New Upper Limit on the Total Neutrino Mass from the 2 Degree Field Galaxy Redshift Survey

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Elgary, O., et al. (2002). New upper limit on the total neutrino mass from the 2 Degree Field Galaxy Redshift Survey.

Originally published in Physical Review Letters, 89(6). Available from: http://dx.doi.org/10.1103/PhysRevLett.89.061301

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C. , evtr,BakodHl,Eigug H H,UK 3HJ, EH9 Edingburgh Hill, Blackford servatory, 0 = 3 .Colless M. , f Λ ot ag,S.Ades ie Y S,UK 9SS, KY6 Fife, Andrews, St. Haugh, North b . . 026( 1 ≈ rit rmidpnetcosmological independent from traints ≡ < od abig B H,UK 0HA, CB3 Cambridge Road, y 6 nns ayn,cl akmatter, dark cold baryons, onents: 0 atmr,M 11-66 USA 21218-2686, MD Baltimore, , Ω n eie rmte2FGalaxy 2dF the from derived ons Ω .Sutherland W. , y aaea A915 USA 91125, CA Pasadena, gy, m . m rmvcu nryo ‘quintessence’ a or energy vacuum from 7 m b yny S 02 Australia 2052, NSW Sydney, ν 3 / .Glazebrook K. , / ≈ < Ω eV) 1 1 tnhmN72D UK 2RD, NG7 ttingham 6 ed,L29T UK 9JT, LS2 Leeds, , m .L Bridle L. S. , 0 .Collins C. , odO13H UK 3RH, OX1 ford 0 hmD13E UK 3LE, DH1 rham . ,adasmn h scalar the assuming and 5, . rmmte.Brosmk up make Baryons matter. from 3 ≈ ver 1 1 Australia 21, / 0 2 n . Ω 5o h atrcontribution matter the of 15 . 1 m ss / 2 2 .Taylor K. , 7 m h .Couch W. , esitSurvey Redshift y 1 5 ν, Mpc .M Baugh M. C. , .Jackson C. , tot < − density 1 1 . t a ith 2] where [22], eV 8 ν 11 nuiso the of units in , 8 .Dalton G. , 6 .Lewis I. , 3 .K. I. , m ν 9 m 9 is , , m tot Ω h2 = ν, , (1) matter) and a cosmological constant. As an illustration, ν 94 eV we show in Fig. 1 the power spectra for Ων = 0, 0.01, and 0.05 (all other parameters are fixed at their ‘con- where mν,tot is the sum of the neutrino mass matrix eigenvalues, and the Hubble parameter H0 at the present cordance model’ values given in the figure caption), after −1 −1 epoch is given in terms of h as H0 = 100 h kms Mpc . they have been convolved with the 2dFGRS window func- Eq. (1) assumes that all three neutrino flavours drop out tion, and their amplitudes fitted to the 2dFGRS power spectrum data. For the 32 data points, the Ων = 0-model of equilibrium at the same temperature, which is a rea- 2 2 sonable approximation. The suppression of the matter had χ = 32.9, Ων = 0.01 gives χ = 33.4, whereas the m model with Ων = 0.05 provides a poor fit to the data power spectrum P (k) on small scales is given approxi- 2 mately by (see e.g. [23]) with χ = 92.2. Clearly, the inference of the neutrino mass depends on ∆Pm our assumptions (‘priors’) on the other parameters. We ≈−8fν, (2) Pm therefore add constraints from other independent cosmological probes. The Hubble parameter has been de- where fν ≡ Ων /Ωm. Therefore even a neutrino mass as termined by the HST Hubble key project to be h = small as 0.1 eV gives a reduction in power of 5-15 %. 0.70 ± 0.07 [29], and Big Bang Nucleosynthesis gives a 2 The 2dFGRS has now measured over 220 000 galaxy constraint Ωbh = 0.020 ± 0.002 on the baryon density redshifts, with a median redshift of zm ≈ 0.11, and is the [30]. For these parameters, we adopt Gaussian priors largest existing galaxy redshift survey [17]. A sample of with the standard deviations given above. this size allows large-scale structure statistics to be mea- Perhaps the least known prior is the total matter den- sured with very small random errors. An initial estimate sity Ωm. The position of the first peak in the CMB power of the convolved, redshift-space power spectrum of the spectrum gives a strong indication that the Universe is 2dFGRS has been determined [18] for a sample of 160 000 spatially flat, i.e. Ωm +ΩΛ = 1 [19,20]. The CMB peak −1 redshifts. On scales 0.02 dark matter. Although the amplitude is increased by redshift-space distortions. the shape of the power spectrum is independent of cur- These data and their associated covariance matrix form vature, the curvature does affect the choice of priors on the basis for our analysis. Ωm, and we choose to consider flat models only. When For each model, we calculate its linear-theory mat- the constraint of a flat universe is combined with surveys ter power spectrum, and for the 2dFGRS power spec- of high redshift Type Ia supernovae [31,32], one finds trum data it is sufficiently accurate to use the fitting Ωm = 0.28 ± 0.14. However, studies of the mass-to- formulae derived in [24]. The relation between the mea- light ratio of galaxy clusters find values of Ωm as low sured galaxy power spectrum and the calculated matter as 0.15 [33], whereas cluster abundances give a range of power spectrum is given by the so-called bias parame- values Ωm ≈ 0.3 − 0.9 [34–39]. Another measurement of 2 ter b ≡ Pg(k)/Pm(k). By definition, b is in principle a Ωm ≈ 0.25 which is independent of the power spectrum function of scale, and several biasing scenarios have been of mass fluctuations and the nature of dark matter has proposed [25]. This issue is of some importance for our been derived from the baryon mass fraction in clusters of 2 analysis. For example, if the galaxy distribution is more galaxies, coupled with priors on Ωbh and h [40,41]. We biased on small scales than on large scales, a non-zero will therefore use two different priors on Ωm. The first best-fit value for fν may be obtained. However, on the is a Gaussian centered at Ωm =0.28 with standard devi- scales we consider there are theoretical reasons to expect ation 0.14, motivated by [31]. As an alternative, we use that b should tend to a constant [26]. For the 2dFGRS, a uniform (‘top hat’) prior in the range 0.1 < Ωm < 0.5. a recent analysis [27] looking for deviations from linear Given that we use the HST Key Project result [29] for h, biasing found no evidence for it. We will therefore in the Ωm < 0.5 is required to be consistent with the age of the following assume that the biasing is scale-independent. Universe [42] being greater than 12 Gyr. In particular two independent analyses [28,27] suggest Finally, the CMB data [20,43,44] are consistent with that that the data are consistent with b ≈ 1 on large the scalar spectral index of the primordial power spec- scales. We choose to avoid the complications in the nor- trum being n = 1, but we also quote results for n = 0.9 malization of the power spectra caused by redshift-space and n = 1.1, and for the case when we marginalize over distortions and the actual value of the bias parameter by n with a Gaussian prior n =1.0 ± 0.1, motivated by the leaving the amplitude of the power spectrum, from here latest data from VSA and CBI [43,44]. on denoted by A, as a free parameter, and then marginal- Results will be presented for the case of Nν = 3 equal- ize over it. mass neutrinos, but the derived upper bound on the total We shall consider here a model with four components: neutrino mass is only marginally different for Nν =1 or baryons, cold dark matter, massive neutrinos (hot dark 2. For each set of parameters, we computed the the-

