No Evidence for Dark Energy Dynamics from a Global Analysis of Cosmological Data
Paolo Serra, Asantha Cooray, Daniel E. Holz, Alessandro Melchiorri, Stefania Pandolfi, Devdeep Sarkar
aa r X i v : . [ a s t r o - ph . C O ] A ug No Evidence for Dark Energy Dynamicsfrom a Global Analysis of Cosmological Data
Paolo Serra ∗ , Asantha Cooray , Daniel E. Holz , AlessandroMelchiorri , , Stefania Pandolfi , , Devdeep Sarkar , Center for Cosmology, Department of Physics and Astronomy, University of California, Irvine, CA 92697 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 Physics Department and Sezione INFN, University of Rome,“La Sapienza,” P.le Aldo Moro 2, 00185 Rome, Italy Physics Department and International Centre for Relativistic Astrophysics,University of Rome, “La Sapienza,” P.le Aldo Moro 2, 00185 Rome, Italy and Physics Department, University of Michigan, Ann Arbor, MI 48109 (Dated: November 4, 2018)We use a variant of principal component analysis to investigate the possible temporal evolutionof the dark energy equation of state, w ( z ). We constrain w ( z ) in multiple redshift bins, utilizingthe most recent data from Type Ia supernovae, the cosmic microwave background, baryon acousticoscillations, the integrated Sachs-Wolfe effect, galaxy clustering, and weak lensing data. Unlike otherrecent analyses, we find no significant evidence for evolving dark energy; the data remains completelyconsistent with a cosmological constant. We also study the extent to which the time-evolution of theequation of state would be constrained by a combination of current- and future-generation surveys,such as Planck and the Joint Dark Energy Mission. I. INTRODUCTION
One of the defining challenges for modern cosmol-ogy is understanding the physical mechanism respon-sible for the accelerating expansion of the Universe [1–3]. The origin of the cosmic acceleration can be dueto a new source of stress-energy, called “dark en-ergy”, a modified theory of gravity, or some mixtureof both [4, 6]. Careful measurement of the expansionhistory of the Universe as a function of cosmic epochis required to elucidate the source of the acceleration.In particular, existing data already allows direct ex-ploration of possible time-variation of the dark energyequation of state.While several recent papers have investigated thepossibility of constraining the temporal evolution ofdark energy (see, e.g., [8]), here we present an analy-sis improving and/or complementing existing work intwo ways: first, we incorporate important recent datareleases, including Type Ia supernovae samples (“Con-stitution” and “Union” datasets) and baryon acousticoscillation data (SDSS Data Release 7). This newdata provide significant improvements in the dark en-ergy constraints. Second, we utilize principal compo-nent analysis techniques to constrain the dark energyin a model independent manner, leading to more ro-bust and unbiased constraints.In the absence of a well-defined and theoreticallymotivated model for dark energy, it is generally as-sumed that the dark energy equation of state (the ra-tio of pressure to energy density) evolves with redshiftwith an arbitrary functional form. Common param-eterizations include a linear variation, w ( z ) = w + ∗ [email protected] w z z [9], or an evolution that asymptotes to a constant w at high redshift, w ( z ) = w + w a z/ (1 + z ) [10, 11].However, given our complete ignorance of the underly-ing physical processes, it is advisable to approach ouranalysis of dark energy with a minimum of assump-tions. Fixing an ad hoc two parameter form could leadto bias in our inference of the dark energy properties.In this paper we measure the evolution history ofthe dark energy using a flexible and almost completelymodel independent approach, based on a variant ofthe principal component analysis (PCA) introducedin [12]. We determine the equation of state param-eter, w ( z ), in five uncorrelated redshift bins, follow-ing the analysis presented in [13, 14, 19, 20]. Tobe conservative, we begin by using data only fromgeometric probes of dark energy, namely the cosmicmicrowave background radiation (CMB), Type Ia su-pernovae (SNe) and baryon acoustic oscillation data(BAO). We perform a full likelihood analysis using theMarkov Chain Monte Carlo approach [34]. We thenconsider constraints on w ( z ) from a larger combina-tion of datasets, including probes of the growth of cos-mological perturbations, such as large scale structure(LSS) data. An important consideration for such ananalysis is to properly