.5 55.8 Figure 4. The theoretical and the bootstrap probability distri- results in this paper should be interpreted with respect to butions for GD07 for the ∆χ2 statistic. Theoretical distribution is to the right as expected by simulated data inFigure 1, which Figure1forGD04andGD07,andFigure2forUnion2data. usesGaussiandeviates suggestingevidencefornon-Gaussianity. Concerns regarding the small number of supernovae in the Gold data and its effect on the efficacy of our method can bee addressed by the fact that overall behavior of Figure 1 is repeated in Figure 2 which is plotted for more than 500 of SNe. This statistic is similar to the one presented in GSL08 andGS10exceptforthefactthatinGS10wehadmarginal- ized over the Hubble parameter. Our results are different from thosepresentedin GSL08 also duetothefact that we have corrected the numerical bug mentioned in the GS10 produced by the fact that the theoretical distribution was producedslightlydifferentlyfromthewaybootstrapmethod thereby creating a greater discrepancy between them then shouldhavebeenthecase.InTable1wegivethebestfitval- ues of ΩM and H0 for all the three data sets. We note that Union2 gives slightly lower value of ΩM and higher value of H0 compared to the Gold data. Reduced χ2 is smallest for GD07 which indicates that errors are overestimated for thisset. Union2is best among all thethreesets in termsof χ2.Thedirectionofmaximumdiscrepancyandthevalueof (∆χ2) is presented in Table 2. GD04:InFig3weplotthebootstrapandthetheoret- ical distribution expected for GD04 and mark the position of GD04. Comparison with Figure 1 shows a signature of Figure 5. The theoretical and the bootstrap probability distri- non-Gaussianitysincethetheoreticaldistributioninsteadof butionsforUnion2forthe∆χ2 statistic.Theoreticaldistribution beingtotherightofthebootstrap isshiftedtotheleft.Po- is not compatible with Figure 2 for the simulated data, which sition of GD04 is about 2 sigma away from the peak of the impliesslightnon-Gaussianityfortheresiduals. bootstrap distribution, indicating direction dependence. GD07: Fig 4 for GD07 is same as for GD04. A com- parisonwithFigure1showsthatourresultsarecompatible Thereforetheoreticalχisareunbounded.However,theboot- with theabsence of non-Gaussianity in thedata. GD07sits strapdistributionisobtainedbyshufflingthroughaspecific at about onesigma away from thepeak of thedistribution, realization of χi, and they have a maximum value for some thus the directional dependencehas got weaker than found supernova. It is clear that this should, on the average, pro- in GD04. duceslightlysmallervaluesof∆χ2 incomparisonwithwhat Union2 In Fig 5 we plot both the distributions for one expects from a Gaussian distributed χi. For reference Union 2, which is to be compared with Fig 2. Comparison we plot the results for simulations in Figure 1 with a total showsaweaksignatureofnon-Gaussianity,sincepositionof of 157 and in Figure 2 with a total of 557 supernovae. Our the bootstrap distribution is not to the left of the theoreti- (cid:13)c 0000RAS,MNRAS000,000–000 High-z Supernova Type Ia Data: non-Gaussianity and Direction Dependence 5 caldistribution.PositionofUnion2isaboutonesigmaaway Jain, P. & Ralston, J.P., 2007, Mod. Phys. Lett. A, 22, from the peak of the bootstrap distribution which is a sign 1153. of direction dependence. Jha, S., Riess, A. G., & Kirshner, R. P. 2007, Astroph. J. ,659, 122. Kendall,M.,&Stuart,A.1977,London:Griffin,1977,4th 4 