Baryon Acoustic Oscillations (BAO) in the Sloan Digital Sky Survey Data Release 7 Galaxy Sample BOMEE LEE 1. Brief Introduction about BAO In our previous class we learned what is the Baryon Acoustic Oscillations(BAO). Today, in this class, I ll explain the state-of-art observation results of BAO using SDSS DR7 and others. The reading for today s class is Percival et al. 2010. BAO occur on relatively large scale, which are predominantly in the linear regime. It is therefore expected that BAO should be seen in the galaxy distribution and detect at low redshift using the 2dF Galaxy Redshift Survey (2dFGRS) and SDSS. BAO is the one of the strongest cosmological probes. The acoustic signatures in the large-scale clustering of galaxies yield three more opportunities to test the cosmological paradigm. They would 1. provide smoking-gun evidence for our theory of gravitational clustering that large-scale fluctuations grow by linear perturbation theory from z 1000 to present. 2. give another confirmation of the existence of dark matter at z 1000 since a fully baryonic model produces an effect much larger than observed. 3. provide a characteristic and reasonably sharp length scale that can be measured at a wide range of redshifts. As we learned previously, BAO present in the power spectrum of matter fluctuations after the epoch of recombination and the correlation function are calculated from the power spectrum. Figure [1] show the large-salce redshift-space correlation function of the SDSS LRG(Luminous Red Galaxies) sample. You can see the bump at 100h 1 Mpc scale. So the wavelength of the BAO is related to the comoving sound horizon at the baryon-drag epoch, r s (z d ) 153.5Mpc with WMAP5 constraints. ( The baryon drag epoch is when baryons are released from the Compton drag of the photon.)
2 Fig. 1. The Baryon Acoustic Peak in the correlation function-the acoustic peak is visible in the clustering of the SDSS LRG galaxy sample. Note that the vertical axis mixes logarithmic and linear scalings. The models are Ω m h 2 = 0.12(top line), 0.13(second line), and 0.14(third line), all with Ω b h 2 = 0.024 and n = 0.98. The bottom line shows a pure CDM model ( Ω m h 2 = 0.105), which lacks the acoustic peak. [from Eisenstein et al. (2005)]
3 2. DATA Today, I ll show BAO results with the spectroscopic SDSS DR7 sample, including both Luminous Red Galaxies (LRG) and main galaxy sample, combined with the 2dFGRS. SDSS-I & SDSS-II : 2.5m telescope in ugriz passband. DR7 sample used in analysis includes 669905 main galaxies with a median redshift z = 0.12, selected to a limiting magnitude at r band, r < 17.77. 80046 LRG which from an extension of the SDSS spectroscopic survey to higher redshifts, 0.2 < z < 0.5. Moreover, 30530 LRGs are selected from the main sample with galactic extinction k-corrected r-band absolute magnitude limit, M 0.1r < 21.8. As a result, the total LRGs are 110576 with median redshift z = 0.31. 143368 2dFGRS are also included in order to increase the volume covered at z < 0.3. The median redshift is z = 0.17. The LRG sample has been optimized for the study of structure on the large scales and show that significantly more biased than average galaxies while it contains fewer galaxies than SDSS main sample and 2dFGRS. ( See Figure [2]) 3. METHOD 1. Splitting into sub-ssamples: In order to probe the distance-redhisft relation in detail, ideally they analyze BAO measured in may independent redshift slices. By doing this, the power spectrum need to be calculated for a single distance-redshift model in a narrow redshift slice. In here, they have chosen the 6 redshift slices presented in 1 to find optimized one. As well as giving the redshift limits of the slices in Table 1, They also give the number of galaxies in each including both the 2dFGRS and the SDSS, and the effective volume, [ ] 2 n(r) P V eff = d 3 r 1 + n(r) P, (1) where n(r) is the observed comoving number density of the sample at location r and P is the expected power spectrum amplitude. To calculate V eff for our redshift slices, distances were calculated assuming a fiducial flat ΛCDM cosmology with Ω m = 0.25. For the numbers given in Table 1, They fix P = 10 4 h 3 Mpc 3, appropriate on scales k 0.15hMpc 1 for a population with bias b = 1.7.