take into account dark energyperturbations, and we make use of the prescription in-troduced in [18]. We also generate mock datasets forfuture experiments, such as the Joint Dark EnergyMission (JDEM) and Planck, to see how much theyimprove the constraints.The paper is organized as follows: in the next sec-tion we describe our methods and the data used inour analysis; in Sec. III we present our results, and inSec. IV we summarize and conclude. -3.5-3-2.5-2-1.5-1-0.5 0 0 0.2 0.4 0.6 0.8 1 w ( z ) redshift z-2-1.5-1-0.5 0 0 0.2 0.4 0.6 0.8 1 w ( z ) redshift z-4-3-2-1 0 1 0 0.2 0.4 0.6 0.8 1 w ( z ) redshift z UnionConstitution FIG. 1: Uncorrelated constraints on the darkequation of state parameters using a “geomet-ric” dataset given by WMAP+UNION+BAO (up-per panel), and a “combined” dataset given byWMAP+UNION+BAO+WL+ISW+LSS (middlepanel); error bars are at 2 σ . The blue line isthe reconstructed w ( z ) using a cubic spline in-terpolation between the nodes. Also shown is acomparison between WMAP+UNION+BAO andWMAP+Constitution+BAO (lower panel); the pointsfor the Constitution dataset have been slightly shifted tofacilitate comparison between the two cases: we find nosignificant difference between UNION and Constitution. II. ANALYSIS
The method we use to constrain the dark energyevolution is based on a modified version of the publiclyavailable Markov Chain Monte Carlo package Cos-moMC [34], with a convergence diagnostics based onthe Gelman-Rubin criterion [35]. We consider a flatcosmological model described by the following set ofparameters: { w i , ω b , ω c , Θ s , τ, n s , log[10 A s ] } , (1)where ω b ( ≡ Ω b h ) and ω c ( ≡ Ω c h ) are the physicalbaryon and cold dark matter densities relative to thecritical density, Θ s is the ratio of the sound horizon tothe angular diameter distance at decoupling, τ is theoptical depth to re-ionization, and A s and n s are theamplitude of the primordial spectrum and the spectralindex, respectively.As discussed above, we bin the dark energyequation of state in five redshift bins, w i ( z ) ( i =1 , , .. z i ∈ [0 . , . , . , . , . w ( z ) to be a smooth, continuous function, since weevaluate w ′ ( z ) in calculating the DE perturbations(and their evolution with redshift). We thus utilize acubic spline interpolation to determine values of w ( z )at redshifts in between the values z i .For z > z = 1 value, since we find that current dataplace only weak constraints on w ( z ) for z >
1. Tosummarize, our parameterization is given by: w ( z ) = w ( z = 1) , z > w i , z ≤ z max , z ∈ { z i } ;spline , z ≤ z max , z / ∈ { z i } . (2)When fitting to the temporal evolution of the darkenergy equation of state using cosmological measure-ments that are sensitive to density perturbations, suchas LSS or weak lensing, one must take into account thepresence of dark energy perturbations. To this end,we make use of a modified version of the publicly avail-able code CAMB [15], with perturbations calculatedfollowing the prescription introduced by [18]. Thismethod implements a Parameterized Post-Friedmann(PPF) prescription for the dark energy perturbationsfollowing [16, 17].Moreover, the dark energy equation of state pa-rameters w = w i are correlated; we follow [13, 14]to determine uncorrelated estimates of the dark en-ergy parameters. We calculate the covariance matrix C = ( w i − h w i i )( w j − h w j i ) T ≡ h ww T i − h w ih w T i ,using CosmoMC; we then diagonalize the resultingFisher matrix F ≡ C − , which can be written as F = O T Λ O , where Λ is the diagonalized inversecovariance of the transformed bins. The vector of un-correlated dark energy parameters, q , is then obtained TABLE I: Mean values and marginalized 68% confidence levels for the cosmological parameters. The set of w ( z ) i represent the measured values of the dark energy equation of state in uncorrelated redshift bins.Parameter WMAP+UNION+BAO WMAP+Constitution+BAO all dataset future datasetsΩ b h . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . Λ . ± .
018 0 . ± .
017 0 . ± .
016 0 . ± . n s . ± .
014 0 . ± .
014 0 . ± .
014 0 . ± . τ . ± .
017 0 . ± .
016 0 . ± .
017 0 . ± . R (2 . ± . · − (2 . ± . · − (2 . ± . · − (2 . ± . · − w ( z = 1 . −− −− −− − . +0 . − . w ( z = 1) − . +0 . − . − . +0 . − . − . +0 . − . − . ± . w ( z = 0 . − . +0 . − . − . +0 . − . − . +0 . − . − . ± . w ( z = 0 . − . +0 . − . − . +0 . − . − . ± . − . ± . w ( z = 0 . − . ± . − . ± . − . ± . − . ± . w ( z = 0) − . ± . − . ± . − . +0 . − . − . ± . σ . ± .
055 0 . ± .
057 0 . ± .
024 0 . ± . m . ± .
