Evidence of Neutrino Enhanced Clustering in a Complete Sample of Sloan Survey Clusters,Implying

∑mν = 0.119 ± 0.034 eV

Razieh Emami,1, 2, ∗ Tom Broadhurst,3, 4 Pablo Jimeno,3 George Smoot,5, 6, 7 Raul Angulo,8

Jeremy Lim,9, 10 Ming Chung Chu,11 Shek Yeung,11 Zhichao Zeng,11, 12 and Ruth Lazkoz3

1Center for Astrophysics, Harvard-Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA2 Institute for Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

3Department of Theoretical Physics, University of the Basque Country UPV-EHU, 48040 Bilbao, Spain4IKERBASQUE, Basque Foundation for Science, Alameda Urquijo, 36-5 48008 Bilbao, Spain

5 Helmut and Anna Pao Sohmen Professor-at-Large, IAS,Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, 999077 Hong Kong, China

6Paris Centre for Cosmological Physics, APC, AstroParticule et Cosmologie, Universite Paris Diderot,CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Universite Sorbonne Paris Cite,

10, rue Alice Domon et Leonie Duquet, 75205 Paris CEDEX 13, France7Physics Department and Lawrence Berkeley National Laboratory, University of California, Berkeley, 94720 CA, USA

8Centro de Estudios de F sica del Cosmos de Arag on, Plaza de San Juan 1, 44001 Terue, spain9Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong

10Laboratory for Space Research, Faculty of Science, The University of Hong Kong, Pokfulam Road, Hong Kon11Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

12Department of Physics, Ohio State University, Columbus, Ohio 43210, USA

The clustering amplitude of 7143 clusters from the Sloan Digital Sky Survey (SDSS) is found to increase withcluster mass, closely agreeing with the Gaussian random field hypothesis for structure formation. The amplitudeof the observed cluster correlation exceeds the predictions from pure cold dark matter (CDM) simulation by' 6% for the standard Planck-based values of the cosmological parameters. We show that this excess can benaturally accounted for by free streaming of light neutrinos, which opposes gravitational growth, so clustersformed at fixed mass are fewer and hence more biased than for a pure CDM density field. An enhancement of thecluster bias by 7% matches the observations, corresponding to a total neutrino mass, mν = 0.119 ± 0.034 eV at67% confidence level, for the standard relic neutrino density. If ongoing laboratory experiments favor a normalneutrino mass hierarchy then we may infer a somewhat larger total mass than the minimum oscillation basedvalue,

∑mν ' 0.056eV , with 90% confidence. Much higher precision can be achieved by applying our method to

a larger sample of more distant clusters with weak lensing derived masses.

PACS numbers:

INTRODUCTION

The standard picture whereby cosmic structure developsgravitationally from a Gaussian random field (GRF) predictsthat the amplitude of clustering should increase steadily withdensity contrast [1–5]. This inherent property of a GRF isknown as the clustering “bias”, and was first employed by [1]to explain the approximately factor of three larger correlationlength of nearby massive galaxy clusters relative to field galax-ies. Subsequent N-body simulations have fully demonstratedthat collapsed halos formed from a GRF should be biased inthis fundamental way [6]. Simulations also predict that theclustering bias should increase with redshift as halo abundancedeclines with redshift, particularly for the most massive clus-ters. The effect of light neutrinos is to smooth the densityfield thereby slowing the growth of structure, thus reducingthe abundance of clusters [7] while enhancing the clusteringamplitude relative to a pure CDM density field.

Measuring this fundamental link between halo mass andclustering bias has not proven feasible using galaxies. Not onlyare the virial masses of galaxies hard to define observationally,but galaxy correlation functions have been found to dependstrongly on galaxy type and luminosity with a complexity thatcannot be simply linked to the growth of structure. In con-

trast, cluster masses can be directly inferred from gravitationallensing and correlate almost linearly with the number of mem-ber galaxies, following a clear mass-richness (MR) relation[8, 10, 15–17].

Large surveys are now underway to realize the anticipatedsensitivity of cluster abundance to the growth of the clustermass function [18], including a predicted small additional sup-pression of their numbers by neutrino free-streaming [19–21].This suppression is claimed to be close to detection in an initialSZ-selected sample of 370 clusters [22] and in combinationwith other methods provides

∑mν . 0.14ev ([25, 26]), tight-

ening the robust 95% upper limit of < 0.25 eV from the purePlanck analysis [27]. This claimed improvement over Planckrests on uncertain assumptions about the inherent spread inmass of strong SZ detected clusters, as cluster collisions com-press the gas thereby boosting the SZ signal [29, 30][28].

