Cosmological Constraints from the Clustering of the Sloan Digital Sky Survey DR7 Luminous Red Galaxies
We present the power spectrum of the reconstructed halo density field derived from a sample of Luminous Red Galaxies (LRGs) from the Sloan Digital Sky Survey Seventh Data Release (DR7). The halo power spectrum has a direct connection to the underlying dark matter power for Mpc, well into the quasi-linear regime. This enables us to use a factor of more modes in the cosmological analysis than an analysis with Mpc, as was adopted in the SDSS team analysis of the DR4 LRG sample (Tegmark et al., 2006). The observed halo power spectrum for Mpc is well-fit by our model: for 40 degrees of freedom for the best-fitting CDM model. We find for a power law primordial power spectrum with spectral index and fixed, consistent with CMB measurements. The halo power spectrum also constrains the ratio of the comoving sound horizon at the baryon-drag epoch to an effective distance to : . Combining the halo power spectrum measurement with the WMAP 5 year results, for the flat CDM model we find and km s Mpc. Allowing for massive neutrinos in CDM, we find eV at the 95% confidence level. If we instead consider the effective number of relativistic species as a free parameter, we find . Combining also with the Kowalski et al. (2008) supernova sample, we find and for an open cosmology with constant dark energy equation of state . The power spectrum and a module to calculate the likelihoods is publicly available at http://lambda.gsfc.nasa.gov/toolbox/lrgdr/.
keywords:cosmology: observations, large-scale structure of Universe, galaxies: haloes, statistics
The past decade has seen a dramatic increase in the quantity and quality of cosmological data, from the discovery of cosmological acceleration using supernovae (Riess et al., 1998; Perlmutter et al., 1999) to the precise mapping of the cosmic microwave background (CMB) with the Wilkinson Microwave Anisotropy Probe (Page et al., 2003; Nolta et al., 2009) to the detection of the imprint of baryon acoustic oscillations (BAO) in the early universe on galaxy clustering (Eisenstein et al., 2005; Cole et al., 2005). Combining the most recent of these three cosmological probes, Komatsu et al. (2009) detect no significant deviation from the minimal flat CDM cosmological model with adiabatic, power law primordial fluctuations, and constrain that model’s parameters to within a few percent.
The broad shape of the power spectrum of density fluctuations in the evolved universe provides a probe of cosmological parameters that is highly complementary to the CMB and to probes of the expansion history (e.g., supernovae, BAO). The last decade has also seen a dramatic increase in the scope of galaxy redshift surveys. The PSCz (Saunders et al., 2000) contains IRAS galaxies out to , the 2dF Galaxy Redshift Survey(2dFGRS; Colless et al. 2001, 2003) collected 221,414 galaxy redshifts with median redshift 0.11, and the Sloan Digital Sky Survey (SDSS; York et al. 2000) is now complete with 929,555 galaxy spectra (Abazajian et al., 2009) including both main galaxies (; Strauss et al. 2002) and Luminous Red Galaxies (LRGs; ; Eisenstein et al. 2001). To harness the improvement in statistical power available now from these surveys requires stringent understanding of modeling uncertainties. The three major components of this uncertainty are the non-linear gravitational evolution of the matter density field (e.g., Zel’dovich 1970; Davis et al. 1977; Davis & Peebles 1977), the relationship between the galaxy and underlying matter density fields (“galaxy bias”, e.g., Kaiser 1984; Rees 1985; Cole & Kaiser 1989), and redshift space distortions (e.g., Kaiser 1987; Davis & Peebles 1983 and Hamilton 1998 for a review).