2 oretical matter power spectrum, and obtained the χ2 tion over n with a prior n = 1.0 ± 0.1, the limit be- for the model given the 2dFGRS power spectrum. We comes mν,tot < 2.2 eV. Previous cosmological bounds on then calculated the joint probability distribution func- neutrino masses come from data on galaxy cluster abun- tion for fν and Γ ≡ Ωmh (which represents the shape dances [14,45], the Lyman α forest [15], and compilations of the CDM power spectrum) by marginalizing over A, h of data including the CMB, peculiar velocities, and large- and fb weighted by the priors given above. For A we scale structure [16]. They give upper bounds on the total used a uniform prior in the interval 0.5

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105 ) 3

4 Mpc 10 −3 (k) (h g P

103 0.01 0.10 k (h Mpc−1)

FIG. 1. Power spectra for Ων = 0 (solid line), Ων = 0.01 (dashed line), and Ων = 0.05 (dot-dashed line) with amplitudes fitted to the 2dFGRS power spectrum data (vertical bars) in redshift space. We have fixed Ωm = 0.3, ΩΛ = 0.7, h = 0.7, 2 Ωbh = 0.02. The vertical dashed lines limit the range in k used in the fits.

4 FIG. 2. 68 (solid line), 95 (dashed line) and 99% (dotted line) conﬁdence contours in the plane of fν ≡ Ων /Ωm and Γ ≡ Ωmh, 2 with marginalization over h and Ωbh using Gaussian priors, and over A using a uniform prior in 0.5

FIG. 3. Probability distributions, normalized so that the area under each curve is equal to one, for fν with marginalization over the other parameters, as explained in the text, for Nν = 3 massive neutrinos and n = 0.9 (dotted line), 1.0 (solid line), and 1.1 (dashed line).

Ωm = 0.28 ± 0.14 (SNIa) 0.1 < Ωm < 0.5 n fν mν,tot (eV) fν mν,tot (eV) 0.9 0.12 1.5 0.11 1.5 1.0 0.14 1.8 0.13 1.8 1.1 0.16 2.1 0.16 2.2

TABLE I. Summary of 95% conﬁdence upper bounds on fν with our two chosen priors on Ωm. The conversion of fν to mν,tot is for h = 0.7 and the central values of Ωm = 0.28 (SNIa case) and Ωm = 0.30 (uniform prior case).