4 Fig. 2. BAO in the SDSS power spectra-the baryon acoustic peak in the previous figure now becomes a series of oscillations in the matter power spectrum of the SDSS sample. The power spectrum is computed for both the main SDSS sample(bottom curve) and the LRG sample( top curve). [from Tegmark et al. (2006)]
5 2. Fit models to three sets of power spectra Single power spectrum for the SDSS LRG sample covering 0.15 < z < 0.5. 3 power spectra for slices, 1,3 and 6. 3 is covering range dominated by SDSS main sample and 6 slice is dominated by LRGs. 6 power spectra for 2 7. They ve measured more power spectra than previous works. 3. Calculation of Power spectra. To measure the power spectra for each catalogue, they use an unclustered random catalogue, which matches the galaxy selection. To calculate this random catalogues, the fitted the redshift distributions of the galaxy samples with a spline fit with noes separated by z = 0.0025, and the angular mask was determined using a routine based on a HEALPIX equal=area pixelization of the sphere. They used a random catalogue containing then times as many points as galaxies. For the sparse LRGs, they used one hundred times as many random points as LRGs. Using this catalogue, they constructed the luminosity dependent galaxy weighted density field in gridded space since more luminous galaxies are stronger tracers of underlying density field, contain more information about the fluctuations. After they assigned the density contrast at each grid, density field can be computed by fast Fourier transform of density contrast. And then, the power spectrum in the Fourier space is obtained. However, to correct both the survey geometry and the differences between our fiducial cosmological model( flat ΛCDM model with WMAP5 constraints.) used to convert redshift to comoving distances and the cosmological model to be tested, power spectrum was convolved with a window function, P(k) obs = dk W(k,k )P(k ) true (2) A model of the BAO was created by fitting a linear matter power spectrum, calculated using CAMB. The theoretical BAO was damped with a Gaussian model as BAO obs = P obs P nw = G damp BAO lin + (1 G damp ). (3) where, G damp = exp( 1 2 k2 D 2 damp ) and P nw is a smooth fit to observed power spectrum, P obs. For our default fits, we assume that the damping scale D damp = 10Mpch 1 at z 0.3.
6 The power spectrum measured from the data was fitted by a model constructed by multiplying this BAO model with a cubic spline. As I said before, each power spectrum model was convolved with a window function. The parametetrisation of D V (z) used to calculate the correct window function. D V (z) is the comoving distance-redshift relation necessary to constrain the comoving distance at the certain redshift defined as [ (1 + z) 2 DA 2 d z = r s (z d )/D V (z),d V (z) = cz ] 1/3. (4) H(z) where, D A is the angular diameter distance and H(z) is the Hubble parameter. The spline nodes were refitted for every cosmology ( or D V (z)) tested. In this study, they considered models for D V (z) with tow modes at z = 0.2 and z = 0.35. This constraints on d z. Figure [3] present the average power spectra for the 6 redshift slices for 70 band powers equally spaced in 0.02 < k < 0.3Mpc 1. One can see that the power spectra from the different redshift intervals are remarkably consistent with P(k) decreasing almost monotonically to small scales. 4. Results BAO are observed in the power spectra recovered from all redshift slices of the SDSS+2dFGRS sample and are shown in figure [4] plot the measured power spectra divided by the spline component of the best-fit model following the definition at equation [3]. Figure [5] describe the likelihood contour plot for fits of two D V (z) cubic spline nodes at z = 0.2 and z = 0.35 for fixed r s (z d ) = 154.7Mpc. This was calculated for default analysis using six power spectra for 6 redshift slices with a fixed damping scale of D damp = 10h 1 Mpc, and for all SDSS and non-overlapping 2dFGRS data. They measured the difference between the maximum likelihood value and the liklihood at parameters of the true cosmological model from 1000 Log-Normal mock catalogue. As one can see, dominant likelihood maximum are close to the parameters of a ΛCDM cosmology. They also show a multi-variate Gaussian fit to this likelihood surface as dashed contour in figure [5]. Using this Gaussian fit, they find that the best-fit model has d 0.2 = 0.1905 ± 0.0061 (3.2%), d 0.35 = 0.1097 ± 0.0036 (3.3%), (5)
7 Fig. 3. Average power spectra recovered from the Log-Normal catalogues(solid lines) compared with the data power spectra (solid circles with 1-]sigma errors) for the six samples in Table!1.
