PHYSICAL REVIEW D Cosmological parameter analysis ...

PHYSICAL REVIEW D 71, 103515 (2005)

Cosmological parameter analysis including SDSS Ly forest and galaxy bias: Constraints on theprimordial spectrum of fluctuations, neutrino mass, and dark energy

Uros Seljak,1,2 Alexey Makarov,1 Patrick McDonald,1 Scott F. Anderson,3 Neta A. Bahcall,4 J. Brinkmann,5 Scott Burles,6

Renyue Cen,4 Mamoru Doi,7 James E. Gunn,4 Zeljko Ivezic,4,3 Stephen Kent,8 Jon Loveday,9 Robert H. Lupton,4

Jeffrey A. Munn,10 Robert C. Nichol,11 Jeremiah P. Ostriker,4,12 David J. Schlegel,4 Donald P. Schneider,13

Max Tegmark,14,6 Daniel E. Vanden Berk,13 David H. Weinberg,15 and Donald G. York16

1Physics Department, Princeton University, Princeton, New Jersey 08544, USA2International Center for Theoretical Physics, Trieste, Italy;

3 Astronomy Department, University of Washington, Seattle, Washington 98195, USA4Princeton University Observatory, Princeton, New Jersey

5Apache Point Observatory, 2001 Apache Point Rd, Sunspot, New Mexico 88349-0059, USA6Dept. of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

7Institute of Astronomy, School of Science, University of Tokyo, Japan8Fermi National Accelerator Laboratory, P.O. Box 500, Batavia,Illinois 60510, USA

9University of Sussex, Sussex, United Kingdom10U.S. Naval Observatory,Flagstaff Station, Flagstaff, Arizona 86002-1149, USA

11Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, United Kingdom12Institute of Astronomy, Cambrdige University, Cambridge, United Kingdom

13Dept. of Astronomy and Astrophysics, Pennsylvania State University, University Park, Pennsylvania 16802, USA14Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

15Department of Astronomy, Ohio State University, Columbus, Ohio 43210, USA16Department of Astronomy & Astrophysics, University of Chicago, Chicago, Illinois 60637, USA

(Received 28 July 2004; published 20 May 2005)

1550-7998=20

We combine the constraints from the recent Ly forest analysis of the Sloan Digital Sky Survey (SDSS)and the SDSS galaxy bias analysis with previous constraints from SDSS galaxy clustering, the latestsupernovae, and 1st year WMAP cosmic microwave background anisotropies. We find significantimprovements on all of the cosmological parameters compared to previous constraints, which highlightsthe importance of combining Ly forest constraints with other probes. Combining WMAP and the Lyforest we find for the primordial slope ns 0:98 0:02. We see no evidence of running, dn=d lnk 0:003 0:010, a factor of 3 improvement over previous constraints. We also find no evidence of tensors,r < 0:36 (95% c.l.). Inflationary models predict the absence of running and many among them satisfythese constraints, particularly negative curvature models such as those based on spontaneous symmetrybreaking. A positive correlation between tensors and primordial slope disfavors chaotic inflation-typemodels with steep slopes: while the V / 2 model is within the 2-sigma contour, V / 4 is outside the 3-sigma contour. For the amplitude we find 8 0:90 0:03 from the Ly forest and WMAP alone. Wefind no evidence of neutrino mass: for the case of 3 massive neutrino families with an inflationary prior,Pm < 0:42 eV and the mass of lightest neutrino is m1 < 0:13 eV at 95% c.l. For the 3 massless 1

massive neutrino case we find m < 0:79 eV for the massive neutrino, excluding at 95% c.l. all neutrinomass solutions compatible with the LSND results. We explore dark energy constraints in models with afairly general time dependence of dark energy equation of state, finding 0:72 0:02, wz 0:3 0:980:10

0:12, the latter changing to wz 0:3 0:920:090:10 if tensors are allowed. We find no evidence

for variation of the equation of state with redshift, wz 1 1:030:210:28. These results rely on the

current understanding of the Ly forest and other probes, which need to be explored further bothobservationally and theoretically, but extensive tests reveal no evidence of inconsistency among differentdata sets used here.

DOI: 10.1103/PhysRevD.71.103515 PACS numbers: 98.80.Es

I. INTRODUCTION

Many different cosmological observations over the pastdecade have helped build what is now called the standardcosmological model. These observations suggest that theuniverse is spatially flat, contains baryons, dark matter anddark energy. The primordial spectrum of fluctuations isapproximately scale invariant and initial fluctuations are

05=71(10)=103515(20)$23.00 103515

Gaussian and adiabatic. This standard cosmological modelcan be described in terms of only a few parameters, whichexplain a large number of observations, such as the cosmicmicrowave background (CMB), galaxy clustering, super-nova data, Hubble parameter determinations, and weaklensing. The latest results come from WilkinsonMicrowave Anisotropy Probe (WMAP) CMB measure-ments [1–3], Sloan Digital Sky Survey (SDSS) and Two

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UROS SELJAK et al. PHYSICAL REVIEW D 71, 103515 (2005)

degree Field (2dF) galaxy clustering analyses [4–6], andfrom the latest Supernovae type Ia (SNIa) data [7,8].

While the standard model is observationally well justi-fied, many theoretical models predict that there should beobservable deviations from it. Perhaps the best motivatedamong these are the predictions of how the universe wasseeded by initial fluctuations. The standard paradigm isinflation, which predicts that the fluctuations should bealmost, but not exactly, scale invariant [9]. A typical de-viation for the slope of the primordial perturbations ispredicted to be of order of a few parts in a hundred awayfrom its scale invariant value ns 1 and could be of eithersign. This should be observable with high precision cos-mological observations. Despite tremendous progress overthe past couple of years the current constraints do not yetdistinguish between different inflationary models [10,11].Alternative models also predict deviations from scale in-variance similar to inflation [12]. Another prediction ofthese models is that the rate of change of slope with scale israther small, s dns=d lnk ns 12 103, whichshould not be observable in the near future. A third pre-diction that can distinguish among the different models isthe amount of tensor perturbations they predict. Somemodels predict no detectable tensor contribution [9,13],while other models predict a tensor contribution to thelarge-scale CMB anisotropies comparable to that fromscalars. It is clear that determining the shape and amplitudeof the scalar and tensor primordial power spectra will beone of the key tests of various models of structureformation.

Current observational constraints on the primordialpower spectrum are mostly limited to scales larger than10h1 Mpc. There are various reasons for this: CMBfluctuations are damped on small scales and their detectionwould require high resolution, low noise detectors, whichare only now being built. Even with sufficient signal-to-noise and angular resolution there may be secondary an-isotropies that may contaminate the signal from primaryanisotropies. On small scales, matter undergoes stronglynonlinear evolution, which erases the initial spectrum offluctuations and prevents galaxy clustering and weak lens-ing surveys from extracting this information. On the otherend, the largest observable scale is the horizon scale seenby CMB fluctuations. The small number of availablemodes on the sky prevents one from accurately determin-ing the primordial spectrum on these scales from the CMB.The largest scales probed by galaxy clustering are evensmaller. As a result, the primordial power spectrum iscurrently probed over a relatively narrow range of scalesand the shape of the primordial power spectrum cannot beaccurately determined.

To improve these constraints one should determine thefluctuation amplitude on smaller scales. Nonlinear evolu-tion prevents one from obtaining useful information at z 0, so one must look for probes at higher redshift. Of the

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current cosmological probes, the Ly forest—the absorp-tion observed in quasar spectra by neutral hydrogen in theintergalactic medium (hereafter IGM)—has the potentialto give the most precise information on small scales [14]. Itprobes fluctuations down to megaparsec scales at redshiftsbetween 2– 4, so nonlinear evolution, while not negligible,has not erased all of the primordial information.

In this paper we combine CMB/LSS constraints with thenew analysis of the Ly forest from SDSS data [15]. TheSloan Digital Sky Survey [16] uses a drift-scanning imag-ing camera [17] and a 640 fiber, double spectrograph on adedicated 2.5 m telescope. The SDSS data sample in datarelease two [18] consists of more than 3000 QSO spectrawith z > 2:2, nearly 2 orders of magnitude larger thanpreviously available [19–21]. This large data set allowsone to determine the amplitude of the flux power spectrumto better than 1%. Theoretical analysis of this flux powerspectrum shows that at the pivot point k 0:009 s=km invelocity coordinates, which is close to k 1h=Mpc incomoving coordinates for standard cosmological parame-ters, the power spectrum amplitude is determined to about15% and the slope to about 0.05, with the error budgetdominated by uncertainties in theoretical modelling[22,23]. This is an accuracy comparable to that achievedby WMAP. More importantly, it is at a much smaller scale,so combining the two leads to a significant improvement inthe constraints on primordial power spectrum shape overwhat can be achieved from each data set individually.

A second theoretical prediction where the basic cosmo-logical model is expected to require modifications is thatneutrinos have mass. Atmospheric mixing and solar neu-trino results suggest that the total minimum neutrino massis about 0.06 eV [24–26]. These observations are onlysensitive to relative neutrino mass differences and not tothe absolute neutrino mass itself. Cosmology on the otherhand can weigh neutrinos directly. Massive neutrinos slowdown the growth of structure on small scales and modifythe amplitude and shape of the matter power spectrum.They also modify the CMB power spectrum. If one mea-sures both the CMB and matter power spectra with highprecision across a wide range of redshifts and scales thenone can determine the neutrino mass with high accuracy[27]. The question of neutrino mass is also interesting inlight of recent Los Alamos Liquid Scintillator NeutrinoDetector (LSND) experimental results, which, if taken at aface value, suggest m > 0:9 eV [28–30], which should beobservable by cosmological neutrino weighing.

A third theoretical prediction of departures from thestandard model, and one whose consequences would beparticularly far reaching, is that dark energy is not simply acosmological constant introduced already by Einstein, butsomething more complicated and dynamical in nature. Inthe case where dark energy is a scalar field one wouldexpect that it has a kinetic energy term in addition to thepotential term, which modifies its equation of state. This is

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COSMOLOGICAL PARAMETER ANALYSIS INCLUDING. . . PHYSICAL REVIEW D 71, 103515 (2005)

expected to evolve with time, but theoretical predictionsare rather uncertain and are suggestive at best. A change inequation of state changes both the rate of growth of struc-ture and the angular size of the acoustic horizon in theCMB. As a result these changes can be observed boththrough the CMB and by comparing the growth of structureat different redshifts.