Here we develop and apply a clustering based method thatis sensitive to the effect of neutrinos, using the correlationlength of optically detected clusters from the thoroughly testedand currently largest and most complete survey of SDSS clus-ters identified by the RedMapper team [31–33]. This largeRedMapper has made redshift complete in our earlier work[9, 10] by cross correlation with the BOSS spectroscopic data,which includes over 7000 clusters for which precise correlation

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functions have been estimated [9, 10]. The correlation functionscales as the square of the bias and, as we show here, is alreadysensitive enough to detect the effects of neutrino mass inferredfrom the oscillation experiments of

∑mν ' 0.056eV .

We rely on the latest Planck determinations of all the cos-mological parameters [46] required here for the theoreticalcalculation of the bias enhancement by neutrinos and forthe comparison with the MXXL cosmological simulations,where for a flat cosmology, ΩΛ = 0.6911, Ωm = 0.3089,Ωb = 0.0497, h = 0.6773 and the amplitude of the primor-dial power-spectrum As = 2.142 × 10−9, and hence the onlyflexible quantity is the cold dark matter density, defined as,Ωc = Ωm − Ωb − Ων, with Ωνh2 ≡

(1.015)3/4mν

94.07 ev . We havIn thisanalysis we examine first one species of dominantly massiveneutrino, corresponding to the standard neutrino hierarchy, andthen we discuss implications for the inverted hierarchy, whichmay be favored by our results. We emphasis here that we aresafe from the simulation to simulation scatter discussed by Tin-ker [37] owing to the relatively narrow halo mass and redshiftrange that we consider.

NEUTRINO BIAS ENHANCEMENT FOR CLUSTERS

We first present a consistent formalism for the biasing effectof light neutrinos on cluster scales, emphasizing that the clustercorrelation provides a relatively sensitive way of detectingthe effect of standard relic neutrinos. We then compare thecluster correlation function predicted by our analysis with thatmeasured for the galaxy clusters from the SDSS selected in themanner described above.

Theoretical Considerations

Here we estimate the effect of light neutrinos on thecorrelation function of clusters using the empirically peakbackground split approximation [5], hereafter Sheth-Tormen(ST).

In the ST approach, the bias is defined as,

bS T ≡aν1 − 1δc

+2p/δc

1 + (aν1)p , (1)

here ν1 ≡ (δc/σ0)2 where δc refers to the critical over-density. In addition, σ2

0 ≡(

12π2

) ∫dkk2P(k)W2(kR) de-

notes the 0-th spectral moment of the matter power-spectrum and we use the top hat filter function, W(kR) =

3 (sin (kR) − kR cos (kR)) /(kR)3, as a smoothing factor. Weuse the usual simulation calibrated free parameter preferences,a = 0.707, p = 0.3. We compute the 0-th spectral moment ofthe matter power-spectrum as well as the critical over-density.The former can be calculated using the matter power spectrum,conveniently given by the CAMB code [38, 55].

To estimate the critical over-density for cluster formation, wedetermine the linearly evolved value of the initial over-destiny

required to have a collapse at red-shift z. Adopting sphericalcollapse, which is a good estimate for heavy halos, we thencalculate the evolution of the CDM + Baryon over-density,δcb ≡ (ρcδc + ρbδb) / (ρc + ρb) as,

δcb + 2Hδcb −43

δ2cb

1 + δcb−

3H20

2(Ωcbδcb + Ωνδν) (1 + δcb) = 0,

R +GMR2 + H2

0

Ωra−4 +ρνpν

3H20 M2

P

−ΩΛ

R +GδMν(< R)

R2 = 0,

(2)

where ρν = 2∫ d3 p

(2π)3

√p2+m2

ν

exp (p/Tν)+1 refers to the neutrino en-

ergy density and pν = 2∫ d3 p

(2π)3p2

√p2+m2

ν

1exp (p/Tν)+1 denotes

the neutrino pressure. Here Tν =(

1.95491a

)K is the phys-

ical temperature. In addition, Ωcb ≡ (Ωc + Ωb) a−3, δν ≡

δMν(< R)/(4πR3H2

0 M2PΩν

)in which δMν(< R) =

mν

∫Vc

d3r∫ d3q

(2π)3 f1(q, r, t) refers to the neutrino mass functioninterior to the radius R. f1 is the linear perturbation to theneutrino distribution function, f = f0 + f1, which can be calcu-lated from the evolution of the radius R, in the absence of theneutrino clustering, [40], as,

f1(r,q, η) = 2mν

Tν

∫ t

t0

dt′

a(t′)exp (q/Tν)

den(1 + exp (q/Tν)