018 0 . ± .
017 0 . ± .
016 0 . ± . H . ± . . ± . . ± . . ± . z reion . ± . . ± . . ± . . ± . t . ± .
14 13 . ± .
15 13 . ± .
13 13 . ± . from q = Ow . If we now define ˜ W so that ˜ W T ˜ W = F then, as emphasized by [43], there are infinitely manychoices for the matrix ˜ W ; following [13], we write theweight transformation matrix as ˜ W = O T Λ O wherethe rows are summed such that the weights from eachband add up to unity, and we apply this transfor-mation matrix to obtain our uncorrelated estimatesof dark energy parameters. Our first analysis consid-ers constraints from “geometric” data: CMB, Type IaSN luminosity distances, and BAO data. We subse-quently include datasets that probe the growth of cos-mic structures, incorporating weak lensing, as well asintegrated Sachs-Wolfe measurements through cross-correlations between CMB and galaxy survey data.We include the latter datasets separately, since ourunderstanding of the cosmic clustering in dark energymodels still suffers from several limitations. TheseLSS uncertainties are mainly related to our poor un-derstanding of both the bias between galaxies andmatter fluctuations (with a possible scale dependenceof the bias itself, see [38–40]) and non-linearities atsmall redshifts (see [36, 41]). For the CMB, we usedata and likelihood code from the WMAP team’s 5-year release [36] (both temperature TT and polariza-tion TE; we will refer to this analysis as WMAP5).In this respect, our approach is more extensive thanthat in [8] and other recent studies, since we fully con-sider the CMB dataset instead of simply using the con-straint on the θ parameter from the analysis of [36].This constraint is model dependent (see, e.g., [42]),and changes with dark energy parameterizations.Supernova data come from the Union data set (UNION) produced by the Supernova CosmologyProject [21]; however, to check the consistency ofour results, we also used the recently released Con-stitution dataset (Constitution) [7] which, with 397Type Ia supernovae, is the largest sample to date.We also used the latest SDSS release (DR7) BAOdistance scale [22, 23]: at z = 0 .
275 we have r s ( z d ) /D V (0 . . ± . r s ( z d ) isthe comoving sound horizon at the baryon drag epoch, D V ≡ [(1 + z ) D A cz/H ( z )] , D A ( z ) is the angulardiameter distance and H ( z ) is the Hubble parame-ter) and the ratio of distances D V (0 . /D V (0 .
20) =1 . ± . w CDM cosmologies. Werefer to [29, 30] for a description of both the method-ology and the datasets used. Finally, we use the re-cent value of the Hubble constant from the SHOES(Supernovae and H for the Equation of State) pro-gram, H = 74 . ± . − Mpc − (1 σ ) [32],which updates the value obtained from the HubbleKey Project [31]. We also incorporate baryon densityinformation from Big Bang Nucleosynthesis Ω b h =0 . ± .
002 (1 σ ) [33], as well as a top-hat prior onthe age of the Universe, 10 Gyr < t <
20 Gyr. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 w e i gh t f un c t i on s redshift z FIG. 2: Weight functions for each of the uncorrelated bins,for the case WMAP+UNION+BAO+WL+ISW+LSS.
III. RESULTS
In Table 1 we show the mean values andmarginalized 68% confidence level limits forthe cosmological parameters considered in thisanalysis for the WMAP+SNe(UNION)+BAOand WMAP+SNe(Constitution)+BAO datasets.We also consider a “global” dataset:WMAP+SNe+BAO+CFHTLS+CMB+WL+ISW+LSS. The w i ( z ) ( i = 1 , , ..
5) entries refer to theuncorrelated values of the dark energy equation ofstate parameters. All values are compatible with acosmological constant ( w = −
1) at the 2 σ level. Aswe can see from Table 1 and from Figure 1, there isno discrepancy between the Union and Constitutiondatasets; moreover, the addition of cosmologicalprobes of cosmic clustering noticeably reduces theuncertainty in the determination of the dark energyparameters, especially at high redshifts. -2.2-2-1.8-1.6-1.4-1.2-1-0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 w ( z ) redshift z FIG. 3: Uncorrelated constraints on the dark energy equa-tion of state parameters, for mock datasets from Planckand JDEM; error bars are at 2 σ . To reinforce our conclusions, we also created severalmock datasets for upcoming and future SN, BAO, andCMB experiments. The quality of future datasets al-lows us to constrain the dark energy evolution beyondredshift z = 1. We thus consider an additional bin at z = 1 .
7, with a similar constraint: w ( z > .
7) = w ( z = 1 . . < z < .