Many different methods have been discussed in theliterature on how to improve the current constraints frommethods such as supernovae type Ia (SNIa), CMB, weaklensing, and cluster abundances. One method to constrainthe nature of dark energy that has not attracted muchattention, yet has the potential to produce results on arelatively short time scale, is comparing measurements ofamplitude of fluctuations at high redshift from the Lyforest and CMB to that at low redshift from galaxy cluster-ing. Dark energy affects the rate of growth of structure,especially for z < 1 where dark energy is dynamicallyimportant. In this paper we combine WMAP and SDSS-Ly forest measurements at high redshifts, where darkenergy is expected to be negligible, with the amplitudedetermination at z 0:1 from the SDSS galaxy bias analy-sis [31]. In general, galaxy clustering is believed to beproportional to matter clustering on large scales up to aconstant of proportionality. This constant, the so calledbias, is a free parameter that cannot be determined fromthe clustering analysis itself. There are many differentmethods for how to determine the bias and thus the ampli-tude of matter fluctuations such as redshift space distor-tions [4,32], the bispectrum [33], or weak lensing [34,35],but the current constraints are weak. A recent analysis ofthe luminosity dependence of galaxy clustering [4], com-bined with a determination of the halo mass distribution forthese galaxies, provides a new constraint on the bias andamplitude of fluctuations in SDSS data [31].

One difference of the current paper in comparison withprevious analyses of this type is that we present 68:32%,95:5% and 99:86% confidence intervals (we denote thesethe 1, 2, and 3- intervals, but note that they do not dependon the assumption of Gaussianity in the error distribution)on all the parameters (or 95% and 99:9% confidence levelupper limits in the case of no detections). Sometimes the3- intervals can be significantly different from 3 times thecorresponding 1- intervals. This can happen if there aredegeneracies in the data that appear to be broken at 1-,but that the 2 or 3 contours allow. In this case the 3-constraints are weaker than the corresponding 1- inter-vals would suggest. The opposite can happen as well,especially if there is a natural boundary that the parametercannot cross (such as a parameter being positive definite).More generally, presenting 1- contours alone is not verymeaningful, since whatever is within 1- is essentially agood fit to the data. One can argue that the goal of obser-vations is to exclude regions of parameter space and this ismuch better represented by reporting 2 and 3- contoursthan the best fit value and its 1- range.

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Another issue that we address in detail is the robustnessof the constraints against the number of parameters one isexploring. Sometimes the constraints change significantlyif new parameters are added to the mix because these newparameters are degenerate with parameters one is inter-ested in. However, often the quality of the fit is not im-proved at all and moreover these new parameters may notbe well motivated from the perspective of fundamentaltheories or other considerations. In this case one is entitledto adopt an Occam’s razor argument against the introduc-tion of these parameters in the estimation. To some extentthis is always a subjective procedure, since what is naturalfor one person may not be for someone else. It has alsobeen argued that one should pay a penalty for each newparameter that is introduced which does not improve thequality of the fit [36]. However, this procedure is alsopoorly defined and there is no unique choice for the pen-alty. In this paper we explore both the solutions with theminimum number of parameters as well as with severaladditional parameters. We believe that there is merit to theapproach which parametrizes the constraints with as fewparameters as possible, so our main results are given forthis case. However, one also wants to know how robust andmodel independent are the constraints, which we exploreby adding several additional parameters to the analysis.

The outline of this paper is as follows. We first presentthe method, then our basic results in several tables and thendiscuss them in detail. We focus particularly on the ques-tion of how have the new results improved upon theprevious constraints and how robust are the conclusionsupon removing one or more of the data ingredients. Thelatter is particularly interesting in light of possible system-atic effects that may be present both in the new analyses ofLy forest and bias as well as in previous analyses ofWMAP, SDSS galaxy clustering, and SNIa.

II. METHOD

We combine the constraints from the SDSS Ly forest[15] with the SDSS galaxy clustering analysis [4], SDSS-bias analysis [31], and CMB power spectrum observationsfrom WMAP [1–3]. We verified that including CBI, VSA,and ACBAR [37–39] makes very little difference in thefinal results and we do not include them in the currentanalysis. Similarly, we verified that including the latest 2dFpower spectrum analysis [5] in addition to SDSS does notmake much difference, so we do not include those con-straints either. We could have used 2dF constraints insteadof SDSS, but we chose not to because for 2dF the biasconstraints are somewhat weaker [33] and we would like tohave an independent verification of results that use the 2dFbias [40]. We will thus refer to CMB constraints as WMAP,to LSS/galaxy clustering constraints as SDSS-gal, toSDSS-bias constraints as SDSS-bias and to SDSS Lyforest constraints as SDSS-ly. We have added earlier

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Ly forest constraints in a weak form [20,41], which havea small, but not negligible effect. We do not include morerecent Ly forest constraints [19,21] since there are signsof systematic discrepancy and/or underestimation of errorswhen compared to SDSS Ly forest data [15]. To this weadd the latest supernova constraints as given in [7]. We donot use this full combination in all calculations, since wewant to emphasize what the new constraints bring to themix and we want to explore the sensitivity of the con-straints to individual data sets. For example, for the inves-tigation of the shape of the primordial power spectrum weperform the analysis using WMAP SDSS-ly alone andshow that this combination in itself suffices to constrain therunning by a factor of 3 better than combining everythingelse together. We also perform several analyses by drop-ping one of the constraints and explore the robustness ofthe conclusions. For example, we explore the constraintson the dark energy equation of state with and without SNIaand with and without SDSS-bias and SDSS-ly.

Our implementation of the Monte Carlo Markov Chain(MCMC) method[42] uses CMBFAST [43] version 4.5.11,outputting both CMB spectra and the corresponding matterpower spectra Pk. We evolve all the matter power spectrato a high k using CMBFAST and we do not employ anyanalytical approximations. We output the transfer func-tions at the redshifts of interest, between 2–4 for SDSS-Ly forest and 0.1 for SDSS-gal. Note that for massiveneutrinos the high precision (HP) option must be used toachieve sufficient accuracy in the transfer function.

A typical run is based on 16–24 independent chains,contains 50 000-200 000 chain elements and requires sev-eral days of running on a computer cluster in a serial modeof CMBFAST. The acceptance rate was of order 30–50%,correlation length 10–30 and the effective chain length oforder 3000-20 000 (see [11] for definitions of these terms).In terms of Gelman and Rubin R-statistics [44] we find thechains are sufficiently converged and mixed, with R <1:05, significantly more conservative than the recom-mended value R < 1:2.

Our most general cosmological parameter space is

p

;!b;!m;

Xm;;w;2

R; ns; s; r; (1)

where is the optical depth, !b bh2, where b isbaryon density in units of the critical density and h is theHubble constant in units of 100 km/s/Mpc, !m mh

2

where m is matter density in units of the critical density,Pm is the sum of massive neutrino masses (assuming

either 3 degenerate neutrino families or 1 massive neutrinofamily in addition to 3 massless), is the dark energydensity today and w its equation of state (which is ingeneral time dependent). Our pivot point for the primordial

1available at cmbfast.org

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power spectrum parametrization is at kpivot 0:05= Mpcand we expand the primordial power spectrum at that point,defining the amplitude of curvature perturbations 2

R,slope ns, and its running s dns=d lnk. The choice ofthe pivot point is somewhat arbitrary, but is meant torepresent the scale somewhere in the middle of the obser-vational range. In this case the largest scales are probed bythe CMB (k 103=Mpc) and the smallest scales areprobed by the Ly forest (k 1=Mpc). In addition, thisscale has been (arbitrarily) chosen as a pivot point inCMBFAST and has been used by previous analyses, whichfacilitates the comparison. Note that there is no Hubbleparameter h in the definition of the pivot point: if CMBdata are used there is no advantage in defining the scale bytaking out the Hubble constant, unlike the case of galaxyclustering and Ly forest.

We parametrize tensors in terms of their amplitude2h, and define the ratio relative to scalars as r T=S

2h=

2R. This is also defined at the pivot point k

0:05=Mpc, just as for the scalar amplitude, slope andrunning. We fix the tensor slope nT using r 8nT . Wedo not allow for nonflat models, since curvature is alreadytightly constrained by CMB and other observations [40]. Inaddition, we will be testing particular classes of models,such as inflation, which predict K 0. For the moregeneral models, such as those with freedom in the darkenergy equation of state, relaxing this assumption can leadto a significant expansion of errors [11]. We are thereforetesting a particular class of inflation inspired models withK 0 and not presenting model independent constraintson the equation of state. Note that this assumption isimplicit in most of the constraints published to date, in-cluding those from the SNIa teams, which often assume aCMB prior on m [7]. This prior is affected by the choiceof parameter space one is working in and a self-consistenttreatment is required. CMB constraints on m using ananalysis where the equation of state or curvature are notvaried need not equal those where these are varied. Wefollow the WMAP team in imposing a < 0:3 constraint.Upcoming polarization data from WMAP will allow averification of this prior.

From this basic set of parameters we can obtainconstraints on several other parameters, such as thebaryon and matter densities b and m, Hubbleparameter h H0=100 km=s=Mpc and amplitude offluctuations 8. Since we do not allow for curvaturewe have 1 m and we use m in all tables. Infact, our primary parameter is the angular scale of theacoustic horizon, which is tightly constrained by theCMB. Similarly, although we use 2

R as the primaryparameter in the MCMC we present the amplitude interms of the more familiar 8. In addition to the cosmo-logical parameters above we also keep track ofseveral parameters related to the specific tracers, describedbelow.

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A. CMB analysis

For the CMB we use the 1st year likelihood routineprovided by WMAP [2,45], but replace l < 12 analysiswith the corresponding full likelihood analysis as givenin [46]. This is important for the running of the spectralindex constraints. As shown in [46], exact analysis in-creases errors on low multipoles compared to the originalWMAP analysis, which leads to less stringent constraintson running: it is typically increased by 1 standard deviationaway from its negative value toward zero, i.e. toward the norunning solution. We find a similar effect in our analysiswhen combined with Ly forest analysis.