)2

GδM(t′)r2

×

(α

qTν− q · r

) [a3(t′)r3

R3 Θ

(r2

(1 + q2/T 2

να2 − 2q/Tνα

)<

R2(t′)a2(t′)

)+

Θ(r2

(1 + q2/T 2

να2 − 2q/Tναq · r

)≥

R2(t′)a2(t′)

)(1 + q2/T 2

να2 − 2q/Tναq · r

)3/2

].

(3)

where δM(t) = M − 4π3 ρcbR3 and Θ denotes the Heaviside step

function. In addition, α is defined as α ≡ Tν (η − η′) /(mνr)and η is defined as, a2dη = dt, note that both r and q are comov-ing quantities. These are the set of the differential equationsthat must be solved together with the evolution of the Hubble

parameter: H2 = H20

(Ωγa−4 + (Ωc + Ωb)a−3 + ΩΛ +

ρν(3M2

PH20 )

)We now describe the initial conditions, for which there are

three independent, relevant quantities, δcb,ini, δcb,ini, aini. Thefirst determines the collapse redshift that relates to second con-dition via: δcb,ini = δcb,ini

(d ln δcb(t)

dt

)|,ini, where the term in the

parenthesis is free of any normalization and can be extractedfrom the CAMB code. Initiating the evolution at z ' 200,we also read off the value of the initial scale factor. Usingthe above quantities, we can obtain Rini = Rini

(1 − 1

3δcb,ini

)and Rini = HiniRini

(1 − 1

3δcb,ini −13 H−1

ini δcb,ini

), where Rini =(

3M4πρcb,ini

)1/3depends on the value of the halo mass.

Next, we compute the evolution of system. The only un-known quantity is the redshift of collapse. We treat this asa perturbation to an Einstein-de (Ed) Sitter universe with aconstant critical over-density [41, 42], δc,Ed = 1.68647, bytruncating the evolution at a redshift once the non-linear over-density for any cosmology reaches this number. We then read

3

off the linearly evolved over-density at this reference redshiftas the critical over-density. We checked the robustness of ourcritical over-density by choosing another method in which wewrite the spherical collapse equations in terms of 1/δ instead ofδ, thus removing the need to specify the actual collapse. Com-paring the results of these two methods, we found a differenceof only 0.01% thus ensuring that the results are independent ofthe method we use. We now have the information necessary tocompute the bias for ST.

5 10 20 30

r[h−1 Mpc

]

101

102

Ξ(r

)

λ1 ∈ [22, 30)

λ2 ∈ [30, 45)

λ3 ∈ [45, 200)

MXXL clustering

MXXL abundances

FIG. 1: Measured correlation functions of ReDMaPPer clusters for 3richness ranges, showing clearly the observed amplitude scales withmass at almost the same rate as the predicted by the MXXL simulation(dotted and dashed curves) confirming directly the fundamental Gaus-sian field hypothesis for the formation of structure. In detail, whenaveraged, the data lies significantly higher than the pure CDM basedprediction based on a fit to the cluster mass function from MXXL, by' 6% (dashed curves).

N-body Simulation

Here we check our analytical predictions against our newcosmological simulations in the presence of massive neutrinos,following the grid-based method described in [11, 12]. Wecombine the GADGET2 N-body code [13] for CDM particlesand simulate relic neutrinos as a grid-based density field, whereonly the long-range force (the PM part) is re-calculated byincorporating the effects of neutrinos: δtot = (1− fν)δc+b + fνδν,where fν is the mass fraction of cosmological neutrinos, andδtot, δc+b and δν are the over-density fields of, respectively, totalmatter, CDM and baryons, and neutrinos. δν follows the linearperturbation theory with the potential provided by all matter,whereas δc+b is calculated in GADGET2 and then replacedby δtot after we take this weighted average. We iterate toadvance δν and δtot using the linear perturbation equation andGADGET2. We have set up four realizations for each neutrinomass, each realization having a box size of 800 Mpc/h and

5123 simulation particles. The halo catalogues are generatedby Amiga Halo Finder [14].