8 Fig. 4. BAO recovered from the data for each of the redshifts slices( solid circles with 1 σ errors). These are compared with BAO in flat ΛCDM model(solid lines).
9 Fig. 5. Likelihood contour plots for fits of two D V (z) at z = 0.2 and z = 0.35. Solid contours are plotted for 2lnL/L true < 2.3, 6.0, 9.3, which for a multi-variate Gaussian distribution correspond to 68%, 95% and 99% confidence intervals. Dashed contours show a multi-variate Gaussian fit to this likelihood surface. The values of D V for a flat ΛCDM cosmology with Ω m = 0.25, h = 0.72, &Ω b h 2 = 0.0223 are shown by the vertical and horizontal solid line.
10 where d z r s (z d )/D V (z). For a cosmological distance redshift model with unit comoving distance ˆd z the likelihood can be well approximated by a multi-variate Gaussian with covariance matrix C ( d0.2 d 0.2 d 0.2 d 0.35 d 0.35 d 0.2 d 0.35 d 0.35, ), (6) where d z d z ˆd z. C has inverse C 1 = ( 30124 17227 17227 86977 ). (7) They diagonalise the covariance matrix of d 0.2 and d 0.35 to get quantities x and y ( ) ( )( ) x 1 1.76 d0.2, (8) y 1 1.67 which gives The distance ratio f D V (0.35)/D V (0.2) is given by d 0.35 x = 0.3836 ± 0.0102 (9) y = 0.0073 ± 0.0070. (10) f = 1.67 1.76y/x 1 + y/x 1.67 8.94y, (11) For the best-fit solution we have d 0.275 = 0.362x, giving d 0.275 = 0.1390 ± 0.0037(2.7%). (12) We also have the statistically independent constraint f D V (0.35)/D V (0.2) = 1.736 ± 0.065. (13) 5. Cosmological Interpretation We now look at how their constraints work well for cosmological parameters. The sound horizon can be approximated using WMAP5 best-fit such as ( ) Ωb h 2 0.134 ( ) Ωm h 2 0.255 r s (z d ) = 153.5 Mpc. (14) 0.02273 0.1326
11 Setting r s,fid = 153.5 Mpc, and using Eq. (12) we have D V (0.275) = (1104 ± 30)[r s (z d )/r s,fid (z d )] Mpc ( ) Ωb h 2 0.134 ( ) Ωm h 2 0.255 = (1104 ± 30) Mpc, (15) 0.02273 0.1326 Using D V (0.275) = 757.4Mpch 1 for a flat Ω m = 0.282 ΛCDM cosmology, we can write h = Ω m h 2 / Ω m, and solve ( ) Ωm h 2 0.49 Ω m = (0.282 ± 0.015) 0.1326 ( ) 2 DV (z = 0.275, Ω m = 0.282), (16) D V (z = 0.275) where we have dropped the dependence of the sound horizon on Ω b h 2, which the WMAP5 data already constrains to 0.5%, 5 times below our statistical error. We can expand the ratio of distances around the best-fit Ω m = 0.282, to