Study of galaxy distributions with SDSS DR14 data and measurement of neutrino masses

We study galaxy distributions with Sloan Digital Sky Survey SDSS DR14 data and with simulations searching for variables that can constrain neutrino masses. To be specific, we consider the scenario of three active neutrino eigenstates with approximately the same mass, so $\sum m_\nu = 3 m_\nu$. Fitting the predictions of the $\Lambda$CDM model to the Sachs-Wolfe effect, $\sigma_8$, the galaxy power spectrum $P_\textrm{gal}(k)$, fluctuations of galaxy counts in spheres of radii ranging from $16/h$ to $128/h$ Mpc, BAO measurements, and $h = 0.678 \pm 0.009$, in various combinations, with free spectral index $n$, and free galaxy bias and galaxy bias slope, we obtain consistent measurements of $\sum m_\nu$. The results depend on $h$, so we have presented confidence contours in the $(\sum m_\nu, h)$ plane. A global fit with $h = 0.678 \pm 0.009$ obtains $\sum m_\nu = 0.719 \pm 0.312 \textrm{ (stat)}^{+0.055}_{-0.028} \textrm{ (syst)}$ eV, and the amplitude and spectral index of the power spectrum of linear density fluctuations $P(k)$: $N^2 = (2.09 \pm 0.33) \times 10^{-10}$, and $n = 1.021 \pm 0.075$. The fit also returns the galaxy bias $b$ including its scale dependence.


I. INTRODUCTION
We measure neutrino masses by comparing the predictions of the ΛCDM model with measurements of the power spectrum of linear density perturbations P (k). We consider three measurements of P (k): (i) the Sachs-Wolfe effect of fluctuations of the Cosmic Microwave Background (CMB) which is a direct measurement of density fluctuations [1,2]; (ii) the relative mass fluctuations σ 8 in randomly placed spheres of radius r s = 8/h Mpc with gravitational lensing and studies of rich galaxy clusters [2,3]; and (iii) measurements of P (k) inferred from galaxy clustering with the Sloan Digital Sky Survey [4][5][6]. Baryon Acoustic Oscillations (BAO) were considered separately [7,8] and are not included in the present study, except for the final combinations.
To be specific, we consider three active neutrino eigenstates with nearly the same mass, so m ν = 3m ν . This is a useful scenario to consider because the current limits on m 2 ν are much larger than the mass-squared-differences ∆m 2 and ∆m 2 21 obtained from neutrino oscillations [3].
Figures 1 to 4 illustrate the problem at hand. Figures 1, 2, and 3 present measurements of the "reconstructed" galaxy power spectrum P gal (k) obtained from the SDSS-III BOSS survey [4], while Fig. 4 presents the corresponding "standard" P gal (k). The "reconstructed" P gal (k) is obtained from the directly measured "standard" P gal (k) by subtracting peculiar motions to obtain the power spectrum prior to non-linear clustering. Also shown are various fits to this data (with floating normalization), and to measurements of the Sachs-Wolfe effect, and σ 8 . The Sachs-Wolfe effect normalizes P (k), within its uncertainty, in the approximate range of log 10 (k/(h Mpc −1 ) from -3.1 to -2.7, while σ 8 is most sensitive to the range -1.3 to -0. 6. Full details will be given in the main body of this article.  [4] ("reconstructed") with b 2 P (k) obtained from a fit of Eq. (5) with mν = 0 eV to the Sachs-Wolfe effect, σ8, and P gal (k). The fit obtains A = 8738 Mpc 3 , keq = 0.068h Mpc −1 , η = 4.5, and b 2 = 1.8, with χ 2 = 24.7 for 19 degrees of freedom. Also shown for comparison is the curve with the same parameters, except mν = 0.6 eV.
The fit in Fig. 1 corresponds to the function where w ≡ k/k eq . Unless otherwise noted, we take the Harrison-Zel'dovich index n = 1 which is close to observations. The parameters A, η, and k eq , as well as the normalization factor b 2 , are free in the fit. The uncertainties of two data points that fall on BAO peaks are multiplied by three (since BAO is not included in P ′ (k)).
Also shown in Fig. 1 is the suppression of P (k) for k greater than k nr = 0.018 · Ω 1/2 m m ν 1eV 1/2 h Mpc −1 (2) due to free-streaming of massive neutrinos that can not cluster on these small scales, and, more importantly, to the slower growth of structure with massive neutrinos [9]. The suppression factor for k ≫ k nr for one massive neutrino, or three degenerate massive neutrinos, is where f ν = Ω ν /Ω m [9]. Ω m is the total (dark plus baryonic plus neutrino) matter density today relative to the critical density, and includes the contribution Ω ν of neutrinos that are non-relativistic today. Ω ν = h −2 m ν /93.04 eV for three left-handed plus righthanded Majorana neutrino eigenstates, or three eigenstates of left-handed Dirac neutrinos plus three righthanded Dirac anti-neutrinos, that are non-relativistic today (right-handed Dirac neutrinos and left-handed Dirac anti-neutrinos are assumed to not have reached thermal and chemical equilibrium with the Standard Model particles). We take f (k, m ν ) = 1 for k < k nr /0.604, and for k > k nr /0.604 and m ν < 1.1 eV, for galaxy formation at a redshift z = 0.5 [9]. Figure 2 is the same as Fig. 1 except that the function with m ν = 0.6 eV is fit. We see that the parameter m ν is largely degenerate with the parameters A, η, and k eq , so that only a weak sensitivity to m ν is obtained unless we are able to constrain k eq . The power spectrum P ′ (k) of Eq. (1) neglects the growth of structure inside the horizon while radiation dominates.
To obtain P (k), we would like to measure the density ρ( r, z) at redshift z, but we only have information on the peaks of ρ( r, z) that have gone non-linear collapsing into visible galaxies. How accurate is the measurement of P (k) with galaxies? The measurement of P gal (k) in Ref. [4] is based on a procedure described in [10] based on "the usual assumption that the galaxies form a Poisson sample [11] of the density field". In other words, the assumption is that the number density of point galaxies n(x) is equal to its expected meann (which depends on the position dependent galaxy selection criteria), modulated by the perturbation of the density field: Both sides of this equation are measured or calculated at the same length scale, and at the same time. The "galaxy bias" b is explicitly assumed to be scale invariant. If we choose a region of space such thatn is constant, then the galaxy power spectrum P gal (k) (derived from n( x)/n) should be proportional, under the above assumption, to the power spectrum of linear density perturbations P (k) (derived from δ c ( x)) up to corrections: It is due to this bias b that we have freed the normalization of the measured P gal (k) in the fits corresponding to Figs. 1 to 4.
In the following Sections we study galaxy distributions with SDSS DR14 data and with simulations, in order to understand their connection with the underlying power spectrum of linear density fluctuations P (k). In the end we return to the measurement of neutrino masses.