B. Galaxy clustering

We use the SDSS galaxy clustering constraints on thegalaxy power spectrum for k < 0:2h=Mpc [4]. We use alinear to nonlinear mapping of the matter power spectrumusing expressions given in [47]. The main nuisance pa-rameter is the linear bias of L galaxies, b, which relatesthe galaxy power spectrum to that of dark matter, P g

k b2P dm

k, where g and dm are the galaxy and darkmatter density fluctuations, respectively, and Pk is theirpower spectrum.

The luminosity dependence of galaxy bias providesadditional cosmological constraints [31]. Observationsshow that bias is relatively constant for galaxies fainterthan L and is rapidly increasing for brighter galaxies [4].Theoretical and simulation predictions of halo bias [48–50] show a similar dependence of bias on halo mass, withthe transition occurring at the so called nonlinear mass,corresponding to the mass within a sphere where the rmsfluctuation level is 1.68. The value of the nonlinear massdepends on cosmological parameters such as the amplitudeand shape of the power spectrum, as well as the matterdensity. A measurement of the halo mass distribution for agiven luminosity class is possible using a weak lensinganalysis around these galaxies, which traces the dark mat-ter distribution directly. This allows a theoretical determi-nation of galaxy bias for a given cosmological model. Onlythose models for which the theoretical predictions agreewith the observations in all luminosity bins are acceptable.This places strong constraints on cosmological models.This constraint is not directly determining the amplitudeof fluctuations and bias, because both the theoretical pre-dictions and observationally inferred values of bias changein a similar way. However, the data suggest that for L,where statistical errors are smallest, the predicted biasvalue is lower than the observed one for standard cosmol-ogy m 0:3 and ns 1. Lowering m or ns reduces thenonlinear mass and increases theoretically predicted bias,bringing it into a better agreement with observations.Additional constraints come from the dependence of biason luminosity, which is constraining the amplitude offluctuations. The method is fairly robust in the sense thateven appreciable changes in halo mass determination do

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not change the bias predictions significantly. The analysisis performed using the bias likelihood code as given in[31].

C. Ly forest

Reference [22] describes in detail our method for ob-taining the Ly forest contribution to !2 for any cosmo-logical model. Rather than attempting to invert PFk toobtain the matter power spectrum, we compare the theo-retical PFk directly to the observed one. In observatio-nally favored models, the Universe is effectively Einstein-de Sitter at z > 2, so the cosmology information relevant tothe Ly forest is completely contained within PLk mea-sured in velocity units. For any given model in the MCMCchain we compute the matter power spectrum in velocityunits and interpolate from a grid of cosmological simula-tions covering a broad range of values to obtain predictionsof the flux power spectrum. We compare these to themeasured SDSS flux power spectrum to derive the like-lihood of the model given the data.

The Ly forest contains several nuisance parameterswhich we are not interested in for the cosmological analy-sis, although some of them are of interest for studies ofIGM evolution. In the standard picture of the Ly forestthe gas in the IGM is in ionization equilibrium. The rate ofionization by the UV background balances the rate ofrecombination of protons and electrons. The recombina-tion rate depends on the temperature of the gas, which is afunction of the gas density. The temperature-density rela-tion can be parametrized by an amplitude, T0, and a slope# 1 d lnT=d ln$. The uncertainties in the intensity ofthe UV background, the mean baryon density, and otherparameters that set the normalization of the relation be-tween optical depth and density can be combined into oneparameter: the mean transmitted flux, Fz. The parametersof the gas model, T0, # 1, and F, must be marginalizedover when computing constraints on cosmology. They are afunction of redshift. Our model for the redshift evolution of F, T0, and # is explained in detail in [22]. We also add

additional nuisance parameters such as the filtering lengthkF [51] and parameters that characterize various physicaleffects [23], described in more detail below. This gives riseto a number of additional nuisance parameters.

Each time there are nuisance parameters that one is notinterested in there are two approaches that one can take.One can either keep these parameters as independent andadd them to the MCMC chain or one can marginalize overthem for each set of cosmological models. The advantageof the first approach is that at the end one can extract thebest fit values of these parameters and their correlationswith other cosmological or nuisance parameters, in caseone is interested in these. The disadvantage is that increas-ing the number of dimensions of the MCMC decreases theacceptance rate of the chains, increasing the computationaltime. Another disadvantage is that in many dimensions the

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MCMC approach often does not find the global minimum,which is of interest if one wants to assess the improvementin !2 with the addition of new parameters.

The second approach is marginalization over the nui-sance parameters. We implement it by maximizing thelikelihood (minimizing !2) over the phase space of theseparameters for each cosmological model. The computa-tional efficiency of this approach depends on the problemat hand and numerical implementation. In our case we findthat the computational time increase is comparable to thepenalty paid in the first approach due to the loweredefficiency of the MCMC sampler, so there is no numericaladvantage in using one over the other. We decided to usethe latter approach because we would like to be able tointerpret the minimum !2 values between different chains:we have found that the marginalization approach gives aminimum !2 within unity of the global minimum for thechain lengths we adopt, while the approach of working in40-dimensional parameter space gave minimum !2 valuesin our MCMC chains that were often significantly higherthan the actual global minimum. This is expected since thelikelihood function is shallow around the maximum andthe large phase space volume of 20 additional dimensionswins over the penalty induced by exp!2 for small!2. We have verified that both approaches lead to thesame probability distributions of cosmological parameters,so this choice is not important for the MCMC distributionsthemselves.

More details of the Ly forest likelihood module aredescribed in [22]. The simulations cover the plausiblyallowed range of F, T0, # 1, kF, 2keff, neff , anddneff=d lnk. Simulations with several box and grid sizesare used to guarantee convergence, which is verified bydetailed convergence studies on smaller box simulations.The grid is based on hydroparticle mesh simulations [51],but these are explicitly calibrated using fully hydrody-namic simulations [51]. The simulation results are com-bined in an interpolation code that produces PFk for any

TABLE I. Constraints on basic 6 parameters and tensors. Mediacombining WMAP, SDSS galaxies (gal), SDSS bias (bias), SDSS Lanalysis. In each case we list individual data sets. Note that WMAP i95% upper limit and (in brackets) 99:9% upper limit. All of the vatheoretical priors and different data sets. The parameters for 6 param

6-p 6-pWMAP gal WMAP gally

102!b 2:380:140:270:390:120:230:33 2:310:090:170:26

0:080:170:24 2:33m 0:2940:0410:0890:143

0:0340:0610:082 0:2990:0370:0820:1330:0320:0610:084 0:2810

0

ns 0:9940:0440:0770:1010:0350:0600:080 0:9710:0230:0480:070

0:0190:0380:055 0:98000

0:1760:0780:1170:1240:0710:1240:161 0:1330:0520:1040:148

0:0450:0870:126 0:16000

8 0:9510:0900:1730:2240:0790:1420:261 0:8900:0340:0650:096

0:0320:0600:089 0:89700

h 0:7060:0370:0680:0970:0340:0650:091 0:6940:0300:0590:092

0:0280:0570:086 0:71000

r 0 0

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relatively smooth (CDM-like) input PLk, F, T0, and #1. We also marginalize over the filtering scale kF, which isrelated to the gas Jeans scale, where pressure balancesgravity, but depends on the full gas temperature historysince reionization rather than just the instantaneous tem-perature T0 [51].

There are several possible systematic effects in the Lyforest that have been investigated in [23]. The most im-portant effect, that from damped systems, can be reliablyremoved using the existing constraints on the abundance ofdamped systems. It leads to an increase in slope by 0.06.We find no evidence of other effects, such as fluctuations inthe UV background or galactic winds. The former effect isconstrained by the expected rapid evolution of the attenu-ation length with redshift, which would cause the effect tobe more significant at high redshift. While current modelsof galactic winds produce no significant effect on the Lyforest flux power spectrum [23], these need to be exploredfurther. The fact that the effective curvature of the matterpower spectrum derived solely from Ly forest analysisagrees with the expected value [22] provides a constrainton any additional contamination. An independent con-straint is provided by the consistency of the matter powerspectrum results as a function of redshift over the range2< z< 4 [22]. Neither of these arguments are conclusiveand we find examples of systematic effects that can escapeone or the other test. Additional analyses, such as correla-tions of the Ly forest with galaxies [52] and quasars[53,54], as well as a bispectrum analysis [55], will beable to test further the current models.

III. RESULTS

The basic results for many different MCMC runs aregiven in Tables I, II, III, and IV. We give results for manydifferent parameter combinations and different experimentcombinations, with the purpose of assessing the robustnessof constraints on both the data and parameter space. Formost of the parameters we quote the median value (50%),

n value, 1, 2 and 3 intervals on cosmological parametersy forest (ly) and SNIa (SN) data as derived from the MCMCs included in all the chains. In the absence of a detection we givelues are obtained from MCMC. The columns compare different

eter models 6-p are ;!b;!m;m 1 ; 8; ns.

6-p 6-pr 6-prall WMAP gally all

0:090:170:260:080:170:25 2:400:120:260:47

0:1050:190:30 2:400:110:230:330:100:190:27

:0230:0460:070:0210:0400:061 0:2780:0360:0760:118

0:0330:0620:094 0:2700:0220:0450:0720:0210:0410:060

:0200:0410:065:0190:0370:051 1:000:0340:0700:124

0:0280:0500:076 1:000:0270:0560:0850:0240:0450:063

:0400:0790:117:0410:0800:120 0:1380:0500:0960:151

0:0450:0850:118 0:1550:0400:0780:1120:0400:0770:114

:0330:0650:097:0310:0580:086 0:9010:0350:0690:107

0:0330:0620:096 0:9040:0350:0690:1060:0310:0590:094

:0210:0440:066:0210:0400:061 0:7190:0360:0760:133

0:0320:0610:091 0:7260:0250:0520:0810:0230:0450:068

0 <0:380:55 <0:360:51

-6

TABLE III. Neutrino mass constraints. Same format as for Table I. All except last column are for the case of 3 degenerate neutrinofamilies. Last column is for 3 massless 1 massive neutrino family.