give D V (z = 0.275) D V (z = 0.275, Ω m = 0.282). Using this approximation, we can manipulate equation [16] to give constraints on Ω m and h. ) 0.58 ( Ωm h 2 Ω m = (0.282 ± 0.018) 0.1326 [1 + 0.25Ω k + 0.23(1 + w)], (17) ( ) Ωm h 2 0.21 h = (0.686 0.022) 0.1326 [1 0.13Ω k 0.12(1 + w)]. (18) 6. Cosmological Parameter Constraints The results of full constraints to a cosmological parameter analysis are listed in these two tables. In here, they consider 4 models, a flat universe with a cosmological constant (ΛCDM), a ΛCDM universe with curvature (oλcdm, a flat universe with a dark energy component with constant equation of state w (wcdm) and a wcdm universe with curvature (owcdm). The best-fit values with the 68/
12 SLICE z min z max N gal V eff n 1 0.0 0.5 895 834 0.42 128.1 2 0.0 0.4 874 330 0.38 131.2 3 0.0 0.3 827 760 0.27 138.3 4 0.1 0.5 505 355 0.40 34.5 5 0.1 0.4 483 851 0.36 35.9 6 0.2 0.5 129 045 0.27 1.92 7 0.3 0.5 68 074 0.15 0.67 Table 1: Parameters of the redshift intervals analyzed. V eff is given in units of h 3 Gpc 3, and was calculated as in Eq. (1) using an effective power spectrum amplitude of P = 10 4 h 3 Mpc, appropriate on scales k 0.15hMpc 1 for a population with bias b = 1.7. The average galaxy number density in each bin n is in units of 10 4 Mpc 3 h 3. parameter ΛCDM oλcdm wcdm owcdm Ω m 0.288 ± 0.018 0.286 ± 0.018 0.290 +0.018 0.019 0.286 ± 0.018 H 0 68.1 +2.2 2.1 68.6 ± 2.2 67.8 ± 2.2 68.2 ± 2.2 Ω k - -0.097 ± 0.081-0.199 +0.080 0.089 w - - -0.97 ± 0.11 0.838 +0.083 0.084 Ω Λ 0.712 ± 0.018 0.811 +0.084 0.085 0.710 +0.019 0.018 0.913 +0.092 0.082 d 0.275 0.1381 ± 0.0034 0.1367 ± 0.0036 0.1384 ± 0.0037 0.1386 ± 0.0037 D V (0.275) 1111 ± 31 1120 ± 33 1109 ± 32 1108 +32 33 f 1.662 ± 0.004 1.675 ± 0.011 1.659 ± 0.011 1.665 ± 0.011 Age (Gyr) 14.02 +0.32 0.31 14.43 ± 0.48 13.95 ± 0.36 14.38 ± 0.44 Table 2: Marginalized one-dimensional constraints (68%) for BAO+SN for flat ΛCDM, ΛCDM with curvature (oλcdm), flat wcdm (wcdm), and wcdm with curvature (owcdm). The non-standard cosmological parameters are d 0.275 r s (z d )/D V (0.275) and f D V (0.35)/D V (0.2). We have assumed priors of Ω c h 2 = 0.1099 ± 0.0063 and Ω b h 2 = 0.02273 ± 0.00061, consistent with WMAP5-only fits to all of the models considered here. We also impose weak flat priors of 0.3 < Ω k < 0.3 and 3 < w < 0.