II. THE HIERARCHICAL FORMATION OF GALAXIES
This Section allows a precise definition of P (k), and an understanding of the connection between P (k) and P gal (k). We generate galaxies as follows (see [12] for full details). The evolution of the Universe in the homogeneous approximation is described by the Friedmann equation The expansion parameter a 1 (t) has been normalized to 1 at the present time t 0 : a 1 (t 0 ) = 1. H 0 has been normalized so E(1) = 1. Therefore H 0 is the present Hubble expansion rate. With these normalizations we have Ω r + Ω m + Ω k + Ω Λ ≡ 1. The matter density is ρ m (t) = Ω m ρ c /a 3 1 , where ρ c = 3H 2 0 /(8πG N ) is the critical density of the Universe. We are interested in the period after the density of matter exceeds the density of radiation. For our simulations we assume flat space, i.e. Ω k = 0, we neglect the radiation density Ω r , take Ω Λ = 0.719 constant [7], and the present Hubble expansion rate H 0 = 100h km s −1 Mpc −1 with h = 0.678 [3]. The solution to Eq. (12) with these parameters is shown by the curve "a 1 " in Fig. 5. The present age of the universe with these parameters is t 0 = 14.1 Gyr.
Setting Ω Λ = 0 we obtain the critical universe with expansion parameter also shown in Fig. 5. We note that a 2 (t 0 ) ≡ a 20 = 0.846. Let us now add density fluctuations to this critical universe and consider a density peak. The growing mode for this density peak is obtained by adding a negative Ω k to the critical Universe. This prescription is exact if the density peak is spherically symmetric. An example with "expansion parameter" a 4 is presented in Fig. 5.
Note that a 4 grows to maximum expansion and then collapses to zero at time , and, in our model [12], a galaxy forms. In the example of Fig.  5 the galaxy forms at redshift z = 0.5. a 3 (t) is the linear approximation to a 4 (t). In the linear approximation for growing modes the density fluctuations relative to ρ 2 grow in proportion to a 2 (t): while δ c ≪ 1. At the time t 4 , when the galaxy forms, δ c ≡ (ρ 3 − ρ 2 )/ρ 2 = 1.69 in the linear approximation (which has already broken down).
In the linear approximation the density due to Fourier components of wavevector | k| ≤ k I is where ϕ k are random phases. The sum of the Fourier series is over comoving wavevectors that satisfy periodic boundary conditions in a rectangular box of volume V = L x L y L z : where L x = n max L 0 , L y = m max L 0 , L z = l max L 0 , n, m, l = 0, ±1, ±2, ±3, · · ·, and where k max = 2π/L 0 , and I = 1, 2, ...I max .
Inverting Eq. (16) obtains where X ≡ x(t)/a 2 (t) is the comoving coordinate in the linear approximation. The power spectrum of density fluctuations is defined in the linear approximation corresponding to a 3 , and is approximately independent of V for large V . Averaging over k in a bin of k ≡ | k| obtains P (k). Note that Each term in this equation is approximately independent of V . The Fourier transform of the power spectrum is the correlation function: The generation of galaxies at time t proceeds as follows. We start with I = 2, calculate δ c ( x, t, I), and search for local maximums of δ c ( x, t, I) inside a comoving volume L x L y L z . If a maximum exceeds 1.69 we generate a galaxy of radius and dark plus baryonic plus neutrino mass if it "fits", i.e. if it does not overlap previously generated galaxies. I is increased by 1 unit to generate galaxies of a smaller generation, until I = I max is reached. See Fig. 6.
x/a(t) The hierarchical formation of galaxies [12]. Three Fourier components of the density in the linear approximation are shown. Note that in the linear approximation δc ≡ (δ3 − δ2)/δ2 ∝ a(t). When δc reaches 1.69 in the linear approximation the exact solution diverges and a galaxy forms. As time goes on, density perturbations grow, and groups of galaxies of one generation coalesce into larger galaxies of a new generation as shown on the right.