CONCLUSIONS ed. Kolatt, T. S. & Lahav, O. 2001, Mon. Not. Roy. Ast. Soc. Wehavepresentedresults fortheGD04, GD07andUnion2 ,323, 859. datausingthestatisticintroducedinGSL08andGS10.Our McClure M. L., & Dyer C. C., 2007, New Astronomy, 12, main conclusions for this part of our work are: 533. Nesseris, S. & Perivolaropoulos, 2004, Phys. Rev. D , 70, (i) GD04showssomeevidencefornon-Gaussianity,how- 043531. ever, GD07 is entirely consistent with a Gaussian distribu- Nesseris, S. & Perivolaropoulos, 2005, Phys. Rev. D , 72, tionofresiduals.Union2dataagainshowssomeevidencefor 123519. non-GaussianityalthoughweakercomparedtoGD04.Thus, Nesseris, S. & Perivolaropoulos, 2007, J. Cosmol. As- intermsofGaussianity,GD07isbestamongthethreedata tropart. Phys., 0702, 025. sets. deOliveira-Costa,A.,&Tegmark,M.,2006, Phys.Rev.D (ii) GD04isdifferentfromthepeakofthebootstrap dis- ,74, 023005. tribution byslightly morethan onesigma; while GD07 and Page, L. et al. 2007, Astroph.J. Suppl., 170, 335. Union2 both are different from the peak by about a sigma. Peebles,P.J.E.1993, Princeton SeriesinPhysics,Prince- There seems a weak but consistent direction dependencein ton,NJ: Princeton University Press, c1993. all threedata sets. This consistency indicates a physical ef- Percival,W.J.,et al. 2002,Mon.Not.Roy.Ast.Soc.,337, fect or a preferred direction. 1068. Perlmutter, S. et al. 1999, Astroph. J. , 517, 565. Acknowledgments: Riess, A.G. et al. 1998, Astron. J. , 116, 1009. Shashikant Gupta thanks Tarun Deep Saini for discus- Riess, A.G. et al. 2002, Astroph.J. , 560, 49. sion; and colleagues of ASASfor constant support. Riess, A.G., et al. 2004, Astroph. J. , 607, 665. Riess, A.G., et al. 2007, Astroph. J. , 659, 98. Sahni,V.,Starobinsky,A.,2006,Int.J.Mod.Phys.D,15, REFERENCES 2105. Alam,U.,SahniV.,Saini,T.D.&StarobinskyA.A.,2006, Silvestri A., Trodden M., 2009, Reports in Progress in Mon. Not. Roy.Ast.Soc. , 354, 275. Physics, 72, 096901. R. Amanullah, et al., 2010, Astroph.J. ,716, 712. Tegmark, M. et al. 2004, Phys. Rev.D., 69, 103501. Amendola Luca, Marra Valerio, Quartin Miguel, 2013, R.Watkins,H.A.FeldmanandM.J.Hudson,2009,Mon. Mon. Not. Roy.Ast.Soc. , 430, 1867. Not.Roy.Ast. Soc. , 392, 743. Antoniou, I., Perivolaropoulos, L., 2010, J. Cosmol. As- Zehavi,I.,Riess,A.G.,Kirshner,R.P.,&Dekel,A.1998, tropart. Phys., 12, 12. Astroph.J. , 503, 483 Bharadwaj,S.,Sethi,S.K.,&Saini,T.D.2009,Phys.Rev. D , 79, 083538. Benoit, A., et al. 2003, Astron. Astrophys., 399, L25. Michael Blomqvist, Edvard Mortsell 2010, J. Cosmol. As- tropart. Phys., 05, 006. Corasaniti,P.S.,2006,Mon.Not.Roy.Ast.Soc.,372,191. Cruz, M., Mart´ınez-González, E., Vielva, P., & Cayoń, L. 2005, Mon. Not.Roy.Ast. Soc. , 356, 29. Cruz, M., Tucci, M., Mart´ınez-González, E., & Vielva, P. 2006, Mon. Not.Roy.Ast. Soc. , 369, 57. Cruz, M., Turok, N., Vielva, P., Mart´ınez-González, E., & Hobson, M. 2007, Science , 318, 1612 Frieman, J. A., Turner, M. S., & Huterer, D. 2008, Ann. Rev. Astron. Astroph., 46, 385. Gumbel E. J., 1965 ”Statistics of extremes”. Dover publi- cation. Gupta, Shashikant; Saini, Tarun Deep; Laskar, Tanmoy, 2008 Mon. Not. Roy.Ast. Soc. 388, 242. Gupta, Shashikant; Saini, Tarun Deep; 2010 Mon. Not. Roy.Ast.Soc. 407, 651. Gupta, Shashikant; Saini, Tarun Deep; 2011 Mon. Not. Roy.Ast.Soc. 415, 2594. Haan L. and Ferreira A., ”Extreme Value Theory: An In- troduction” Springer2006. Jain, P. & Ralston, J.P., 2006, Astroph.J. , 637, 91. (cid:13)c 0000RAS,MNRAS000,000–000