Several major advances have enabled previous analyses of 2dFGRS and SDSS to begin to address these complications. Progress in -body simulations (e.g., Heitmann et al. 2008), analytical methods (see Carlson et al. 2009 for an overview and comparison of many recent methods), and combinations thereof (e.g., Smith et al. 2003; Eisenstein et al. 2007b) have allowed significant progress in the study of the non-linear real space matter power spectrum. Recent power spectrum analyses have accounted for the luminosity dependence of a scale independent galaxy bias (Tegmark et al., 2004a; Cole et al., 2005), which can introduce an artificial tilt in in surveys which are not volume-limited (Percival et al., 2004). Cresswell & Percival (2009) have recently examined the scale dependence of galaxy bias as a function of luminosity and color. Tegmark et al. (2004a) applied a matrix-based method using pseudo-Karhunen-Loève eigenmodes to measure three power spectra from the SDSS galaxy distribution, allowing a quantification of the clustering anisotropy and a more accurate reconstruction of the real-space power spectrum than can be obtained from the angle-averaged redshift space power spectrum. Non-linear redshift space distortions, caused in part by the virialized motions of galaxies in their host dark matter haloes, create features known as Fingers-of-God (FOGs) along the line of sight in the redshift space galaxy density field (Davis & Peebles, 1983; Gramann et al., 1994). Both Tegmark et al. (2004a) and Cole et al. (2005) apply cluster-collapsing algorithms to mitigate the effects of FOGs before computing power spectra. Previous analyses have fit galaxy power spectra to linear (Percival et al., 2001, 2007) or non-linear matter models (Spergel et al., 2003; Tegmark et al., 2004), but did not attempt to model the scale dependence of the galaxy bias. Cole et al. (2005) introduced a phenomenological model to account for both matter non-linearity and the non-trivial relation between the galaxy power spectrum and matter power spectrum:
where denotes the underlying linear matter power spectrum. For the 2dFGRS analysis, Cole et al. (2005) fit using mock galaxy catalogues and derive expected central values of . In the fit to the observed galaxy power spectrum, they allow to vary up to twice the expected value, which is supported by halo model calculations of the cosmological dependence of the galaxy . This approach appears to work well for the case of 2dFGRS galaxies because it was calibrated on mock catalogues designed to match the properties of this galaxy population; however, its application to the LRG sample in Tegmark et al. (2006), where the best-fitting was much larger than for 2dFGRS galaxies, is questionable (see Reid et al. 2008 and Yoo et al. 2009, but also Sánchez & Cole 2008).
In this paper we focus our efforts on accurately modeling the relationship between the galaxy and matter density fields for the SDSS LRG sample. Several authors have studied this relation using the small and intermediate scale clustering in the SDSS LRG sample (Masjedi et al., 2006; Zehavi et al., 2005; Kulkarni et al., 2007; Wake et al., 2008; Zheng et al., 2008; Reid & Spergel, 2009) and galaxy-galaxy lensing (Mandelbaum et al., 2006). The LRG selection algorithm in the SDSS (Eisenstein et al., 2001) was designed to provide a homogenous galaxy sample probing a large volume with a number density, , which maximizes the effective survey volume on the large scales of interest, Mpc. is given by (Feldman et al., 1994; Tegmark, 1997):
where denotes the measured galaxy power spectrum, the average galaxy number density in the sample at position , and the integral is over the survey volume. The total error on is minimized (i.e., is maximized) when , which optimally balances cosmic variance and shot noise for a fixed number of galaxies. The LRG sample has proven its statistical power through the detection of the BAO (Eisenstein et al., 2005; Percival et al., 2007). However, parameterizing the LRG power spectrum with a heuristic model for the non-linearity (Eqn. 1) and marginalizing over fitting parameters limits our ability to extract the full cosmological information available from the power spectrum shape and can introduce systematic biases (Sánchez & Cole, 2008; Dunkley et al., 2009; Verde & Peiris, 2008; Reid et al., 2008).
On sufficiently large scales, we expect galaxies to be linearly biased with respect to the underlying matter density field (Mo & White, 1996; Scherrer & Weinberg, 1998). However, an often overlooked consequence of a sample with is that errors in the treatment of the shot noise can introduce significant changes in the measured shape of and can be interpreted as a scale dependent galaxy bias. In the halo model picture, the LRGs occupy massive dark matter haloes, which themselves may not be Poisson tracers of the underlying matter density field, as they form at the high peaks of the initial Gaussian density distribution (e.g., Bardeen et al. 1986). Moreover, an additional shot noise-like term is generated when multiple LRGs occupy individual dark matter haloes (Peacock & Smith, 2000; Cooray & Sheth, 2002). Our approach is to first eliminate the one-halo contribution to the power spectrum by identifying groups of galaxies occupying the same dark matter halo, and then to calibrate the relation between the power spectrum of the reconstructed halo density field, , and the underlying matter power spectrum, , using the -body simulation results presented in Reid et al. (2008). As a result, the effects of non-linear redshift space distortions caused by pairs of galaxies occupying the same halo are diminished. However, a further complication is that LRGs occupy the massive end of the halo mass function, and velocities of isolated LRGs within their host haloes could still be quite large. The details of the relation between LRGs and the underlying matter distribution can then have a significant impact on the non-linear corrections to the power spectrum.
The DR7 LRG sample has sufficient statistical power that the details of the relation between LRGs and the underlying matter density field become important and need to be reliably modeled before attempting a cosmological interpretation of the data. This paper offers three sequential key improvements to the modeling of LRG clustering compared with Eisenstein et al. (2005) and Tegmark et al. (2006):
We reconstruct the underlying halo density field traced by the LRGs before computing the power spectrum, while Tegmark et al. (2006) apply an aggressive FOG compression algorithm. The reconstructed halo density field power spectrum deviates from the underlying matter power spectrum by at , while the Tegmark et al. (2006) power spectrum differs by at (Reid et al., 2008).
We produce a large set of mock LRG catalogues drawn from -body simulations of sufficient resolution to trace a halo mass range relevant to LRGs without significant errors in the small-scale halo clustering and velocity statistics (see Appendix A of Reid et al. 2008). We present novel consistency checks between the mock and observed LRG density fields in halo-scale higher order clustering, FOG features, and the effective shot noise.
We use these tests along with the halo model framework to determine tight bounds on the remaining modeling uncertainties, and marginalize over these in our likelihood calculation. In contrast, Eisenstein et al. (2005) assume no uncertainty in their model LRG correlation function, and Tegmark et al. (2006) marginalize over in Eqn. 1 with only an extremely weak prior on .
This paper represents a first attempt to analyse a galaxy redshift survey with a model that accounts for the non-linear galaxy bias and its uncertainty; other approaches that utilize the galaxy distribution rather than the halo density field are in development (Yoo et al., 2009).
|measured angle averaged redshift-space power spectrum of the LRGs||-|
|measured power spectrum of reconstructed halo density field||-|
|linear power spectrum computed by CAMB||Lewis et al. (2000)|
|theoretical real-space non-linear power spectrum of dark matter||-|
|theoretical linear power spectrum without BAO (“no wiggles”)||Eisenstein & Hu (1998)|
|theoretical linear power spectrum with damped BAO (Eqn. 10)||Eisenstein et al. (2007b)|
|model for the reconstructed halo power spectrum for cosmological parameters||Reid et al. (2008)|
|convolved with survey window function (Eqn. 5)||Percival et al. (2007)|
|and directly compared with in the likelihood calculation (Eqn. 6)|
In this paper we present and analyse a measurement of the power spectrum of the reconstructed halo density field from the SDSS DR7 LRG sample. DR7 represents a factor of increase in effective volume over the analyses presented in Eisenstein et al. (2005) and Tegmark et al. (2006), and covers a coherent region of the sky. Section 2 describes the measurement of the reconstructed halo density field power spectrum, , along with the window and covariance matrices used in our likelihood analysis. Section 3 describes the details of our model for the reconstructed halo power spectrum, . In Section 4 we summarize the tests we have performed for various systematics in our modeling of the relation between the galaxy and dark matter density field. We quantify the expected level of uncertainty through two nuisance parameters and present several consistency checks between the model and observed reconstructed halo density field. In Section 5 we discuss the cosmological constraints from alone as well as in combination with WMAP5 (Dunkley et al., 2009) and the Union supernova dataset (Kowalski et al., 2008). Section 6 compares our findings with the results of previous analyses of galaxy clustering, and Section 7 summarizes our conclusions.
In a companion paper (Percival et al. prep; hereafter, P09) we measure and analyse BAO in the SDSS DR7 sample, of which the LRG sample considered here is a subset. BAO are detected in seven redshift shells, leading to a 2.7% distance measure at redshift , and a measurement of the gradient of the distance-redshift relation, this quantified by the distance ratio between and . We show in Section 5 that the results from these measurements are in agreement with our combined results from BAO and the shape of the power spectrum calculated using just the LRGs. The results from these different analyses will be correlated because of the overlapping data used, so they should not be combined in cosmological analyses. The best data set to be used will depend on the cosmological model to be tested. While the inclusion of 2dFGRS and main SDSS galaxies in P09 provides a higher significance detection of the BAO, we show in Section 5.4 that the full power spectrum information provides tighter constraints on both massive neutrinos and the number of relativistic species.
Throughout the paper we make use of two specific cosmological models. The simulation set described in Reid et al. (2008) and used to calibrate the model adopts the WMAP5 recommended CDM values: (, , , , , ) = (0.2792, 0.0462, 0.7208, 0.960, 0.817, 0.701). We refer to this model throughout the paper as our ‘fiducial cosmological model.’ To convert redshifts to distances in the computation of the , we adopt a flat CDM cosmology with and . Throughout we refer to the power spectrum of several different density fields and several theoretical spectra. Table 1 summarizes their definitions.
2.1 LRG sample
The SDSS (York et al., 2000) is the largest galaxy survey ever produced; it used a 2.5m telescope (Gunn et al., 2006) to obtain imaging data in 5 passbands , , , and (Fukugita et al., 1996; Gunn et al., 2006). The images were reduced (Stoughton et al., 2002; Pier et al., 2003; Ivezić et al., 2004) and calibrated (Hogg et al., 2001; Smith et al., 2002; Tucker et al., 2006; Padmanabhan et al., 2008), and galaxies were selected for follow-up spectroscopy. The second phase of the SDSS, known as SDSS-II, has recently finished, and the DR7 (Abazajian et al., 2009) sample has recently been made public. The SDSS project is now continuing with SDSS-III where the extragalactic component, the Baryon Oscillation Spectroscopic Survey (BOSS; Schlegel et al. 2009), has a different galaxy targeting algorithm. DR7 therefore represents the final data set that will be released with the original targeting and galaxy selection (Eisenstein et al., 2001; Strauss et al., 2002).
In this paper we analyse a subsample containing Luminous Red Galaxies (LRGs: Eisenstein et al. 2001), which were selected from the SDSS imaging based on , and colours, to give approximately galaxies per square degree. The SDSS also targeted a magnitude limited sample of galaxies for spectroscopic follow-up (Strauss et al., 2002). The LRGs extend this main galaxy sample to , covering a greater volume. Our DR7 sample covers deg (including a 7190 deg contiguous region in the North Galactic Cap), with an effective volume of Gpch, calculated with a model power spectrum amplitude of Mpc. This power spectrum amplitude is approximately correct for the LRGs at . For comparison, the effective volume of the sample used by Eisenstein et al. (2005) was Gpch, and Gpch in Tegmark et al. (2006); this work represents a factor of 2 increase in sample size over these analyses. The sample is the same as that used in P09, and its construction follows that of Percival et al. (2007), albeit with a few improvements.
We use SDSS Galactic extinction-corrected Petrosian magnitudes calibrated using the “übercalibration” method (Padmanabhan et al., 2008). However, we find that the power spectrum does not change significantly when one adopts the old standard calibration instead (Tucker et al., 2006). Luminosities are K-corrected using the methodology of Blanton et al. (2003); Blanton et al. (2003b). We remove LRGs that are not intrinsically luminous by applying a cut , where is our estimate of the absolute magnitude in the -band for a galaxy at .
Spectroscopic LRG targets were selected using two color-magnitude cuts (Eisenstein et al., 2001). The tiling algorithm ensures nearly complete samples (Blanton et al., 2003). However, spectroscopic fiber collisions prohibit simultaneous spectroscopy for objects separated by , leaving of targeted objects without redshifts (Masjedi et al., 2006). We correct for this effect as in Percival et al. (2007): for an LRG lacking a spectrum but 55” from an LRG with a redshift, we assign both galaxies the measured redshift. If the LRG lacking a redshift neighbors only a galaxy from the low redshift SDSS main sample, we do not assign it a redshift. These galaxies are assumed to be randomly distributed, and simply contribute to the analysis by altering the completeness, the fraction of targeted galaxies with good redshifts, in a particular region. The impact of the fiber collision correction is addressed in Appendix B.3 and Appendix B.4.
Fig. 1 compares the number density as a function of redshift for the LRG selection used in this paper (Percival et al., 2007) and the one used in Tegmark et al. (2006) and presented in Zehavi et al. (2005). The main differences are that our selection includes a small number of galaxies at , and our cut on the intrinsic luminosity of the LRGs slightly reduces the number density of galaxies at high . The different selections produce a similar number of galaxies per unit volume, and we expect no difference between the samples on the large scale structure statistics of interest here.
2.2 Recovering the halo density field
In real space, the impact of more than one LRG per halo on the large scale power spectrum can be accurately modeled as an additional shot noise term (Cooray & Sheth, 2002; Reid et al., 2008). However, this picture is much more complicated in redshift-space because of the velocity dispersion of the LRGs shifts them along the line of sight by Mpc (Reid et al., 2008), and the distribution of intrahalo velocities has long tails. This shifting causes power to be shuffled between scales and causes even the largest scale modes along the line of sight to be damped by these FOG features (Davis & Peebles, 1983; Peacock & Dodds, 1994; Seljak, 2001). We substantially reduce the impact of these effects by using the power spectrum of the reconstructed halo density field.
We follow the Counts-In-Cylinders (CiC) technique in Reid et al. (2008) to identify LRGs occupying the same halo and thereby estimate the halo density field. Two galaxies are considered neighbors when their transverse comoving separation satisfies Mpc and their redshifts satisfy ( km s). A cylinder should be a good approximation to the density contours of satellites surrounding central galaxies in redshift space, as long as the satellite velocity is uncorrelated with its distance from the halo centre and the relative velocity dominates the separation of central and satellite objects along the line of sight. Galaxies are then grouped with their neighbors by a Friends-of-Friends (FoF) algorithm. The reconstructed halo density field is defined by the superposition of the centres of mass of the CiC groups. We refer to the power spectrum of the reconstructed halo density field as ; it is our best estimate of the power spectrum of the haloes traced by the LRGs. For comparison we also compute the power spectrum without applying any cluster-collapsing algorithm, .
Our reconstructed halo density field contains haloes derived from LRGs.
2.3 Calculating power spectra, window functions and covariances
In this paper we focus on using the angle-averaged power spectrum to derive constraints on the underlying linear theory power spectrum. On linear scales the redshift space power spectrum is proportional to the real space power spectrum (Kaiser, 1987; Hamilton, 1998). Our halo density field reconstruction mitigates the effects of FOGs from objects occupying the same halo. Though we do not explore it here, we expect that our halo density field reconstruction will be useful to an analysis of redshift-space anisotropies (e.g., Hatton & Cole 1999).
The methodology for calculating the power spectrum of the reconstructed halo density field, , is based on the Fourier method of Feldman et al. (1994). The halo density is calculated by throwing away all but the brightest galaxy where we have located a set of galaxies within a single halo. This field is converted to an over-density field by placing the haloes on a grid and subtracting an unclustered “random catalogue”, which matches the halo selection. To calculate this random catalogue, we fit the redshift distributions of the halo sample with a spline model (Press et al., 1992) (shown in Fig. 1), and the angular mask was determined using a routine based on a HEALPIX (Górski et al., 2005) equal-area pixelization of the sphere as in (Percival et al., 2007). This procedure allows for the variation in radial selection seen at , which is caused by the spectroscopic features of the LRGs moving across the wavebands used in the target selection. The haloes and randoms are weighted using a luminosity-dependent bias model that normalizes the fluctuations to the amplitude of galaxies (Percival et al., 2004). To do this we assume that each galaxy used to locate a halo is biased with a linear deterministic bias model, and that this bias depends on according to Tegmark et al. (2004a) and Zehavi et al. (2005), where is the Galactic extinction and K-corrected -band absolute galaxy magnitude. This procedure is similar to that adopted by P09.
The power spectrum was calculated using a grid in a series of cubic boxes. A box of length was used initially, but we then sequentially divide the box length in half and apply periodic boundary conditions to map galaxies that lie outside the box. For each box and power spectrum calculation, we include modes that lie between and the Nyquist frequency (similar to the method described by Cole et al. 2005), and correct for the smoothing effect of the cloud-in-cell assignment used to locate galaxies on the grid (e.g. Hockney & Eastwood, 1981, chap. 5). The power spectrum is then spherically averaged, leaving an estimate of the “redshift-space” power. The upper panel of Fig. 2 shows the shot-noise subtracted bandpowers measured from the halo density field, calculated in bands linearly separated by . This spacing is sufficient to retain all of the cosmological information.
The calculation of the likelihood for a cosmological model given the measured bandpowers requires three additional components determined by the survey geometry and the properties of the galaxy sample: the covariance matrix of measured bandpowers , the window function , and the model power spectrum as a function of the underlying cosmological parameters, . The calculation of model power spectra is considered in Section 3.
The covariance matrix and corresponding correlation coefficients between bandpowers and are defined as
The covariance matrix was calculated from Log-Normal (LN) catalogues (Coles & Jones, 1991; Cole et al., 2005). Catalogues were calculated on a grid with box length as in P09, where LN catalogues were similarly used to estimate covariance matrices. Unlike -body simulations, these mock catalogues do not model the growth of structure, but instead return a density field with a log-normal distribution, similar to that seen in the real data. The window functions for these catalogues were matched to that of the halo catalogue. The input power spectrum was a cubic spline fit matched to the data power spectra, multiplied by a damped CDM BAO model calculated using CAMB (Lewis et al., 2000). The recovered LN power spectra were clipped at 5 to remove extreme outliers which contribute less than 0.05% of the simulated power spectra, and are clearly non-Gaussian. This covariance matrix calculation matches the procedure adopted by P09. The middle panel of Fig. 2 shows the correlations expected between band-powers calculated using this procedure.
As described in Cole et al. (2005), the window function can be expressed as a matrix relating the theory power spectrum for cosmological parameters and evaluated at wavenumbers , , to the central wavenumbers of the observed bandpowers :
The term arises because we estimate the average halo density from the sample, and is related to the integral constraint in the correlation function (Percival et al., 2007). The window function allows for the mode-coupling induced by the survey geometry. Window functions for the measured power spectrum (Eqn. 15 of Percival et al. 2004) were calculated as described in Percival et al. (2001), Cole et al. (2005), and Percival et al. (2007): an unclustered random catalogue with the same selection function as that of the haloes was Fourier transformed using the same procedure adopted for our halo overdensity field described above. The shot noise was subtracted, and the power spectrum for this catalogue was spherically averaged, and then fitted with a cubic spline, giving a model for . For ease of use this is translated into a matrix by splitting input and output power spectra into band powers as in Eqn. 5.
The window functions and the corresponding correlation coefficients for every other bandpower are shown in the lower panel of Fig. 2. In addition to the window coupling for nearby wavenumbers, there is a beat-coupling to survey-scale modes (Hamilton et al., 2006; Reid et al., 2008). That is, density fluctuations on the scale of the survey couple to the modes we can measure from the survey. However, this effect predominantly changes only the amplitude of , which is marginalized over through the bias parameter in Eqn. 15 below. Fig. 2 can be compared with Fig. 10 in Percival et al. (2007), where the windows and correlations were presented for the SDSS DR5 data. For the DR5 plot, variations in the amplitude were removed leaving only the small- difference couplings. The power spectrum, window functions, and inverse covariance matrix are electronically available with the likelihood code we publicly release (see Section 5).