The peculiar velocity of the generated galaxies is
and their peculiar displacement is x + x pec is the proper coordinate of a galaxy at the time t of its generation, and a 2 ≡ a 2 (t). The comoving coordinate of this galaxy, i.e. its position extrapolated to the present time, is the corresponding ( x+ x pec )/a 1 (t).
x pec causes the difference between the data points P gal (k) in Figs. 3 and 4 at large k. Fig. 6 that the formation of galaxies is hierarchical: small galaxies form first, and, as time goes on, density perturbations grow, and groups of galaxies coalesce into larger galaxies in an ongoing process until dark energy dominates and the hierarchical formation of galaxies comes to an end. The distribution of galaxies of generation I depend only on P (k) for k < 2πI/L 0 . Also, luminous galaxies occupy a total volume (luminous plus dark) less than 1/2.69 of space.

Note in
Neutrinos with 0 < m ν < 1.17 eV become nonrelativistic after the densities of radiation and matter become equal, as illustrated in Fig. 7.

III. FLUCTUATION AMPLITUDE σ8
σ 8 is the root-mean-square fluctuation of total mass relative to the mean in randomly placed volumes of radius r s = 8h −1 Mpc. We use a "gaussian window function" which smoothly defines a volume Note that The Fourier transform of W (r) is Then An alternative window function is the "top hat" function f (r) = 3/(4πr 3 s ) for r < r s , and f (r) = 0 for r > r s . Then Direct measurements obtain [3] (33) 80% of σ 2 8 is due to k/h in the range 0.05 to 0.25 Mpc −1 . For comparison, from the 6-parameter ΛCDM fit [3], σ 8 = 0.815 ± 0.009.