7, as expected from JDEM orsimilar future surveys [44]. The error in the distancemodulus for each SN is given by the intrinsic error, σ int = 0 . D V ), respectively, 4 BAO constraintsat z = [0 . , . , . , .
2] with corresponding fidu-cial survey precisions (in D V ) of [1 . , . , . , . . , . , . , . , . , . , . z = 1 . z = 1 .
65 in steps of 0.1 [47].We simulate Planck data using a fiducial ΛCDMmodel, with the best fit parameters from WMAP5,and noise properties consistent with a combination ofthe Planck 100–143–217 GHz channels of the HFI [48],and fitting for temperature and polarization using thefull-sky likelihood function given in [49]. In addition,we use the same priors on the Hubble parameter andon the baryon density as considered above. As canbe seen from Table 1 and Figure 3, future data willreduce the uncertainties in w i by a factor of at least2, with the relative uncertainty below 10% in all butthe last bin (at z = 1 . IV. CONCLUSIONS
One of the main tasks for present and future darkenergy surveys is to determine whether or not thedark energy density is evolving with time. We haveperformed a global analysis of the latest cosmologi-cal datasets, and have constrained the dark energyequation of state using a very flexible and almostmodel independent parameterization. We determinethe equation of state w ( z ) in five independent redshiftbins, incorporating the effects of dark energy pertur-bations. We find no evidence for a temporal evolu-tion of dark energy— the data is completely consistentwith a cosmological constant . This agrees with mostprevious results, but significantly improves the overallconstraints [13, 14, 19, 20].Bayesian evidence models strongly suggests that thedark energy is a cosmological constant, given that thecosmological constant remains a very good fit to thedata as the number of dark energy parameters in-creases (see e.g. [50] and references therein). We showthat future experiments, such as Planck or JDEM,will be able to reduce the uncertainty on w ( z ) to lessthan 10% in multiple redshift bins, thereby mappingany temporal evolution of dark energy with high pre-cision. With this data it will be possible to measurethe temporal derivative of the equation of state pa-rameters, dw/dz , useful in discriminating between twobroad classes of “thawing” and “freezing” models [5]. Note : As we were completing this paper we becameaware of the work reported in [51], which considersa similar analysis of cosmological data to constrain w ( z ). While those authors find weak evidence for evo- lution of the EOS, we find no such evidence. The twoanalyses differ in the way w ( z ) is interpolated (we usea spline, while they employ a tanh function), as wellas different calculations of the effects of DE perturba-tions. Furthermore, we analyze different datasets; inthis paper we have utilized both the latest BAO mea-surements [22, 23], and the latest value of the Hubbleconstant from the SHOES program [32].PS acknowledges Alexandre Amblard for useful dis-cussions and Shirley Ho for help with the ISW like-lihood code. This research was funded by NSF CA-REER AST-0605427 and by LANL IGPP-08-505. [1] A. Albrecht et al. , [arXiv:astro-astro-ph/0609591] .[2] A. G. Riess et al. [Supernova Search TeamCollaboration], Astron. J. , 1009 (1998), [arXiv:astro-ph/9805201] .[3] S. Perlmutter et al. [Supernova Cosmology ProjectCollaboration], Astrophys. J. , 565 (1999), [arXiv:astro-ph/9812133] .[4] J. P. Uzan, Gen. Rel. Grav. , 307 (2007), [arXiv:astro-ph/0605313] .[5] R. R. Caldwell and M. Kamionkowski, [arXiv:astro-ph/0903.0866 ] .[6] E. Copeland, M. Sami and S. Tsujikawa,Int. J. Mod. Phys.D :1753-1936,2006, [arXiv:astro-ph/0603057] .[7] M. Hicken et al. , [arXiv:astro-ph/0901.4787] .