Comparison with Observations

The carefully constructed and well tested sample of SDSSclusters as defined by the RedMapper team [31–33] has pro-vided many insights, including accurate lensing and SZ basedmass-richness relations [10, 17, 43]. Redmapper clusters aredetermined to be complete to z = 0.33 in terms of detectabilitywithin the SDSS above a minimum richness of λ > 20 overthis redshift range [31–33]. This claim is strongly supportedby [9], where the observed numbers of RedMapper clusters areproportional to the cosmological volume in the redshift rangez < 0.33, but above which their numbers markedly declinerelative to the available volume. Importantly, this volume com-plete sample of Redmapper clusters has been made redshiftcomplete by [9] for the first time, at a level of 97% within theabove RedMapper limit z < 0.33, by cross correlation with thelatest SDSS/BOSS spectroscopic redshift surveys, allowingcorrelation functions to be accurately measured on small scalesusing these accurate redshifts [9].

The 2-point cluster correlation function of this completesample is shown in Figure 1, where it is clear that the ampli-tude of this correlation function increases with cluster richness.The correlation function has been integrated on small scalesalong the line of sight to account for the well known velocityeffects that otherwise affect pairwise distance separations. Wecompare this data with the predicted correlation functions fromthe large, MXXL simulation of pure CDM [44]. This simu-lation is scaled by the latest Planck cosmological parameters,following the prescription of [44], for the same redshift rangeas the complete RedMapper sample, and integrated along theline of sight in the same way as the data. Two different sets ofsimulation based predictions are calculated, where the lowercurves shown in Figure 1 are determined by a joint fit to theabundance of RedMapper clusters as a function of richnesswhereas the upper curves are fitted to the clustering. For bothof the above simulations a standard power-law function forthe mass-richness relation is used for this transformation withthree free parameters, slope, scatter and the pivot point normal-ization (see [9] for details), that are solved for simultaneouslyin comparing the measurements with the MXXL simulation.

It is clear from the comparison in Figure 1 that the ob-served cluster correlations are close to the MXXL predictionsin slope and amplitude and the scaling with richness shownin Figure 1 for the 3 independent richness bins, implying ex-cellent agreement with the GRF hypothesis for the origin ofcosmic structure. In detail we see the abundance-based pre-diction (solid curves) is systematically below the data andbelow the cluster-based fit (dashed curves). This mismatchwas highlighted by [9] without any satisfactory resolution,with the difference between these two mass-richness relationsfound to be mainly in the normalization, with the correlation-function based predicted cluster being ' 56% higher in mass,

4

M200m = 4.7 × 1014M/h, than the abundance-based meancluster mass (3.02 ± 0.11) × 1014M/h. This is a large differ-ence in mass and reflects the relatively shallow dependence ofclustering amplitude with cluster mass, so that a significantlyhigher mean mass is required to match the observed clusteringlevel but corresponds to a much lower cluster abundance asthe cluster mass function is inherently steep. Furthermore,the larger mean mass of the correlation function fit (shown inFigure 1) is excluded by the independent weak-lensing basedmass-richness relation derived by [43] for the same RedMappersample, where the mean mass is just (2.70 ± 0.2) × 1014M/hwith a 7% estimated uncertainty [43], and therefore consistentwithin 1.3σ with our aforementioned abundance-based zero-point mass. This WL based mass is also supported by a newmass-richness related analysis performed in [45] who derivea mean mass of 3.0 × 1014M/h using WL measurements forRedMapper clusters. This agreement of independent weaklensing based masses with our abundance-based best fittingmean mass further motivates our exploration of the effect oflight neutrinos: since, as we derived in the previous section,an enhancement in bias is expected to be significant in termsof reconciling our observed mass function with the observedcorrelation function without the need for increasing clustermasses.

We now show how our excess correlation may be explainedby the neutrino induced bias derived above. The relation be-tween the 2-point mass correlation function of collapsed halosrelates simply to the bias of a GRF via: ξHH(r) ≡ b2ξMM(r),where ξHH denotes the correlation of the collapsed halos, and bis the bias. This relation may be linked to the mass power spec-trum via ξMM(r) ≡

(1

2π2

) ∫dkk2P(k)W(kr), where W(kr) refers

to the top hat filter function. Observationally, we estimatethe above correlation function with the following power-lawscaling: ξ(r)obs = (r/r0)−γ, of correlation length ro and slopeof γ = 1.7 ± 0.05.