IV. THE SACHS-WOLFE EFFECT
The spherical harmonic expansion of the CMB temperature fluctuation is Averaging over m obtains C l ≡ |a lm | 2 . The variable that is measured is [2] ∆T (n 1 )∆T ( For 7 < l < 20 the dominant contribution to C l is from the Sachs-Wolfe effect [1][2][3]. This range corresponds to 0.0007 Mpc −1 < k/h < 0.002 Mpc −1 . The Sachs-Wolfe effect relates temperature fluctuations of the CMB to perturbations of the gravitational potential φ [2]: When expressed as a function of comoving coordinates, φ( X) is independent of time when matter dominates. The primordial power spectrum of gravitational potential fluctuations is assumed to have the form [2] The relation between N 2 φ and N 2 is N 2 φ = 9N 2 /25 [2]. In the present analysis, unless otherwise stated, we assume the Harrison-Zel'dovich power spectrum with n = 1, which is close to observations [3]. For 7 l 20, [2] where the "quadrupole moment" Q is measured to be from the 1996 COBE results (see list of references in [2]). Then, for P ′ (k), and for P ′′ (k), independently of m ν . Detailed integration obtains results within 10% for 5 < l < 18.

V. DATA AND SIMULATIONS
The data are obtained from the publicly available SDSS DR14 catalog [5,6], see acknowledgement. We consider objects classified as GALAXY, with redshift z in the range 0.4 to 0.6, with redshift error zErr < 0.002, passing quality selection flags. We further select galaxies in the northern galactic cap, in a "rectangular" volume with L x = 400 Mpc along the line of sight (corresponding to redshift z ≈ 0.5 ± 0.046), L y = 3800 Mpc (corresponding to an angle 86 0 across the sky), and L z = 1400 Mpc (corresponding to an angle 32 0 ). In total 222470 galaxies pass these selections. The distributions of these galaxies are shown in Fig. 8.
Some definitions are in order. For data we define the absolute red magnitude of a galaxy MAGr at redshift z as the SDSS DR14 variable -modelMag r corrected to the reference redshift 0.35. Similarly, we define the absolute green magnitude of a galaxy MAGg at redshift z as the SDSS DR14 variable -modelMag g corrected to the reference redshift 0.35. For a simulated galaxy we define the absolute magnitude MAG ≡ −19 + 2.5 log 10 (M/10 16 M ⊙ ), where M is defined by Eq. (24). Note that MAGr and MAGg are derived from observed luminosities, while MAG is derived from the total (baryonic plus dark plus neutrino) mass of the simulation. These quantities can only be compared if the luminosity-to-mass ratio is known. The number of galaxies per unit volume depends on the limiting magnitude of the survey, or on I max of the simulation.

VI. DISTRIBUTIONS OF GALAXIES IN SDSS DR14 DATA AND IN SIMULATION
We would like to obtain P (k) from Eqs. (19) and (20). Unfortunately we do not have access to the relative density fluctuation δ c ( x, t). Instead we have access to the positions of galaxies and their luminosities. The relation between luminosity and mass of galaxies depends on many variables and is largely unknown, so we focus on the information contained in the positions of galaxies. Let be the number density of point galaxies at redshift z as a function of the comoving coordinate X ≡ x(t)/a(t). We have applied periodic boundary conditions in a comoving volume V = L x L y L z , so k ′ has the discrete values of Eq. (17). n( X) is real, so ∆ − k ′ = ∆ * k ′ . The number of galaxies in V is N gal =nV . To invert Eq. (42), we  multiply it by exp (−i k · X), integrate over V , and obtain a sum over galaxies j: The first term on the right hand side of Eq. (43) is the result of a coherent sum of terms corresponding to mode k. The second term is the result of an incoherent sum which we have approximated to N   φ is arbitrary. We define the "galaxy power spectrum" and obtain   The transition between signal and noise occurs at log 10 (P gal (k)/h −3 Mpc 3 ) ≈ 3.47 for our data sample, and ≈ 3.49 for our reference simulation. To test these ideas we can select a narrow range of MAGr, MAGg, or MAG to shift the noise upwards, compare Figs. 9, 10, and 11 (which plot the first term on the right hand side of Eq. (45) and include the noise at large k).
Averaging over k in a bin of k ≡ | k| obtains P gal (k). The factor V is inserted so that P gal (k) becomes independent of the arbitrary choice of V for large V . The function P gal (k) defines statistically the distribution of galaxies. The variables k in Eqs. (16) and (45) should not be confused: there is not necessarily a one-to-one relation between them.
Results for data are presented in Figs. 9, 10, and 11. We note that the galaxy bias b depends on MAGr and MAGg. Even tho N gal ≫ N k , N k > N 1/2 gal at small k.
For this reason P gal (k) in Fig. 9 extends to higher k than in Figs. 10 and 11 before saturating with noise. Figure  12 presents the noise subtracted galaxy power spectrum P gal (k), obtained from Fig. 9, compared with P (k) calculated with the indicated parameters. Their ratio is the bias b 2 .
Results for the simulations are presented in Figs. 13, 14, and 15. In Fig. 15 we compare the reference simulation with P ′ (k), with simulations with P ′ (k) · (− log 10 (k/h Mpc −1 )) ("steeper slope"), or P ′ (k)/(− log 10 (k/h Mpc −1 )) ("less slope"). Note that the function − log 10 (k/h Mpc −1 ) varies between ≈ 1.3 to ≈ 0.5 in the region of interest. We observe, qualitatively, that the slope of P (k) has a larger effect on P gal (k) than the amplitude A. A comparison of the simulations in Fig.  15 with P gal (k) from data in Fig. 9 favors a power spectrum P (k) "steeper" than in the reference simulation. The reference simulation has parameters of P (k) similar to the ones obtained from the fit in Fig. 1 which assumes scale invariant b, and m ν = 0 eV. The reference simulation is also similar to the fit " m ν = 0.014 eV" in Fig. 12 (taken from Fig. 3 which assumes scale invariant b). A steeper P (k) implies m ν > 0 as shown in Fig.  12 by the curve " m ν = 0.719 eV", and corresponds to a bias b with positive slope as in Eq. (55) below.