6-p3 m 6-p3 m 6-p3 m 6-p3 m s r 6-p1 m

WMAP gal ly WMAP gal all all all

102!b 2:410:160:310:460:140:250:37 2:340:080:170:25

0:080:170:25 2:360:090:190:320:090:180:29 2:470:130:260:38

0:120:220:32 2:350:120:250:360:100:190:28

m 0:3520:1310:2410:3340:0800:1200:149 0:3160:0290:0670:124

0:0270:0520:080 0:2840:0250:050:0790:0230:0440:060 0:2770:0250:0510:086

0:0230:0450:064 0:2870:0280:0600:1030:0250:0480:069

ns 1:000:0510:0980:1310:0410:0710:095 0:9780:0230:0510:069

0:0200:0390:055 0:9890:0260:0530:0760:0230:0420:060 1:0200:0330:0660:094

0:0330:0610:082 1:000:0320:0610:0830:0250:0470:067

0:1330:0810:1440:1650:0600:1010:128 0:1530:0550:1070:140

0:0420:0750:101 0:1850:0520:0990:1140:0460:0890:125 0:2060:0590:0880:093

0:0580:1050:143 0:1950:0590:0960:1040:0550:1020:147

8 0:7860:1190:2300:3010:1000:1720:230 0:8730:0350:0660:099

0:0320:0650:093 0:8900:0350:0710:0980:0330:0640:092 0:8820:0320:0690:107

0:0300:0570:087 0:8950:0350:0670:100:0330:0630:094

h 0:6630:0700:1170:1640:0760:1130:146 0:6840:0230:0470:070

0:0220:0470:083 0:7100:0230:0470:0750:0220:0440:067 0:7230:0270:0540:082

0:0250:0470:080 0:7440:0240:0500:0780:0230:0470:072

r 0 0 0 <0:470:63 0102s 0 0 0 0:181:232:463:78

1:242:503:62 0Pm 1.54 (2.26) eV 0.54 (0.86) eV 0.42 (0.67) eV 0.66 (0.93) eV 0.84(1.61) eV

TABLE II. Constraints on running. Same format as for Table I.

6-ps 6-ps 6-ps 6-ps 6-ps rWMAP WMAP gal WMAP ly all WMAP gally

102!b 2:330:160:330:500:160:320:47 2:300:140:290:45

0:140:270:38 2:360:110:220:320:100:190:27 2:330:090:180:28

0:090:170:25 2:420:120:240:390:120:220:31

m 0:2460:0720:1590:2630:0570:1030:140 0:2690:0410:0910:156

0:0330:0620:095 0:2570:0550:1050:1510:0480:0730:092 0:2810:0220:0450:067

0:0210:0430:062 0:2730:0370:0770:1190:0330:0590:089

ns 0:9770:0610:1220:1810:0610:1230:190 0:9590:0520:1040:164

0:0530:1070:161 0:9900:0320:0630:0900:0290:0530:076 0:9770:0250:0520:083

0:0210:0400:058 1:000:0340:0700:1020:0320:0600:085

0:2040:0700:0920:09570:0860:1490:192 0:1950:0650:0970:103

0:0680:1230:165 0:1880:0780:1080:1110:0750:1300:171 0:1630:0410:0830:123

0:0410:0780:111 0:1420:04930:09790:1430:04650:08790:117

8 0:8730:1150:240:3810:1070:2010:297 0:8970:0590:1080:189

0:0590:1040:137 0:8950:0340:0680:1020:0320:0640:094 0:8990:0340:0700:107

0:0300:0580:085 0:9000:0340:0690:1000:0320:0630:094

h 0:7360:0610:1270:2040:0540:1030:146 0:7160:0390:0790:135

0:0400:0800:121 0:7300:0530:0920:1280:0460:0800:107 0:7090:0220:0460:072

0:0210:0400:059 0:7250:0370:0740:1230:0350:0660:094

r 0 0 0 0 <0:450:64102s 1:243:757:6311:8

3:637:2311:1 2:413:076:249:453:106:149:20 0:2631:272:664:15

1:132:213:22 0:291:082:353:631:001:842:61 0:571:212:493:48

1:142:263:39


[15:84%, 84:16%] interval ( 1), [2:3%, 97:7%] interval( 2) and [0:13%, 99:87%] interval ( 3). These arecalculated from the cumulative one-point distributions ofMCMC values for each parameter and do not depend onthe Gaussian assumption. For the parameters without adetection we quote a 95% confidence upper limit and a99:9% confidence upper limit. We have found that our

TABLE IV. Dark energy constraints. Same format as for Table I. ALast column gives constraints for the case where dark energy is tim

6-pw 6-pw 6WMAP galSN all WMAP

102!b 2:360:130:260:380:110:210:31 2:330:100:200:32

0:090:180:27 2:3400

m 0:3030:0290:0610:0930:0280:0520:072 0:2820:0230:0470:074

0:0230:0440:067 0:2640:0:

ns 0:9870:0410:0770:1050:0300:0540:075 0:9810:0270:0550:080

0:0230:0420:062 0:9800:0:

0:1600:0820:1300:1390:0670:1160:153 0:1630:0640:1210:135

0:0570:1030:146 0:1450:0:

8 0:9450:0890:1870:2900:0800:1500:212 0:8950:0330:0670:104

0:0310:0590:089 0:9200:0:

h 0:6990:0270:0540:0800:0260:0500:073 0:7080:0230:0460:069

0:0220:0440:064 0:7360:0:

r 0 0102s 0 0

w 1:0090:0960:180:260:1120:240:38 0:9900:0860:160:22

0:0930:200:35 1:080w1 0 0

103515

MCMC gives a reliable estimate of 3-sigma contours forone-dimensional projections. The corresponding 2-d pro-jections are however very noisy and we do not plot 3-sigmacontours in our 2-d plots.

All of the restricted parameter space fits are acceptablebased on !2 values, starting from the basic 6-parametermodel p ;!b;!m; 1 m;

2R; ns. We de-

ll columns except last one assume constant equation of state w.e dependent as w w0 w1a21.

-pw 6-pw s r 6-pw0 w1

galbiasly all all:090:190:28:090:160:25 2:480:150:290:43

0:130:240:34 2:330:100:200:320:090:170:25

0280:0560:1090250:0460:062 0:2600:0240:0500:077

0:0220:0400:056 0:2850:0240:0470:0700:0230:0450:066

0260:0510:0680200:0380:059 1:0200:0410:0800:114

0:0370:0680:096 0:9780:0280:0580:0840:0220:0410:059

0660:1250:1520560:1090:142 0:2010:0570:0910:098

0:0630:1170:163 0:1520:0670:1270:1460:0560:1010:136

0400:0840:120410:0720:093 0:8900:0300:0630:099

0:0280:0560:089 0:8970:0330:0680:1040:0310:0590:088

0390:0800:1190380:0690:112 0:7260:0250:0500:078

0:0240:0480:072 0:7070:0240:0490:0740:0230:0460:066

0 <0:510:67 00 1:071:242:644:15

1:162:263:31 00:1490:240:310:1930:370:54 0:9080:0770:140:19

0:0910:190:32 0:9810:1930:380:570:1930:370:52

0 0 0:050:831:922:880:651:131:38

-7


note this as 6-p in the tables. Introducing additional pa-rameters such as tensors, running, equation of state, orneutrino mass does not improve the fits. We do not reportthe values of nuisance parameters such as the galaxy biasor Ly forest mean flux, temperature-density relation, orfiltering length. Some of these are discussed elsewhere[22,31]. When comparing the improvements over previousanalyses we try to compare the results to our own MCMCanalysis of previous data. This is because small changes inthe treatment, such as assumed priors, can affect the pa-rameters and so the constraints between different groupsare not directly comparable. When comparing our analysisto [11] we find in general a very good agreement betweenthe two, even though our MCMC implementation is inde-pendent. Our primary goal is to determine how much thenew data improve over the previous situation and to answerthis it is best to perform identical analyses with and withoutthe new data. Below we discuss the results from thesetables in more detail.

A. Amplitude of fluctuations

From Tables I, II, III, and IVone can see that the value of8 is remarkably tight. For 6-p models (Table I) we find

m

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0 0.05 0.1 0.15 0.2 0.25 0.3τ

Ω8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

0 0.05 0.1 0.15 0.2 0.25 0.3

σ

τ

FIG. 1 (color online). 68% (inner, blue) and 95% (outer, red) contoall measurements.

103515

8 0:8970:0330:0650:0970:0310:0580:088: (2)

This value does not change significantly when running,tensors and massive neutrinos are added to the mix, whichshows that the constraint is model independent. In contrast,in an analysis without the Ly forest and bias 8 changesfrom 8 0:9510:90

0:079 (Table I) to 8 0:7860:1190:100

(Table III) when massive neutrinos are added as a parame-ter (see also [11]), so previous constraints were signifi-cantly more model dependent.

It is useful to analyze what drives the 8 determination.WMAP alone cannot provide a very tight determination,nor can the Ly forest alone. But combining the two isextremely powerful: from Table I we see that just these twodata sets alone give 8 0:8950:034

0:032 even with running.So this combination in itself provides nearly all of theinformation on 8; galaxy clustering and bias do not con-strain this parameter any further when added to the mix.They are however consistent with it: using WMAP andSDSS galaxy clustering with bias and without Ly forestgives 8 0:89 0:06 [31], in remarkable agreementwith the analysis of WMAP SDSS-ly. Assuming thatWMAP data are valid this implies that two independent

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0 0.05 0.1 0.15 0.2 0.25 0.3

h

τ

s

0.93 0.94 0.95 0.96 0.97 0.98 0.99

1 1.01 1.02 1.03 1.04

0 0.05 0.1 0.15 0.2 0.25 0.3τ

n

urs in the plane of versus m, h, 8 and ns, respectively, using

-8


analyses of different data, SDSS-galbias and SDSS-ly,lead to essentially the same value. Both improve uponprevious constraints, by a factor of 1.5–2 forWMAP+SDSS-gal+bias and a factor of 3–4 forWMAP SDSS-ly. These new constraints remove al-most all of the degeneracy between 8 and optical depth (Fig. 1).

There are many recent determinations of 8 in theliterature, which vary between 0.6 and 1.1. Recent discus-sion of some of these methods and results, such as weaklensing, cluster abundance, galaxy bias determination, andSZ power spectrum can be found in [11,31]. The valuefound here is in good agreement with most of these con-straints: it is on the low end of the SZ constraints and on theupper end of some of the cluster abundance constraints. Itis also in good agreement with the 2dF bias constraints andwith several weak lensing constraints.

While in the tables we do not present results for theamplitude of metric (described here with curvature fluc-tuation R) fluctuations at the pivot point we find it is alsotightly constrained to

2Rkpivot 0:05=Mpc 2:45 0:23 109: (3)

This is the constraint with the Ly forest, CMB and galaxyclustering data in 6-parameter models, as in Table I.