To compare the RedMapper observations with theory, we arerequired to use the mass-richness relation. This link is obtainedsimultaneously in the fits, as described in Ref. [9] - where wefit the halo mass function from the simulations leaving freethe 3 parameters of the MR relation, namely the zero point,the slope and the dispersion, as shown in figure 14 of [9]. Thevalues of these 3 parameters are in very good agreement withthose independently derived using lensing, X-ray, and SZ basedMR relations in Refs. [10, 43]. These independent derivationswere made subsequent to our work, providing verification thatadds great confidence to our best-fitting neutrino mass.

We can make a rough consistency check of this result that isindependent of the simulation based comparison used above bylooking at the absolute value of r0 from a weak-lensing basedmass-richness relation for this same cluster sample derived in[43]. In order to calculate this quantity, we use the fact that atr = r0 the halo-halo correlation function is, by definition, unity,and seek b2

S T

(1

2π2

∫dkk2P(k)W(kr)

)= 1 for which we use the

ST bias as the input. The results are given in Fig. 2. In the leftpanel, we present the relation between the fitted correlation

length r0 and the halo mass in these sets of simulation, averagedover four realisations. To be consistent with the RedMapperdata, the shown results are also averaged over the redshiftrange of 0.1 to 0.3. In the right panel, we show the analyticalresults the r0. Both the analytical and simulation predictionsare in close accord. The impact of the massive neutrinos onenhancing the halo clustering is clearly seen from these plots.The mean observational value (marked in red point) is slightlyhigher than the pure CDM predictions corresponding to therange 0.08eV <

∑mν < 0.13eV .

Discussion and Conclusions

We have shown here that even the minimum mass densityof standard relic neutrinos is expected to enhance the cluster-ing length of galaxy clusters by at least 3%, relative to purecold dark matter. This bias induced boost to the clustering ofclusters is more than an order of magnitude larger than theeffect of neutrinos on the general power spectrum of galaxiesbecause the clustering bias of clusters is approximately threetimes that for galaxies and the correlation function amplitudescales as the square of this bias. Furthermore, the sign of thisboost in the clustering of halos has the opposite sign to theusually sought suppression of the matter power spectrum of themass density field, which is predicted to “step” down on scalesbelow the predicted free streaming scale of . 100Mpc. Thisenhancement of the halo correlation function is noted in thesimulations of [59] and also visible in Figure 3 of [60] whenincorporating light neutrinos.

The carefully defined RedMaPPer clusters from the SDSSwith full spectroscopic redshifts by [10] provides a large, com-plete sample of clusters with accurate redshifts. Using thisdata we have claimed a 6% bias enhancement of the correla-tion length of clusters, corresponding to total neutrino mass of∑

mν = 0.119 ± 0.034eV at 67% confidence level, for the stan-dard relic mass density and standard Cosmological parameters.This lies well below the robust bound of < 0.24eV from thePlanck collaboration (2015), and below subsequent claimedimprovements on this upper limit that incorporate additionalconstraining data from cluster counts [22] and the Lyman-αforest [23].

Finally, in order to make our predictions more robust, wehave explored the relatively sensitive dependence of neutrinomass on Ωm by modifying CosmoMC [54, 55] to include ourmeasured correlation length data points with their errors shownin Figure 2. The data we used are the ratios between r0 in auniverse with and without neutrinos. In the case without neutri-nos, r0 is calculated assuming independent values of As (whereAs refers to the amplitude of the primordial power-spectrum)and Ωch2 than the case with massive neutrinos. This naturallyincreases the error bars compared with the case with similarvalues of As and Ωch2 for the numerator and denominator of r0ratio. The likelihood is assumed to be Gaussian. We also usethe Planck CMB measurements including low-` and high-`temperature and polarization data. For convenience, the mass

5

14.2 14.3 14.4 14.5 14.6Log(Mh/M h 1)

14

16

18

20

22

24r 0

(Mpc

/h)

(Simulation-Observation)experimental datafit,lcdmfit, m = 0.05eVfit, m = 0.1eVfit, m = 0.15eV

mean datasim,lcdmsim, m = 0.05eVsim, m = 0.1eVsim, m = 0.15eV

14.2 14.3 14.4 14.5 14.6Log(Mh/M h 1)

14

16

18

20

22

24

r 0(M

pc/h

)

(Linear theory-Observation)

experimental datalcdmm = 0.05eV

m = 0.1eVm = 0.15eVmean data

FIG. 2: (Left) comparison between the correlation length r0 from the simulation and from 3 independent richness bins (green points) and thedata mean (purple point) as a function of halo mass. The simulation points are shown with different colors. (Right) comparison between thelinear theory and the RedMapper results. The predicted neutrino mass from the linear theory is similar to that from the non-linear simulation,with a somewhat weaker mass dependence.