VII. LUMINOSITY AND MASS DISTRIBUTIONS OF GALAXIES
Distributions of MAGr and MAGg from data, and MAG from several simulations are presented in Figs. 16 and 17. From these figures it is possible to obtain the "mean" luminosity-to-mass ratios. We note that these figures do not show useful sensitivity to m ν .

VIII. TEST OF SCALE INVARIANCE OF THE GALAXY BIAS b
In this Section we test the scale invariance of the bias b defined in Eq. (11). To do so, we count galaxies in an array of N s = N x × N y spheres of radii r s , and obtain their meanN , and their root-mean-square (rms). All spheres have their center at redshift z = 0.5 to ensure the homogeneity of the galaxy selections. The results for   Table I. The (rms) 2 has a contribution σ 2 from P (k), and a contributionN from statistical fluctuations: We compare σ/N obtained from galaxy counts, with the relative mass fluctuations σ rs/h obtained from Eqs. (6) and (31). The ratio of these two quantities divided by a I: Mean galaxy countsN in spheres of radius rs. All spheres have their center at redshift z = 0.5. The number of spheres is Ns = Ny × Nz. Note that the observed root-mean-square (rms) fluctuation relative to the meanN is larger than the corresponding statistical fluctuation, i.e. rms/N > 1/ √N . σ rs/h is calculated with Eqs. (6) and (31) with N 2 chosen so σ8 = 0.770 to set the scale for b (e.g. N 2 = 0.8472 × 10 −10 for mν = 0 eV, or N 2 = 1.3575 × 10 −10 for mν = 0.6 eV). Both the galaxy counts and σ rs/h are obtained with the top-hat window function. The true standard deviation is obtained from σ 2 = rms 2 −N ± 2σ N /Ns. The measured "bias" is defined as b ≡ (σ/N )/(0.779 · σ rs/h ). The last column is the χ 2 of the five b's of spheres withN > 1, assuming these b's are scale invariant. h = 0.678, n = 1.0, and Ωm = 0.281. correction factor C(Ω Λ /(Ω m (1 + z) 3 ))/(C(Ω Λ /Ω m )(1 + z)) = 0.779 [2] is the bias b.
The measured bias b is a function of r s , m ν , h and the spectral index n. Results for h = 0.678 and n = 1 are presented in Table I. The last column is the χ 2 of the five b's of spheres withN > 1, assuming these b's are scale invariant with respect to their weighted average. Additional measurements of χ 2 are presented in Fig. 18. Assuming that b is scale invariant we obtain m ν = 0.939 + 0.035 · δh + 0.089 · δn ± 0.008 eV, (47) with minimum χ 2 = 3.2 for four degrees of freedom. We have defined δh ≡ (h − 0.678)/0.009, and δn ≡ (n − 1)/0.038.
In conclusion, the galaxy bias b is scale invariant within the statistical uncertainties of b presented in Table I, provided m ν satisfies Eq. (47), else scale invariance is broken. Note in Table I that the variation of b with scale depends on m ν .