B. Optical depth

The optical depth due to reionization is a parameter thathas a strong effect on the CMB. It suppresses the CMB onsmall scales and thus leads to a strong degeneracy withamplitude. This degeneracy can be lifted by the polariza-tion observations [56], but for WMAP 1st year these arenoisy and may contain significant contamination fromforegrounds. The current analysis based on 1st year datais rather unsatisfactory, since it is based on the existingtemperature-polarization cross-correlation analysis, whichon large scales may suffer from similar problems as thetemperature autocorrelation analysis [46]. The upcoming2nd year data release of WMAP should provide polariza-tion maps and the corresponding analysis may help im-prove the situation. Until then we will use the currentWMAP provided likelihood code [45], but this should betaken as preliminary and the constraints on optical depthfrom polarization, both the best fitted value and the asso-ciated errors, may change.

With the addition of new constraints from the Ly forestand SDSS-bias there remain correlations between opticaldepth and several other parameters from 6-parameteranalysis on all data in Table I. Results are shown inFig. 1. The degeneracies are significantly less severe thanbefore, since the parameters are better determined with thenew data. Still, there is room to improve the constraintswith a better determination of the optical depth. For ex-ample, if the optical depth ends up being at the lower end ofits allowed range this would lead to a decrease in the best

103515

fitted value of ns, h and 8 and to an increase in the bestfitted value of m. Note that the values of do not extendup to the cutoff value 0:3 for the 95% contours, sothese distributions are not affected by the choice of theprior < 0:3. However, in chains with more parameters,such as dark energy equation of state w, this is no longer thecase. At the moment the only argument for adopting thisprior is that if > 0:3 this would possibly have led todetectable autocorrelation of polarization in the WMAPdata, but this argument is inconclusive since the polariza-tion maps are not available and such analysis has not beenpublished yet. In the absence of any published results wefollow the WMAP team approach and adopt < 0:3.

C. Neutrino mass

Both the CMB and LSS are important as tracers ofneutrino mass. At the time of decoupling, neutrinos arestill relativistic, but become nonrelativistic later in theevolution of the universe if their mass is sufficientlyhigh. Neutrinos free-stream out of their potential wells,erasing their own perturbations on smaller scales. Belowthis suppression scale the power spectrum shape is thesame as in regular CDM models, so on small scales theonly consequence is the suppression of the amplituderelative to large scales. In the matter power spectrumneutrinos leave a characteristic feature at the transitionscale. The actual shape of the transition depends on theindividual masses of neutrinos and not just on their sum.For masses of interest today the transition is occurringaround k 0:1h=Mpc, which are the scales measured bySDSS-gal. Neutrinos with mass below 2 eV are still rela-tivistic when they enter the horizon for scales around k 0:1h=Mpc and are either relativistic or quasirelativistic atthe time of recombination, z 1100. As a result neutrinoscannot be treated as a nonrelativistic component withregard to the CMB and are not completely degeneratewith the other relativistic components in the CMB.

From the joint analysis we find for the sum of all masses(Table III)

Xm < 0:42 eV0:67 eV3 families; (4)

at 95% (99:9%) c.l. for a single component and assumingno running, as was done in all of the work to date. Ourconstraints improve upon WMAPSDSS-gal, where wefind m < 1:54 eV and upon WMAP 2dF constraints,where m < 0:69 eV was found by combining WMAP and2dF with the bias determination from the bispectrumanalysis [33].

If running and tensors are allowed, the parameter spaceexpands. In this case, we find m < 0:660:93 eV. Muchof this is caused by running: as discussed in [31] runningand neutrino mass are anticorrelated. Negative runnings aslarge as -0.04 and neutrino masses as high as 1.5 eV areallowed at 2-sigma. Running is poorly motivated by infla-

-9


tionary models and there is no evidence for it in the currentdata, so adopting the inflationary prior with no running isreasonable, but one should be aware that the limits aremodel dependent.

The constraint from Eq. (4) is remarkably tight andimplies the upper limit on neutrino mass assuming degen-eracy is 0.14 eV at 95% c.l. Our constraint has beenobtained assuming 3 degenerate mass neutrino families,but if the neutrino mass splittings are small the constraintson the sum are almost the same even if individual massesare not identical. If the masses are very large compared tomass splittings then the neutrino masses are close to de-generate. However, our upper limit is so low that includingmass splittings is necessary. Super-Kamionkande (SK)results find neutrino mass squared difference m2

23 2:5 103 eV2 [24,26], while solar neutrino constraintsfind neutrino mass squared difference m2

12 8 105 eV2 [25,57,58]. This gives one neutrino family withminimum mass around 0.05 eV and another with minimummass close to 0.007 eV. Since only the mass square differ-ence is measured, it is in principle possible that the actualneutrino masses are larger than that. Our constraints incombination with SK and solar neutrino constraints limitthe mass of the neutrino families to

m1 < 0:13 eV; m2 < 0:13 eV; m3 < 0:14 eV;

(5)

all at 95% c.l. These limits essentially exclude the range ofmasses argued by the Heidelberg-Moscow experiment ofneutrinoless double beta decay if neutrinos are Majoranaparticles [59], although the two results may still be com-patible given all the uncertainties in nuclear matrix elementcalculations. From m m2=2m with m 0:13 eV wefind the neutrino masses are not degenerate at the level:

m3

m1> 1:1; 1:1<

m3

m2< 7; (6)

all at 95% c.l., where the upper limit on m3=m2 is deter-mined solely from SK and solar neutrino constraints.

The mass limits presented above are based on 3 degen-erate massive neutrino families. If one assumes a modelwith 3 massless families and 1 massive family (such as asterile neutrino model), as motivated by LSND results [28],then the mass limits on the sum change, since both theCMB and the matter power spectrum change (see Figure 6in [31]). These limits are improved as well with the addi-tion of SDSS-ly and SDSS-bias. We find

m < 0:79 eV1:55 eV3 1 families; (7)

at 95% (99:9%), assuming all neutrinos are thermallyproduced. This can be compared to the WMAP 2dFanalysis without bias where the 95% confidence limit is1.4 eV [29] and to the SDSS WMAP analysis where thelimit is 1.37 eV [31]. We have subtracted from the totalsum in Table III the masses of the active neutrinos to obtain

103515

the limit in Eq. (7). These limits are improved by almost afactor of 2 compared to previous analyses. These limits aremore model independent, as there is little correlation withrunning and/or tensors in this model: for the chains withrunning and tensors we find m < 0:88 eV1:40 eV at95% (99:9%) c.l.

From the LSND experiment the allowed regions are fourislands with the lowest mass m 0:9 eV and the nextlowest 1.4 eV [28–30,60]. Thus the lowest island allowedby LSND results is excluded at 95% c.l. and all the othersat 99:9%. Our derived limits will be tested directly withMiniBoone Experiment at Fermilab [61].

D. Tensors

Gravity waves (tensors) are predicted in many models ofinflation. The simplest single field models of inflationpredict a tight relation between tensor amplitude and slope,which we assume here. We choose to parametrize them atthe pivot point k 0:05=Mpc, just as for the amplitude,slope and running. This pivot differs from that in theWMAP analysis [10]. While tensors have their largesteffect on large scales, within the single field model adoptedhere the slope is assumed to be determined from the tensoramplitude. Thus there is no need to have a separate pa-rameter for the shape of the tensor power spectrum.

For 7-parameter model without running or neutrinomass, the limit on tensors is (Table I)

TS< 0:360:51 (8)

at 95% (99:9%) c.l. This does not change significantly ifneutrinos or running are added to the mix (Tables II andIII), in the latter case we find r < 0:450:64. This con-straint is nearly a factor of 2 better than from WMAPanalysis, a consequence of tighter constraint on runningfrom the Ly forest. We return to these constraints belowwhere we discuss inflation.

E. Spectral index

Constraints on the scalar spectral index are primarilydriven by the WMAP and SDSS-ly combination. Usingthese two experiments alone one finds ns 0:9900:032

0:029 forthe chains with running, compared to ns 0:9620:054

0:056 forWMAPSDSS-galSDSS-bias without SDSS-ly and tons 0:9750:028

0:024 for the case where all observations areincluded (Table II). The inclusion of the SDSS Ly forestthus reduces the error on the primordial slope by a factor of2. In the absence of running and with bias and SNIa, thisconstraint improves further to

ns 0:9810:0190:0400:0610:0180:0370:053: (9)

Note that the scale invariant model ns 1 is only 1-sigmaaway from the best fit. It is remarkable that such a vastrange of observational constraints can be reproduced with ascale invariant power spectrum with 4 parameters only, b,

-10

s

s

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.8 0.85 0.9 0.95 1 1.05 1.1

WMAP+lyaWMAP+gal+bias

n

α

FIG. 3 (color online). 68% (inner, dark) and 95% (outer, light)contours in the s; ns plane using WMAP SDSS-ly versusWMAPSDSS-galbias. Adding the SDSS Ly forest dra-matically reduces the allowed region of parameter space inthis plane. Note that the simplest model with ns 1 and s 0 is within 68% interval.

s

φ

φ4

2V~

V~

0

0.1

0.2

0.3

0.4

0.5

0.6

0.9 0.95 1 1.05 1.1 1.15 1.2n

WMAP+gal+lyaWMAP+gal+bias

r

FIG. 2 (color online). 68% (inner, dark) and 95% (outer, light)contours in the r T=S; ns plane with and without SDSS-ly.There is a correlation between tensors and slope ns. Inclusion ofthe Ly forest significantly reduces the allowed region in thisplane. Also shown are the positions of two chaotic inflationmodels, V / 2 with N 50 and V / 4 with N 60.


m, h and amplitude 2R (plus possibly optical depth to

explain the polarization data).Tensors are positively correlated with the slope (Fig. 2)

and their inclusion increases the best fit slope value to ns 1:000:034

0:028. All of these are consistent with a scale invariantspectrum and are in a good agreement with theWMAPext 2dF constraint ns 0:97 0:03 [40].While 2dF gives a slightly redder spectrum than SDSSthe differences in different values quoted in the literaturereflect mostly the differences in the assumed parameterspace, as shown here for the example of tensors.