13 6.1. SN + BAO + CMB prior likelihood fits The best-fit value of Ω m ranges from 0.286 to 0.290, with the 68% confidence interval, ±0.018, while the mean value of H 0 varies between 67.8km/s/Mpc and 68.6km/s/Mpc, and the 68% confidence interval remains ±2.2km/s/M pc throughout the four models. The small difference between the errors in Table 2 and those expected is caused by the supernova data helping to constrain Ω m and H 0 slightly. For the owcdm model, the weak prior on Ω k leads to an apparent constraint on w, but these errors depend strongly on the prior. The data are compared with the best-fit ΛCDM model in Figure [6]. In the bottom panel, they plot D V (z) over D V (0.2). in the middle panel, they plot r s (z d )/D V (z), where we now have to model the comoving sound horizon at the drag epoch. In the top panel we include a constraint on the sound horizon projected at the last-scattering surface as observed in the CMB. Marginalising over the set of flat ΛCDM models constrained only by the WMAP5 data gives r s (z d )/S k (z d ) = 0.010824 ± 0.000023, where S k (z d ) is the proper distance to the baryon-drag redshift z d = 1020.5, as measured by WMAP5 team. 6.2. CMB + BAO likelihood fits Next one is the constraints from BAO measurements combined with the full WMAP5 likelihood (calculation with COSMOMC). Results for the four models are presented in Table 3. For the ΛCDM model, we find Ω m = 0.278 ± 0.018 and H 0 = 70.1 ± 1.5Km/s/Mpc, with errors significantly reduced compared to the WMAP5 alone analysis (Ω m = 0.258±0.03 and H 0 = 70.5 +2.6 2.7Km/s/M pc). Figure 7 shows WMAP5+BAO constraints on cosmological parameter for 4 models. The WMAP5 results alone tightly constrain Ω m h 2 in all of these models (dashed lines). Allowing w 1 relaxes the constraint on Ω m from the BAO measurement, and in addition allowing Ω k 0 relaxes the constraint even further. The impact on the constraints on Ω m and H 0 is shown in the lower right panel. All of the contours lie along the banana with Ω m h 2 fixed from the CMB. In the oλcdm model, the combination of scales measured by the CMB and the BAO tightly constrain the curvature of the universe: Ω k = 0.007 +0.006 0.007. The constraints on Ω m and H 0 in this model are well described by Eqns. (17) & (18), while in the wcdm cosmology they degrade because w is not well-constrained by the low redshift BAO information alone. When the parameter space is opened to both curvature and w, the WMAP5 data are not
14 Fig. 6. The BAO constraints (solid circles with 1σ errors), compared with the best-fit ΛCDM model. The three panels show different methods of using the data to constrain models.
15 Fig. 7. WMAP5+BAO constraints on Ω m h 2, Ω m, and H 0 for ΛCDM (solid black contours), oλcdm (shaded green contours), wcdm (shaded red contours), and owcdm (shaded blue contours) models. Throughout, the solid contours show WMAP5+LRG ΛCDM constraints. The first three panels show WMAP5 only constraints (dashed contours) and WMAP5+BAO constraints (colored contours) in the Ω m h 2 - Ω m plane as the model is varied. In the lower right we show all constraints from WMAP5+BAO for all four models in the Ω m H 0 plane, which lie within the tight Ω m h 2 0.133 ± 0.006 WMAP5-only constraints.
16 Fig. 8. For the owcdm model we compare the constraints from WMAP5+BAO (blue contours), WMAP5+SN (green contours), and WMAP5+BAO+SN (red contours). Dashed and solid contours highlight the 68% confidence intervals for the WMAP5+BAO and WMAP5+SN models respectively.
17 able to eliminate the degeneracy between Ω m and w in the BAO constraint. The constraints relax to Ω m = 0.240 +0.044 0.043 and H 0 = 75.3 ± 7.1km/s/Mpc; Ω k = 0.013 ± 0.007 is still well-constrained but w is not (see Fig. 7). Figure [8] show us that we can recover the tight constraints on Ω m and H o for owλcdm universe by combining SN+WMAP5+BAO. Eisenstein D.J., et al., 2005, ApJ, 633, 560 Percival W.J., et al., 2007a, ApJ, 657, 51 Percival W.J., et al., 2007b, ApJ, 657, 645 REFERENCES Percival W.J., Cole S., Eisenstein D., Nichol R., Peacock J.A., Pope A., Szalay A., 2007c, MNRAS, 381, 1053 Tegmark, M., et al., 2006, PRD, 74, 123507 This preprint was prepared with the AAS L A TEX macros v5.2.