[8] D. Rubin et al. , Astrophys. J. (2009) 391, [arXiv:astro-ph/0807.1108] ; W. L. Freed-man et al. , [arXiv:astro-ph/0907.4524] ;M. Hicken et al. , Astrophys. J. (2009) 1097, [arXiv:astro-ph/0901.4804] .[9] A. .R. Cooray and D. Huterer, Astrophys. J. , L95(1999), [arXiv:astro-ph/9901097] .[10] M. Chevallier and D. Polarski, Int. J. Mod. Phys. D, , 213 (2001), [arXiv:gr-qc/0009008] .[11] E. V. Linder, Phys. Rev. Lett. , 091301 (2003), [arXiv:astro-ph/0208512] .[12] D. Huterer and G. Starkman, Phys. Rev. Lett. ,031301 (2003), [arXiv:astro-ph/0207517] .[13] D. Huterer and A. .R. Cooray, Phys. Rev. D :023506, (2005), [arXiv:astro-ph/0404062] .[14] D. Sarkar et al. , Phys. Rev. Lett. , 241302 (2008), [arXiv:astro-ph/0709.1150] [15] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. , 473 (2000) [arXiv:astro-ph/9911177] .[16] W. Hu and I. Sawicki, Phys. Rev. D :104043,2007, [arXiv:astro-ph/0708.1190] .[17] W. Hu, Phys. Rev. D :103524,2008, [arXiv:astro-ph/0801.2433] .[18] W. Fang, W. Hu and A. Lewis, Phys. Rev. D :087303 (2008), [arXiv:astro-ph/0808.3125] .[19] G. B. Zhao, D. Huterer, X. Zhang, Phys. Rev. D :121302 (2008), [arXiv:astro-ph/0712.2277] .[20] S. Sullivan, A. Cooray and D. E. Holz, JCAP ,004 (2007) [arXiv:astro-ph/0706.3730] .[21] M. Kowalski et al. , [arXiv:astro-ph/0804.4142] .[22] B. A. Reid et al. ,[SDSS collaboration], submitted toMNRAS, [arXiv:astro-ph/0907.1659] . [23] W. J. Percival et al. [SDSS collaboration], submittedto MNRAS, [arXiv:astro-ph/0907.1660] .[24] R. Massey et al. , [arXiv:astro-ph/0701480] .[25] J. Lesgourgues et al. JCAP , 008 (2007), arXiv:astro-ph/0705.0533 .[26] M. Kilbinger, private communication (2008).[27] L. Fu et al. , [arXiv:astro-ph/0712.0884] .[28] see http : W L.html .[29] S. Ho et al. , Phys. Rev. D :043519, (2008), [arXiv:0801.0642v2] .[30] C. Hirata et al. , Phys. Rev. D :043520, (2008), [arXiv:0801.0644v2] .[31] W. L. Freedmann et al. , Astrophys. J. : 47, 2001, [arXiv:astro-ph/0202006v1] .[32] A. G. Reiss et al. , Astrophys. J., , 539 (2009). [arXiv:astro-ph/09050695] .[33] S. Burles, K. M. Nollett and M. S. Turner, Astro-phys. J. : L1, 2001, [arXiv:astro-ph/0010171] .[34] A. Lewis and S. Bridle, Phys. Rev. D , 103511(2002), [arXiv:astro-ph/0205436] , Available at cosmologist.info .[35] A. Gelman and D. B. Rubin, “Inference from itera-tive simulation using multiple sequencies”. StatisticalScience, 7, 457-472 (1992).[36] J. Dunkley et al . [WMAP collaboration], [arXiv:astro-ph/0803.0586] , in press on APJS.[37] E. Komatsu et al . [WMAP collaboration], [arXiv:astro-ph/0803.0547] , in press on APJS.[38] P. McDonald, Phys. Rev. D , (2006)103512; Erratum-ibid. D (2006) 129901, . [arXiv:astro-ph/0609413] .[39] R. E. Smith, R. Scoccimarro andR. K. Sheth, Phys. Rev. D , (2007) 063512, [arXiv:astro-ph/0609547] .[40] S. Joudaki, A. Cooray and D. E. Holz, Phys. Rev. D , 023003 (2009), [arXiv:astro-ph/0904.4697] .[41] J. Hamann, et al , JCAP 0807:017,2008, [arXiv:astro-ph/0804.1789] .[42] P. S. Corasaniti and A. Melchiorri, Phys. Rev. D (2008) 103507, [arXiv:astro-ph/0711.4119] .[43] A. J. S. Hamilton and M. Tegmark, MNRAS, , 285-294 (2000), [arXiv:astro-ph/9905192] ;M. Tegmark, Phys. Rev. D, , 5895-5897,(1997), [arXiv:astro-ph/9611174] .[44] A. G. Kim et al. et al. [arXiv:astro-ph/0710.4143] . [46] H-J. Seo and D. J. Eisenstein, Astrophys. J. , 720(2003), [arXiv:astro-ph/0307460] .[47] A. Riess (private communication).[48] Planck collaboration, [arXiv:astro-ph/0604069] .[49] A. Lewis, Phys. Rev. D : 083008, (2005), [arXiv:astro-ph/0502469] . [50] P. Serra, A. Heavens and A. Melchiorri, Mon.Not. Roy. Astron. Soc. , 169 (2007), [arXiv:astro-ph/0701338] .[51] G. B. Zhao, X. Zhang, [arXiv:astro-ph/0908.1568][arXiv:astro-ph/0908.1568]