F. Running of the spectral index

The issue of the running of the primordial slope hasgenerated a lot of interest lately. WMAP argued for someweak evidence for negative running in their combinedanalysis, but some of that evidence was based on Lymanalpha constraints by previous workers [19,62], which wereshown to underestimate the errors [42]. It was argued thateven from WMAP alone, or WMAP 2dF, there is someevidence for running, and the WMAPSDSS-gal analysiswithout bias information gave s 0:071 0:044 [11].Similar values have been found from the recent analysesincluding CBI [37] and VSA [38] data. However, much ofthis effect comes from low l multipoles and a full like-lihood analysis of WMAPSDSS-gal, together with amore conservative treatment of foregrounds, changes thisvalue to 0:022 0:033 [46]. Other foreground treat-ments give very similar values. Including the biasing con-straints does not really change this result. In the absence ofmassive neutrinos and tensors we find s 0:0220:030

0:032,so s 0 is within one sigma of 0 and the error has notbeen reduced.

Including SDSS-ly reduces the errors dramatically.The constraint on running from WMAP SDSS-ly

103515

alone is s 0:00260:0130:011. Including everything this

changes slightly to

s 0:00290:0110:0230:0360:0100:0180:026; (10)

which is a factor of 3 improvement over previous con-straints. Even with this significant improvement we find nohint of running in the joint analysis. The result is in perfectagreement with no running and 95% of chain elementshave s >0:015. This should be compared to values aslow as s 0:10 in Fig. 3. Similarly low values havebeen found in recent analyses [37,38]. Figure 3 shows oldand new constraints in the s; ns plane, highlighting thedramatic reduction of available parameter space whenCMB and Ly forest data are combined together. Theimplications of this result for inflation are discussed inthe next section.

If tensors are also included they induce weak anticorre-lation with running, so the best fit value becomes s 0:0060:012

0:011, which is still perfectly consistent with norunning. This is shown in Fig. 4, where we see that addingSDSS-ly to the mix dramatically reduces the allowedregion of parameter space. Specifically, withoutSDSS-ly, runnings as negative as 0:15 are in the 95%confidence region, a consequence of strong correlationbetween running and tensors. Our joint analysis eliminatesthese large negative running solutions. We find no evidencefor running in the current data, with or without tensors,despite a factor of 3 reduction in the errors.

Running is correlated with some of the ‘‘nuisance’’parameters we marginalize over in the analysis and addi-tional observations constraining these could lead to a fur-ther reduction of errors on the primordial slope and itsrunning even with no additional improvements in the ob-servations. For example, in our current treatment of the

-11

m

8

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42

σ

WMAP+galWMAP+gal+bias+SN+lya

Ω

FIG. 5 (color online). 68% and 95% contours in the m; 8plane showing previous constraints from WMAP and galaxyclustering with the new data.

m

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38

h

Ω

WMAP+gal+SN+biasWMAP+gal+SN+bias+lya

FIG. 6 (color online). 68% (inner, dark) and 95% (outer, light)contours in the m; h plane with and without SDSS-ly. Thereis a strong correlation between the two parameters becauseWMAP constrains best the combination mh

2.

s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

−0.2 −0.15 −0.1 −0.05 0 0.05α

r

WMAP+gal+lyaWMAP+gal+bias

FIG. 4 (color online). 68% (inner, dark) and 95% (outer, light)contours in the s; r plane using WMAP SDSS-ly versusWMAPSDSS-galbias. Adding the SDSS Ly forest dra-matically reduces the allowed region of parameter space inthis plane. Note that the simplest model with s 0 and r 0 is within the 68% interval.


filtering parameter kF (a generalization of the Jeanslength), we assume that the minimum reionization redshiftis around 10 with a reheating temperature of 25 000 K. Ifwe change the redshift to 7, this leads to an increase in themaximum value of kF allowed. In this case we find forWMAP SDSS-ly analysis the running changes froms 0:0017 to 0:0045, with an error around 0.01 (seeTable II). If we change this redshift to 4, below its theo-retically allowed lower limit of 6.5, to allow for anyresidual resolution issues in numerical simulations, wefind s 0:009 with comparable errors. All the otherparameters change much less. While these changes aresmall and do not qualitatively change our conclusions,they may be important for the future analyses wheresmaller errors may be obtained. In all these cases thedata prefer a high value of kF, i.e. a late epoch of reioniza-tion. Independent constraints on the temperature evolutionof IGM would be helpful to constrain this further.

G. Matter density and Hubble parameter

The matter density parameter m has contributions fromcold dark matter, baryons, and neutrinos. We assume spa-tially flat universe, so matter density m is related to darkenergy density m 1 . As emphasized in [63], thematter density is still allowed to cover a wide range ofvalues from the present data: in 7-parameter models withrunning WMAPSDSS-gal gives m 0:2690:041

0:033.WMAP SDSS-ly gives a slightly lower value withcomparable error, m 0:2570:055

0:048 in models with run-ning. Combining WMAP, SDSS-gal and SDSS-ly givesm 0:2990:037

0:032. Including the bias and SNIa and ignor-ing running brings the value to

0:2820:0210:0430:0660:0200:0430:067 (11)

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which is a factor of 2 improvement over previous con-straints. The matter density is correlated with r and inclu-sion of tensors in the parameter space slightly reduces thedensity parameter. There is a significant improvement in8;m plane with the addition of new data (Fig. 5).

Despite the improvements the matter density remainsstrongly correlated with the Hubble parameter h, as ex-pected from the fact that mh

2 is better determined fromthe CMB than each parameter separately. This is shown inFig. 6 for 6-parameter models for the analysis with andwithout inclusion of SDSS-ly.

For the Hubble parameter the best fit value and its erroris h 0:71 0:02 in 6-parameter space. In 9-parameterspace with tensors, massive neutrinos and running we findh 0:74 0:05. All of these fits are statistically accept-able and are in good agreement with the HST key projectvalue h 0:72 0:08 [64], although a different groupusing almost the same data continues to find a significantlylower value h 0:58 0:06 [65].

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The new data also improve significantly the age of theuniverse constraint. We find t0 13:60:19

0:19 Gyr, comparedto 14:11:0

0:9Gyr found from the WMAPSDSS-gal analysis[11].

H. Dark energy

So far we have assumed dark energy in the form of acosmological constant, w 1. We now relax this as-sumption and explore the constraints on w. To maximizethe constraints we add to some of the analyses the ‘‘gold’’SNIa data [7]. Because we do not want to limit ourselves tow>1 we assume dark energy does not cluster (ndyn 3 option in CMBFAST4.5). Note that clustering of darkenergy vanishes for w 1 and so if w is close to 1 thenit makes very little difference if clustering is included ornot. Figure 7 shows the constraints in the w;m plane.We find

w 0:9900:0860:160:2220:0930:2010:351: (12)

We see that w 1 is an acceptable solution. This shouldbe compared to w 1:010:097

0:12 we find in the absence ofbias and Ly forest constraint, to w 0:910:13

0:15 usingthe new SNIa data but just some of the LSS constraints[66], to w 1:020:13

0:19 using a simple m prior [7], andto w 0:980:12

0:12 from the WMAP 1st year analysis [40].It is worth emphasizing the agreement and complementar-ity of the LSS, CMB, and SNIa constraints: in the absenceof SNIa data the constraint is w 1:020:15

0:19 and w ispositively correlated with m (Fig. 7). These solutionsallow phantom energy models (w<1) with w as lowas 1:5 for low matter density values. On the other handthe two are anticorrelated for the WMAPSDSS-galSNIa data constraints, and phantom energy solutionsare allowed for high values of the matter density. Combingthe two sets of constraints significantly reduces the pa-

m

−1.6

−1.5

−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38

w

Ω

WMAP+gal+bias+lyaWMAP+gal+SN

WMAP+gal+SN+bias+lya

FIG. 7 (color online). 68% and 95% contours in the m;wplane showing previous constraints without SDSS-ly and bias,constraints without SNIa, and combined constraints. In allcases the data are consistent with a cosmological constant model(w 1).

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rameter space of allowed solutions. All of these differentcombinations give very consistent results and the medianvalue hardly changes at all and is in all cases very close tow 1. Our constraints are a factor of 1.5–2 better thanpreviously published constraints on the dark energy equa-tion of state. Some of the improvement comes from ourmore sophisticated analysis which includes all of the in-formation previously available and some from the newconstraints from the bias and Ly forest, which furtherreduce the errors. This is an example of how combiningdifferent data sets leads not only to a significant improve-ment in the accuracy of cosmological parameters, but alsohow consistency among the different methods gives con-fidence in the resulting constraints.

The results are weakly model dependent, in the sensethat they are sensitive to the parameter space over whichone is projecting. If we include tensors and running in theanalysis we find

w 0:9080:0770:1430:1920:0910:1970:324; (13)

roughly a 1-sigma change in the central value compared tothe case without tensors in Eq. (12). Figure 8 shows thattensors and the equation of state are correlated. The shift inthe best fitted value of w reflects a large volume of pa-rameter space associated with r > 0 models and not any fitimprovement when adding tensors and running: !2

changes only by 1 and there is no need to introduce tensors(or w 1) to improve the fit to the data. We also find nocorrelation between the equation of state and running.

Our constraints eliminate a significant fraction of pre-viously allowed parameter space, with 95% contours at1:19<w<0:83 without tensors and at 1:11<w<0:77 with tensors. Thus a large fraction of the parameterspace of ‘‘phantom energy’’ models with w<1 [67] andtracker quintessence models with w 0:7 [68] appears

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

−1.15 −1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7

r

w

FIG. 8 (color online). 68% and 95% contours in the w; rplane using the WMAPSDSS-galbias ly SNIa con-straints. The presence of tensors favors a slightly lower valueof w, but the quality of the fit is only marginally improved andthe cosmological constant model (w 1) is near the 68%contours.

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−0.8

−0.6


to be excluded. Other dark energy models which predictw 1 remain acceptable. It is interesting to note thatsimplest quintessence solutions with w>1 are moreacceptable if tensors are present at a level predicted bysome inflationary models (r 0:2).