IX. MEASUREMENT OF NEUTRINO MASSES WITH THE SACHS-WOLFE EFFECT AND σ8
We return to the measurement of neutrino masses. Since the galaxy bias b may be scale dependent, in this  Section we exclude measurements of P gal (k) with galaxies. The ΛCDM model is described by Eq. (6) that has three free parameters: N 2 , n, and m ν . We keep n fixed. We vary the two parameters N 2 and m ν to minimize a χ 2 with two terms corresponding to two observables: the Sachs-Wolfe effect (N 2 from Eq. (41)), and σ 8 given by Eq. (33). We therefore have zero degrees of freedom. The result is a function of h, Ω m , and the spectral index n, so we define δh ≡ (h − 0.678)/0.009 [3], δΩ m ≡ (Ω m −0.281)/0.003 [7], and δn ≡ (n−1)/0.038 [3], and obtain m ν = 0.595 + 0.047 · δh + 0.226 · δn + 0.022 · δΩ m ±0.225 (stat) Note that in the "6 parameter ΛCDM fit" [3], which assumes m ν = 0.06 eV, n = 0.968 ± 0.006. Here, and below, the systematic uncertainties are obtained by repeating the fits with the top-hat window function instead of the gaussian window function for σ 8 (and for σ/N if applicable), and also with σ 8 = 0.815 ± 0.009 obtained with the "6 parameter ΛCDM fit" [3], instead of σ 8 from direct measurements, Eq. (33).
The fit of Eq. (48) is compared with measurements of P gal (k) obtained from the SDSS-III BOSS survey [4] in Fig. 19. It is interesting to note that the discrepancy, i.e. the drop of P (k) in the range −1.6 < log 10 (k/h Mpc) < −1.3, is also observed in Fig. 12.
X. MEASUREMENT OF NEUTRINO MASSES WITH THE SACHS-WOLFE EFFECT, σ8 AND P gal (k) We repeat the fit of Fig. 3, which includes the "reconstructed" SDSS-III BOSS P gal (k) measurements [4], but this time we allow the galaxy bias b to depend on scale: b ≡ b 0 + b 1 log 10 (k/h Mpc −1 ). Minimizing the χ 2 with respect to m ν , N 2 , n, h = 0.678 ± 0.009, b 0 , and b 1 , we obtain m ν = 0.80 ± 0.23 eV, We repeat the measurements of m ν of Section IX but add 4 more experimental constraints: σ/N of galaxy counts in spheres of radius r s = 16/h, 32/h, 64/h, and 128/h Mpc, which are listed in Table I. Spheres of radius 8/h Mpc were not considered because they haveN < 1. Spheres of radius 256/h Mpc were excluded because there are only 4 spheres of this radius, and the difference between the rms for the top-hat and gaussian window functions turns out to be large (while consistent results are obtained for the other radii). We add two more parameters to be fit: b 0 and b s which define the bias b = b 0 −i s b s , with i s = 0, 1, 2, 3 for r s = 16/h, 32/h, 64/h, and 128/h Mpc, respectively. Note that we do not obtain a good fit with fixed bias b = b 0 , and so have introduced a "bias slope" b s .

XII. CONCLUSIONS
We have studied galaxy distributions with Sloan Digital Sky Survey SDSS DR14 data and with simulations searching for variables that can constrain neutrino masses. Fitting the predictions of the ΛCDM model to the Sachs-Wolfe effect, σ 8 , P gal (k), fluctuations of galaxy counts in spheres of radii ranging from 16/h to 128/h Mpc, BAO measurements, and h = 0.678 ± 0.009, in various combinations, with free spectral index n, and free galaxy bias and galaxy bias slope, we obtain consistent measurements of m ν . The uncertainty of m ν is dominated by the uncertainty of h, so we have presented confidence contours in the ( m ν , h) plane.

XIII. ACKNOWLEDGEMENT
Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington.