Features in the matter power spectrum, such as baryon acoustic oscillations (Peebles & Yu 1970; bond & Efstathiou 1984; Holtzman 1989; Hu & Sugiyama 1996; see Figure 1), can be used as CMB-calibrated standard rulers for determining the angular-diameter distance r(z) and constraining dark energy (Eisenstein, Hu, & Tegmark 1998; Linder 2003; Seo & Eisenstein 2003, hereafter SE03; Angulo et al. 2005). The lowest k peaks in the series of baryonic oscillations have been observed in spectroscopic redshift surveys (Eisenstein et al. 2005; Cole et al. 2005; Huetsi 2005), but the samples are not large enough to provide sufficient precision for significant constraints on r(z).
Figure 1 Baryon acoustic oscillations. Left panel: Expected results for 4 redshift bins from the LSST survey. We divide the survey into 7 slices from z = 0.2 to 3. Each slice is 450-570 h-1Mpc thick. We use a galaxy distribution of n(z) = 640 z2 e-z / 0.35 arcmin-2, which gives rise to a projected galaxy counts of 54.4 per arcmin2. To avoid nonlinear effects, we only consider k ranges where fluctuations in logarithmic k-bins are less than 0.25, i.e. Δ2(k) < 0.25. We divide the power spectra by a reference power spectrum that bears no baryon oscillations and shift them upward in increments of 0.1 for better readability. The rms photometric redshift error takes the form of σz = σz0 (1 + z). The statistical errors are dominated by the sample variance on large scales, which is more pronounced at low redshift because of smaller volumes. The shot noise is more important at high redshift where galaxy number density is low, but it is compensated by larger volumes. Right panel: The same as the left panel, but for a spectroscopic survey over 1000 deg2.
Several papers have studied the prospects of measuring the baryon acoustic oscillation from photometric surveys (SE03; Dolney, Jain, & Takada 2004; Blake & Bridle 2005; Glazebrook & Blake 2005; Linder 2005). The main advantage of a photometric redshift survey, such as the LSST survey, is the wide coverage, which reduces the sample variance error, and deep photometry, which leads to more galaxies and lower shot noise. One may quantify this advantage with the effective survey volume Veff = ∫d3r n2(r)P2(k) [1 + n(r)P(k)]-2 (Feldman, Kaiser, & Peacock 1994; Tegmark 1997), where P(k) is the power spectrum, and n(r) is the galaxy number density. Figure 2 compares the effective volumes of several existing surveys with that of LSST. One sees immediately the unparalleled large effective volume LSST probes.
Figure 2 Effective survey volumes. The survey data, except that of LSST, are from Eisenstein et al. (2005). LSST survey parameters are the same as in Figure 1. The curve labeled with 0.01n(r) applies to the case where a sub sample of galaxies is selected for statistics.
Challenges for photometric redshift surveys include redshift errors, dust extinction, galaxy bias, redshift distortion, and nonlinear evolution (Zhan et al. 2005). These uncertainties do not produce strong oscillating features in the power spectrum as the baryon oscillations and can be controlled, despite some degradation to the measurements (Seo & Eisenstein 2005; White 2005).
We divide the LSST photometric redshift galaxy survey into 7 bins of roughly equal width from z = 0.2 to 3.0, and then combine the 12-parameter [including r(z)] Fisher matrix of each bin with the 7-parameter CMB Fisher matrix to infer the errors on the angular diameter distance and dark energy equation-of-state parameters w0 & wa (Zhan & Knox 2005). Additional parameters are also introduced to account for the rms photometric redshift error and bias. The errors on r(z) are shown in Figure 3; they are typically around one percent.
Figure 3 Error forecasts for the angular diameter distance r(z). Following SE03, we combine CMB priors from the Planck mission in the Fisher matrix analysis. The resulting distance errors in 7 redshift bins from LSST are shown for different survey qualities: the same as that in Figure 1 (solid line), twice the rms photometric redshift error (dotted line), and a tenth of the galaxy density (dash-dotted line). The error is roughly proportional to σz01/2, in agreement with SE03. For comparison, we include errors for the spectroscopic survey in the right panel of Figure 1 (S1000, dashed line).
We then project the Fisher matrix of a smaller set of marginalized parameters in the dark energy parameter space to obtain the errors on the dark energy equation-of-state parameters, w0 and wa, where w(z) = w0 + wa(1 - a). Assuming that the rms photometric redshift error σz0 is known to 0.5% and the bias is less than 0.1 σz0, we achieve σΩDE = 0.013, σw0 = 0.10, and σwa = 0.25 for a fiducial model with w0 = -1 and wa=0 (see Figure 4).
Figure 4 Error forecasts for w0 & wa in two cosmologies. One sigma error contours in the w0-wa plane are shown for LSST simple shear-shear power spectrum tomography (dashed contours), LSST baryon acoustic oscillations (solid contours), and JDEM 2000 SNe (dash-dotted contour). LSST results combine CMB priors from the Planck mission in Fisher matrices. One may significantly increase the constraining power in this plane and on w(z) generally by combining CMB, baryon oscillations, weak lensing (shear-shear and also bispectrum), and SNe together, though investigations must be carried out to address the cross-correlation between baryon oscillations and weak lensing statistics before quantitative conclusions can be reached.