We also ran a MCMC simulation exploring a nonconst-ant equation of state. We use a second order expansion

w w0 a 1w1 a 12w2; (14)

where a 1=1 z is the expansion factor [69]. Theadvantage of this expansion is that it is well behavedthroughout the history of the universe from early times,when a 0, to today (a 1). This is in contrast to theoften adopted expansion in terms of the redshift, w w0 w0z, which diverges at high redshift and so cangive artificially tight constraints on w0 if CMB (or evenBBN) constraints at high redshift are used, without actuallysaying much about the time dependence of w in the rele-vant regime 0< z< 1. In contrast, using our expansion0< z< 1 covers half of the full range of w so w1 is beingconstrained in the regime of interest. If we impose w2 0then the best fit values and errors we find using all the dataare

w0 0:9810:1930:3840:5680:1930:3730:521 w1 0:050:831:922:88

0:651:131:38:

(15)

We find that w0 1, w1 0 is well within 1- contourand very close to the best fit model (Fig. 9).

The parameters w0, w1 and w2 are strongly correlated, asshown in Fig. 9 for the first two, so the error on w0 hasexpanded by a factor of 2 compared to the constant equa-tion of state case. We can explore less model dependentconstraints on wz by computing the median and 1, 2-intervals from MCMC outputs at any redshift. Over anarrow range of redshift these contours will be nearlymodel independent as long as the equation of state is a

1

0

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5

w

w

FIG. 9 (color online). 68% (inner, blue) and 95% (outer, red)contours in the w0;w1 plane using WMAPSDSS-gal-ly SNIa measurements. We find that the simplest solution,w0 1, w1 0 (marked by a cross), fits the data best.

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relatively smooth function of redshift. We find that the dataconstrain best the equation of state w at z 0:3, where wefind wz 0:3 1:0110:0950:1760:264

0:0990:2150:357: Thus z 0:3is the pivot point for the current measurements of equationof state and the constraint here is nearly model indepen-dent. This is confirmed by our analysis with w2. In this casewe find severe degeneracies among the 3 parameters, butthe value at z 0:3 is

w z 0:3 0:9810:1060:2050:2690:1200:2490:386; (16)

which is nearly the same as for the two parameter analysiswith w2 0. These constraints are shown in Fig. 10.

The corresponding constraint at z 1 for two parameter(w0, w1) analysis is wz 1 1:000:170:270:33

0:280:661:00.Adding w2 we find

w z 1 1:030:210:390:520:280:580:85; (17)

so 1- contours are nearly the same, while 2 and 3-contours expand in the positive direction and shrink inthe negative direction compared to 2-parameter analysis.This value is thus also relatively independent ofparametrization.

Adding tensors and running to the 3-parameter expan-sion of w gives,

w z 0:3 0:9140:0890:1690:2290:1060:2250:343 (18)

and

w z 1:0 0:930:210:350:480:250:560:90: (19)

This is shown in Fig. 11. Thus, in either case, there is noevidence for any time dependence of the equation of stateand its value is remarkably close to 1 even at z 1. As

−2

−1.8

−1.6

−1.4

−1.2

−1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

w

z

FIG. 10 (color online). Median (central line), 68% (inner, red)and 95% (outer, yellow) intervals of wz using all the data in thechains without tensors and with a 3 parameter expansion ofequation of state with respect to the expansion factor. Verysimilar results are found for the 2 parameter expansion of w,so the constraints are reasonably model independent as long as wis a smooth function of redshift. We find that the simplestsolution, w 1, fits the data at all redshifts.

-14

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

w

z

FIG. 11 (color online). Same as Fig. 10 but for MCMC withtensors.


for a constant w analysis we find that tensors increase thepreferred value of w by about 0.1. These constraints on thetime dependence of w are significantly better compared tothe 0.8–0.9 allowed variation between z 0 and z 1found previously [7]. Ly forest analysis measures thegrowth of structure in the range 2< z < 4 and so helpsin constraining models with a significant component ofdark energy present at z > 2 [55].

IV. IMPLICATIONS FOR INFLATION

Inflation is currently the leading paradigm for explainingthe generation of structure in the universe. Inflation, anepoch of accelerated expansion in the universe, explainswhy the universe is approximately homogeneous and iso-tropic and why it is flat [70–73]. During this acceleratedexpansion quantum fluctuations are transformed into clas-sical fluctuations when they cross the horizon (i.e., theirwavelength exceeds the Hubble length during inflation)and can subsequently be observed as perturbations in thegravitational metric [73–77]. A generic prediction of asingle field inflation models is that the perturbations areadiabatic (meaning that all the species in the universe areunperturbed on large scales except for the overall shiftcaused by the perturbation in the metric) and Gaussian.These predictions, together with flatness (K 0), havebeen explicitly assumed in our analysis.

Here we will explore a class of single field inflationmodels, in which there is a single field responsible forthe dynamics of inflation (even though additional fieldsmay be present or even required to end inflation, as in thecase of hybrid inflation [78]). We will assume the earlyuniverse is dominated by a minimally coupled scalar field, which we will express in Planck mass units setting8(G 1. During inflation the energy density is dominatedby potential V. The Hubble parameter H2 V=3 is nearlyconstant and the equation of state is w p=$1. SinceH d lna=dt it follows that the expansion factor is ex-ponentially increasing with time, a aendeHttend. One

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can introduce the number of e-folds before the end ofinflation at time t0 as

N lnaend=a0 Z tend

t0Htdt

Z end

0

VV 0d; (20)

which can be computed for any specific form of the poten-tial. Here we will define it to be the number of e-foldsbefore the end of inflation when the pivot point,kpivot 0:05= Mpc, crosses the horizon. Note that theusual definition is with respect to the largest observablescale, k 103=Mpc, which corresponds to N 4larger number of e-folds. The latter number is expectedto be between 50–60 e-folds for standard inflation (64 forV / 4), but could be as low as 20 or as high as 100 inspecial cases [79,80]. For our pivot point choice we willthus adopt N 50 as the standard value (60 for V / 4),but also explore more general constraints on it.

If the kinetic energy density were negligible all the timethe universe would keep exponentially expanding and therewould be no end to inflation. Typically therefore one musthave deviations from the pure w 1 case. These devia-tions lead not only to a finite number of e-folds, but alsobreak the scale invariance of the primordial power spec-trum. Since we know from current observational con-straints that r < 1 and ns 1 we can adopt the slow-rollapproximation to relate the form of the potential to theobserved quantities r, ns, s, and 2

R. The slow-roll pa-rameters are defined as [9]

,V 1

2

V 0

V

2

-V V00

V.V

V 0V000

V2 : (21)

Note that in some early literature the 3rd slow-roll parame-ter . was denoted as .2 to emphasize the point that it isgenerically of second order in , or - [81]. We will not usethis notation since . can be positive or negative and since itdoes not have to be of second order in the slow-rollexpansion.

The relations between the slow-roll parameters and ob-servables are [9]

2R

V

24(2,Vr 16,V

ns 1 6,V 2-V

s 16,V-V 24,2V 2.V:

(22)

As mentioned in the previous section, we assume r 8nT and do not consider the running of the tensor spectralindex, both of which should be valid for single field in-flation in the relevant regime.

Traditionally the inflationary models are divided intoseparate classes depending on the value of first two slow-roll parameters [9,82,83]. Figure 12 shows the distributionin the ,V; -V plane. We see that both positive and nega-

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V

V

0<η<2ε

η<0

η>2ε

−0.04

−0.02

0

0.02

0.04

0.08

0.1

0.12

0.14

0 0.005 0.01 0.015 0.02 0.025 0.03ε

0.06η

FIG. 12 (color online). 68% (inner, blue) and 95% (outer, red)contours in the ,V; -V plane using all the measurementswithout running (Table. I, 5th column). Also shown are theregions occupied by the 3 classes of inflationary models. All 3classes of models are allowed, but individual models within eachclass are constrained. Note that the solutions disfavor low energymodels (,V 0) with large positive curvature (-V > 0), typicalof hybrid inflation models, as well as models where both ,V islarge and -V < ,V=2, typical of chaotic inflation models withsteep potentials.


tive values of - are allowed and that there is a strongcorrelation between the two from the observational con-straints, a consequence of positive correlation betweentensors and primordial slope. Figure 13 shows the distri-bution in the -V; .V plane. Both parameters are consis-tent with 0. The basic constraints are , < 0:03,0:04<-< 0:12 and 0:015 <.V < 0:035, so allslow-roll parameters are small.

A. Large field models

The simplest inflationary models are the monomial po-tentials, V V0

p, for which the first two parameters are

V

V

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14η

ξ

FIG. 13 (color online). 68% (inner, blue) and 95% (outer, red)contours in the -V; .V plane from MCMC with running andtensors (Table II, 5th column).

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comparable, , -, and the curvature is positive, -> 0.These potentials occur in chaotic inflation models [84]. Inthese models a deviation from scale invariance, ns 1 2 p=2N, also implies a significant tensor contribu-tion, r 4p=N, while running is negligible, s 2ns 12=p 2 p 2=2N2. Because bothslow-roll parameters are of order p=2 these chaoticinflation-type potentials require a large field, > 1, tosatisfy observationally required r < 1 and ns 1. Forthis reason these models are sometimes called large fieldmodels. While this may limit their particle physics moti-vation there are brane inspired models where this propertycan be justified [85]. More generic parametrization of thesemodels in terms of curvature is 0<-V < 2,V .

With the exception of p 2, chaotic models are notparticularly favored from our analysis. Figure 2 shows theposition in the r; ns plane for two representative cases,p 2 and p 4. We find that the V / 2 model (ns 0:96, r 0:16 for N 50) is within the 2-sigma contour,while the V / 4 model (ns 0:95, r 0:27 for N 60)is outside the 3-sigma contour, since it predicts moretensors and a redder spectrum for that tensor amplitudethan observed. Figure 14 shows all chain elements withns < 1 converted to p;N values using the expressionsabove. For standard inflation we require N < 60 and thislimits us to p < 3. Similarly, Fig. 12 shows that ,V > -V=2with large ,V models are disfavored.

For specific models we also minimized !2 by exploringall of the parameter space of the remaining parameters andcompared that to the global minimum in !2. We find!2 5 for the V / 2 model and !2 13 for the V /4 model. These results are in agreement with the MCMCresults and show that the latter case is excluded at morethan 3 confidence.

0

20

40

60

80

100

120

140

0 1 2 3 4 5 6 7 8

N

p

FIG. 14 (color online). Scatter plot of MCMC solutions withns < 1 converted into the p;N plane assuming relations validfor chaotic inflation models. Here p is the slope of the infla-tionary potential and N is the number of e-folds. For N < 60 datarequire p < 3.