18 parameter ΛCDM oλcdm wcdm owcdm owcdm+sn owcdm+h 0 owcdm+sn+h 0 Ω m 0.278 ± 0.018 0.283 ± 0.019 0.283 ± 0.026 0.240 +0.044 0.043 0.290 ± 0.019 0.240 +0.025 0.024 0.279 ± 0.016 H 0 70.1 ± 1.5 68.3 +2.2 2.1 69.3 ± 3.9 75.3 ± 7.1 67.6 ± 2.2 74.8 ± 3.6 69.5 ± 2.0 Ω k - 0.007 +0.006 0.007 - -0.013 ± 0.007-0.006 ± 0.008-0.014 ± 0.007-0.003 ± 0.007 w - - -0.97 ± 0.17 1.53 +0.51 0.50-0.97 ± 0.10 1.49 +0.32 0.31-1.00 ± 0.10 Ω Λ 0.722 ± 0.018 0.724 ± 0.019 0.717 ± 0.026 0.772 ± 0.048 0.716 ± 0.019 0.773 ± 0.029 0.724 ± 0.018 100Ω b h 2 2.267 ± 0.058 2.269 ± 0.060 2.275 ± 0.061 2.254 +0.062 0.061 2.271 ± 0.061 2.254 +0.061 0.062 2.284 ± 0.061 τ 0.086 ± 0.016 0.089 ± 0.017 0.087 ± 0.017 0.088 ± 0.017 0.089 ± 0.017 0.088 ± 0.017 0.089 +0.017 0.018 n s 0.961 ± 0.013 0.963 ± 0.014 0.963 ± 0.015 0.958 ± 0.014 0.963 ± 0.014 0.957 ± 0.014 0.964 ± 0.014 ln(10 10 A 05 ) 3.074 +0.040 0.039 3.060 ± 0.042 3.070 ± 0.041 3.062 +0.042 0.043 d 0.275 0.1411 ± 0.0030 0.1387 ± 0.0036 0.1404 +0.0036 0.0035 D V (0.275) 1080 ± 18 1110 +32 31 3.062 +0.041 0.042 0.1382 ± 0.0037 0.1379 ± 0.0036 0.1387 +0.0036 0.0037 3.062 ± 0.042 3.072 ± 0.042 0.1402 +0.0033 0.0034 1089 ± 31 1111 ± 33 1115 ± 32 1107 ± 31 1091 +27 28 f 1.6645 ± 0.0043 1.6643 ± 0.0045 1.661 ± 0.019 1.72 ± 0.056 1.660 ± 0.011 1.7187 +0.0337 0.0334 Age (Gyr) 13.73 ± 0.12 14.08 ± 0.33 13.76 +0.15 0.14 Ω ch 2 0.1139 ± 0.0041 0.1090 +0.0060 0.0061 Ω tot - 1.007 +0.006 0.007 σ 8 0.813 ± 0.028 0.787 ± 0.037 0.792 +0.081 0.082 1.6645 ± 0.0107 14.49 ± 0.52 14.04 ± 0.36 14.48 ± 0.48 13.86 +0.34 0.33 0.1122 +0.0068 0.0069 0.1107 +0.0063 0.0062 0.1096 +0.0061 0.0062 0.1108 +0.0060 0.0061 0.1115 ± 0.0061-1.013 ± 0.007 1.006 ± 0.008 1.014 ± 0.007 1.003 ± 0.007 0.907 ± 0.117 0.780 +0.052 0.053 0.904 ± 0.074 0.801 +0.053 0.052 Table 3: Marginalized one-dimensional constraints (68%) for WMAP5+BAO for flat ΛCDM, ΛCDM with curvature (oλcdm), flat wcdm (wcdm), wcdm with curvature (owcdm), and owcdm including constraints from supernovae. The non-standard cosmological parameters constrained by the BAO measurements are d 0.275 r s (z d )/D V (0.275) and f D V (0.35)/D V (0.2).