of the neutrinos is assumed to be entirely one species. Flatpriors are used for Σmν, Ωch2, and ln(1010As). The other cos-mological parameters are fixed: Ωbh2 = 0.02280, τ = 0.066,ns = 0.9667, H0 = 67.73 km s−1 Mpc−1. The Gelman andRubin statistics for convergence R − 1 is 0.01138. Figure 3presents the results of this analysis. Here we present 1σ and2σ level contours of the neutrino mass wrt Ωm. The inferredneutrino mass is Σmν = 0.119 ± 0.034 eV at 67% confidencelevel.

At face value, our result is in best agreement with the in-verted based hierarchy minimum mass of '0.1eV based onneutrino oscillation work, and in some tension with the mini-mum value for the normal hierarchy of ∼0.056eV at the 95%confidence level. If ongoing laboratory results from the NOvAand T2K collaborations continue to favor a normal neutrinomass hierarchy [53, 56], however, then our result may implya somewhat higher neutrino mass than minimum oscillationbased value of mν ' 0.056eV . Alternatively, the standard relicdensity underestimates the total cosmological neutrino den-sity by a factor of two, implying an additional light neutrinocontribution. This possibility, however, is not supported bymeasurements of the CMB anisotropy that strictly imply onlythree relic neutrino species [62] as in the Standard Model ofparticle physics.

We aim to improve upon our result by jointly fitting thecluster correlation function enhancement and the cluster abun-dance suppression. We will also use the simulations to make anassessment of the preferred mass range for constraining relicneutrinos, as the halo mass function is so steep (proportional

to M−4) in the cluster regime that selecting groups > 1013Mmay prove more fruitful than expanding the survey volume.An elegant group definition for the SDSS has been devisedby Zhao et al [57] for which the mean bias is b ' 2, simi-lar to the luminous red galaxy (LRG) mass scale, so that wemay use these large samples with redshift measurements for asignificant improvement.

We can also improve upon our precision for∑

mν by defin-ing a more accurate weak lensing based mass-richness relationfor a representative subset of our complete cluster sample.Such work will be possible with the upcoming wide field J-PAS survey [61], which can go beyond the careful SDSS basedwork of Simet et al (2017) to greater depth and higher angularresolution over the Northern sky.

We can see that pursuing the above practical improvementsis really well motivated given how close we are already toachieving the accuracy required to definitively distinguish be-tween the inverted and normal hierarchies and the fully massdegenerate minimum of

∑mν ' 0.15eV . Furthermore, with

future data reaching higher redshift we may also examine theredshift dependence of the combined correlation amplitude andcluster abundance evolution to test whether the lightest relicneutrino eigenstate remains relativistic until today.

Acknowledgments

We are grateful for useful conversations with Neta Bahcall,Andrew Cohen, Daniel J. Eisenstein, Lars Hernquist, Lam

6

0.306 0.308 0.310 0.312 0.314m

0

0.05

0.1

0.15

0.2

0.25m

(eV)

3.05 3.06 3.07 3.08 3.09ln(1010As)

0

0.05

0.1

0.15

0.2

0.25

m(e

V)FIG. 3: Left: the 1 ans 2σ level contours of Σmν versus Ωm, marginalising over the other relevant cosmological parameters, from our jointMCMC analysis. Right: same as left, but with Σmν versus ln(1010As) instead. The inferred neutrino mass is mν = 0.119 ± 0.034 eV at 67%confidence level.

Hui, Francisco-Shu Kitaura, Marilena Loverde, Airam Marcos-Caballero, Abraham Loeb, Enrique Martinez, Alberto Rubino,Martin Schmaltz, David Spergel, Henry Tye, Jun Qing Xiaand Francisco Villaescusa. We also thank two anonymousreferees for their insightful comments. The work of R.E. waspreviously supported by Hong Kong University through theCRF Grants of the Government of the Hong Kong SAR underHKUST4/CRF/13 and is currently supported by the Institutefor Theory and Computation at the Center for Astrophysics atHarvard University. GFS acknowledges the IAS at HKUSTand the Laboratoire APC-PCCP, Universite Paris Diderot andSorbonne Paris Cite (DXCACHEXGS) and also the finan-cial support of the UnivEarthS Labex program at SorbonneParis Cite (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02). TJB thanks IAS fr generous hospitality.

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