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B. Large positive curvature models

We turn next to models with positive large curvature,-> 2,. A generic potential of this type can be obtained byadding a constant to the monomial potential, V V01 cp, where c is a positive dimensionless constant. Thesemodels allow small field solutions to inflation, 1, andso are popular for model building in the context of super-symmetry. In this limit, and if dimensionless c is not toolarge, one has , 1. In such models, inflation never ends(since the potential never drops to zero), so another fieldmust be brought in to accomplish this. Hybrid inflation isan example of such a mechanism [78]. If , is small thenthese models predict r 0 and ns > 1 (Eqs. (22), the lattercondition requires , < -=3). For p 2 the slope is con-stant, ns 1 2c and there is no running, while for p > 2running is negative and is given by s p 2=p1 ns 12=2. This is always small since a large devia-tion in the ns > 1 direction is strongly disfavored, so s 0. Some of these models are disfavored: for r 0 and inthe absence of running we find ns < 1:0 at 90% confidenceand ns < 1:04 at 99:9% confidence, so if ,V 0 then -V >0:02 is excluded at 3 sigma. Thus the deviations from scaleinvariance have to be very small for these models to beacceptable.

C. Large negative curvature models

The most promising models from the observational per-spective are negative curvature models, -< 0. As notedabove, the main reason that large positive curvature modelsare disfavored is that in the absence of tensors the datafavor ns < 1, while small positive curvature models aredisfavored because they predict large tensors and a redspectrum at the same time, whereas the data are moreconsistent with blue spectrum if tensors are significant. Ageneric potential of negative curvature models can beobtained by switching the sign on the hybrid potentialform, V V01 cp, where c is a positive dimension-less constant. In these models the field is slowly rollingfrom low to high values until reaching the point where thepotential vanishes at cp 1, at which point inflationstops. This is a generic scenario of spontaneous symmetrybreaking models as in the first working inflation model,that of new inflation [72,73]. For p 2 the slope is againconstant at ns 1 2c and there is no running.

In these models one has ns 1 2p 1=p2=N and s p 2=p 1 ns 12=2. The run-ning is of order ns 12=2 and the prefactor is unity atbest, so running is negligible. The slope ns ranges between0.96 (in the limit of jpj ! 1, where ns 1 2=N) and1, in excellent agreement with observational constraints.

One finds good agreement using other potentials pro-posed in the literature, such as the potential based on one-loop correction in a spontaneous symmetry broken SUSY[86]. The potential is of the form V V0 1 ln=Q.In this model the number of e-folds is of the order N

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2=2 (this expression works best if 1). This modelpredicts ns 1 2 1 3=2=2 and s ns 12 2 32=2 1=2= 1 3=22. Running is againnegligible. Solutions with 1 require 2 1,in which case the slope becomes ns 1 1=N 0:02 for N 50, in excellent agreement with the ob-served value ns 0:9710:023

0:019.Many other models in this class also work. A model

often mentioned as an example of allowing a large runningis the softly broken SUSY model with V V01 c2ln= 1=2=2. This model has a large 3rd de-rivative for small field , V 000=V0 c=, so it can leadto large . and large runnings. For this model there is aninequality relation between slope and running of the forms > ns12

4 >2 103, so a large negative runningcannot be accommodated in this model for the allowedvalues of ns. Our solutions do not favor large negativerunnings anyways, unless one is willing to consider modelswith massive neutrinos whose mass exceeds 0.3 eV, so thismodel is acceptable, but it can overpredict the running onthe positive side.

There are also examples of models which can changefrom one inflationary case to the other, such as hybridmodel with one-loop correction [87], V V0 1

ln=Q c4

0p, which under specially arranged

conditions causes the slope to change from ns > 1 onlarge-scale to ns < 1 on small scale. Again, there is noevidence for such a transition in the data, so there is noneed to consider these special cases.

Finally, there are models that predict the simplest pos-sible case of r 0, ns 1 and s 0 [88]. These modelsare perfectly acceptable from our data.

While we only surveyed a small subset of inflationarymodels here, it is clear that their generic prediction is anearly scale invariant spectrum, jns 1j< 0:05, little orno tensors, r < 1 and small running, s 103. All ofthese predictions agree with our constraints. Running is aparticularly powerful test of standard inflationary (andcyclic) models in the sense that if running turned out tobe large, a large class of inflationary models would havebeen eliminated. The original suggestions of running in theWMAP data sparked a lot of theoretical interest in infla-tionary models with running [89,90], but such models areunnatural in the sense that they require a feature in thepotential at exactly the scale of observations today. Ourresults suggest that the natural prediction of inflation, smallrunning, is confirmed by observations.

V. CONCLUSIONS

In this paper we performed a joint cosmological analysisof WMAP, the SDSS galaxy power spectrum and its bias,the SDSS Ly forest power spectrum, and the latest super-novae SNIa sample. We work in the context of currentstructure formation models, such as inflation or cyclic

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models, so we assume spatially flat universe and adiabaticinitial conditions. We also ignore more exotic componentssuch as warm dark matter. The new ingredients, SDSS Lyforest and SDSS-bias, lead to a significant reduction of theerrors on all the parameters. Many parameters are im-proved in accuracy by factors of two or more. For example,for the amplitude of fluctuations we find 8 0:90 0:03and for the matter density we find m 0:28 0:02, botha significant improvement over previous constraints. Fromthe fundamental physics perspective the highlights of thenew constraints are:

(1) T
he scale invariant primordial power spectrum is aremarkably good fit to the data and there is noevidence that the spectral index deviates from thescale invariant value ns 1, nor is there any evi-dence of its running with scale. We also find noevidence of tensors in the joint analysis. The con-straints on running have improved by a factor of 3compared to an analysis without the new Ly forestconstraints. These provide a data point at 2< z < 4and k 1=Mpc, a significantly smaller scale thanscales traced by the CMB and galaxies.
(2) T
here is no cosmological evidence of neutrino massyet. In the standard models with 3 neutrino familieswe find for the total neutrino mass
Pm < 0:42 eV

(95% c.l.). When our analysis is combined withatmospheric and solar neutrino experiments[25,26] we find that neutrino masses are not degen-erate: the most massive neutrino family has to be atleast 10% more massive than the least massivefamily, m3=m1 > 1:1: the mass of the least massiveneutrino family has to be m1 < 0:13 eV, and that ofthe most massive neutrino family m3 < 0:15 eV,both at 95% c.l. In alternative models with a 4thmassive neutrino family in addition to 3 (nearly)massless ones we find m < 0:79 eV, excluding allof the allowed LSND islands at 95% c.l.

(3) D
ark energy continues to be best characterized as astandard cosmological constant with constant en-ergy density and equation of state w 1. Whenall the data is combined together the error on w is0.09, a reduction compared to previously publishedvalues [7,40,66]. A cosmological constant with w 1 is remarkably close to the best fit value for avariety of different subsamples of the data. A sig-nificant region of phantom energy parameter spacewith w<1 is excluded, as are some of the trackerquintessence models with w 0:7. The currentdata do not support any time dependence of theequation of state.
As the statistical errors are being reduced the requiredlevel at which systematics must be controlled increases aswell. Our limits on cosmological parameters assume thatthe errors from the SDSS Ly forest SDSS power spec-trum shape, SDSS-bias, WMAP CMB power spectrum,and the SNIa data are all properly characterized by the

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authors and that there are no additional sources of system-atic error. Each one of these ingredients has to be tested andredundancy is necessary for the results to be believable. Inour extensive tests we find no evidence of a disagreementbetween the different observational inputs, but further testswith these and other data sets are needed to verify andconfirm our results. In addition, the upcoming 2 yearanalysis of WMAP polarization will improve the con-straints on the optical depth and reduce the errors onparameters correlated with it.

Tests of the basic model are particularly important forLy forest , which is responsible for most of the improve-ment on the primordial power spectrum shape and ampli-tude. Despite the extensive tests presented in [23], morework is needed to investigate all possible physical effectsthat can modify its distribution and to see how these mayaffect the conclusions reached in this paper. Some of thesetests will come from the ongoing work on SDSS data, suchas the bispectrum analysis. Similarly, more work is neededto verify the accuracy of simulations with independenthydrodynamic codes. The present analysis, together withits sister papers [22,23], is not the final word on thissubject, but merely a first attempt to take advantage ofthe enormous increase in statistical power given by theSDSS data [15]. Current analysis marginalizes over manyphysical processes that have little or no external constraintsand as a result the statistical power of cosmological con-straints from the Ly forest is weakened. Better theoreticalunderstanding of these processes together with externalconstraints from additional observational tests could leadto a significant reduction of observational errors on theprimordial slope and its running even with no additionalimprovements in the observations.

In summary, adding SDSS Ly forest and SDSS-biasconstraints to cosmological parameter estimation leads to asignificant improvement in the precision with which thecosmological parameters can be determined. Despite theseimprovements we find no surprises. Many of these resultsare not unexpected, but the tightness of the constraints israpidly eliminating many of the alternative models ofstructure formation, neutrinos and dark energy. Futurecosmological observations and improvements in theoreti-cal modelling will allow us to verify the constraints foundhere and improve them further. As the constraints becometighter there may be additional surprises awaiting us in thefuture.

ACKNOWLEDGMENTS

We thank the WMAP and SNIa teams for creating thedata sets used in present analysis. Our MCMC simulationswere run on a Beowulf cluster at Princeton University,supported in part by NSF grant AST-0216105. U. S. issupported by the David and Lucile Packard Foundation,NASA grants NAG5-1993, NASA NAG5-11489 and NSFgrant CAREER-0132953.

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Funding for the creation and distribution of the SDSSArchive has been provided by the Alfred P. SloanFoundation, the Participating Institutions, the NationalAeronautics and Space Administration, the NationalScience Foundation, the U. S. Department of Energy, theJapanese Monbukagakusho, and the Max Planck Society.The SDSS Web site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical ResearchConsortium (ARC) for the Participating Institutions. The

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Participating Institutions are The University of Chicago,Fermilab, the Institute for Advanced Study, the JapanParticipation Group, The Johns Hopkins University, LosAlamos National Laboratory, the Max-Planck-Institute forAstronomy (MPIA), the Max-Planck-Institute forAstrophysics (MPA), New Mexico State University,University of Pittsburgh, Princeton University, the UnitedStates Naval Observatory, and the University ofWashington.

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PHYSICAL REVIEW D Cosmological parameter analysis ...

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