We assume that the likelihood distribution of the power spectrum band powers is close to a standard multi-variate Gaussian; by the central limit theorem, this should be a good approximation in the limit of many modes per band. The final expression for the likelihood for cosmology is then
A single comoving distance-redshift relation , that of a flat, cosmology, is assumed to assign positions to the galaxies in our sample before computing . Rather than recomputing for each comoving distance-redshift relation to be tested, Percival et al. (2007) and P09 account for this when evaluating the likelihood of other cosmological models by altering the window function. (Eisenstein et al., 2005) quantifies the model dependence of the conversion between (ra, dec, z) and comoving spatial coordinates when galaxy pairs are distributed isotropically:
where is the physical angular diameter distance. Following Tegmark et al. (2006) we partially correct for the discrepancy between the fiducial model and the of the model to be tested by introducing a single dilation of scale. To first order, changes in the cosmological distance–redshift model alter the scale of the measured power spectrum through , so we introduce a scale parameter that depends on this quantity,
Strictly, we should allow for variations in across the redshift range of the survey, as in P09. However, to first approximation we can simply allow for a single scale change at an effective redshift for the survey . When comparing , computed using , with a model comoving distance-redshift relation , in practice we use 111This correction was incorrectly applied in previous versions of cosmomc, and is corrected in the code we release. This correction is primarly important for constraining the BAO scale rather than the turnover scale, and so previous analyses with cosmomc should be minimally affected.
In Appendix A.2 we verify that this approximation is valid for our sample with .
In our cosmological analysis, we include modes up to , where the model power spectrum deviates from the input linear power spectrum by . We also impose a conservative lower bound at , above which galactic extinction corrections (see the analysis in Percival et al. 2007), galaxy number density modeling, and window function errors should be negligible.
P09 present a detailed analysis demonstrating that the BAO contribution to the likelihood surface is non-Gaussian; this is in large part due to the relatively low signal-to-noise ratio of the BAO signature in our sample. Therefore, to match expected and recovered confidence intervals, P09 find that the covariance matrix of the LRG-only sample must be inflated by a factor . Though our likelihood surface incorporates constraints from the shape of the power spectrum, for which the original covariance matrix should be accurate, we conservatively multiply the entire covariance matrix by this factor required for the BAO constraints throughout the analysis. Therefore our constraints likely slightly underestimate the true constraints available from the data. This factor is already included in the electronic version we release with the full likelihood code.
3 Modeling the Halo Power Spectrum
We consider three effects that cause the shape of to deviate from the linear power spectrum, , for cosmological parameters . We will assume that these modifications of the linear power spectrum can be treated independently. These effects are the damping of the BAO, the change in the broad shape of the power spectrum because of non-linear structure formation, and the bias because we observe galaxies in haloes in redshift space rather than the real space matter distribution. We also need to consider the evolution of these effects with redshift.
Reid et al. (2008) construct a large set of mock LRG catalogues based on -body simulations evaluated at a single cosmological model . We use these catalogues to calibrate the model halo power spectrum, and make detailed comparisons between the observed and mock density fields in Appendix B.
3.1 BAO damping
The primary effect of non-linear structure formation and peculiar velocities on the BAOs is to damp them at large . Eisenstein et al. (2007b) showed that this can be accurately modelled as a Gaussian smoothing, where
Here is the linear matter power spectrum computed by CAMB (Lewis et al., 2000) and shown in the upper left panel of Fig. 3 for our fiducial cosmological model. , defined by Eqn. 29 of Eisenstein & Hu (1998), is a smooth version of with the baryon oscillations removed. The upper right panel of Fig. 3 shows the ratio for our fiducial cosmology. The amplitude of the damping is set by and depends on the cosmological parameters, whether the power spectrum is in real or redshift space, and whether we are considering the matter or a tracer like the LRGs. We fix , i.e., the value of appropriate for the reconstructed halo density field, using fits to the reconstructed halo density field power spectrum in the mock LRG catalogues presented in Reid et al. (2008) and shown here in Fig. 4. We performed tests in the CDM case which demonstrate that cosmological constraints are not altered when is allowed to vary with cosmology p according to the dependence given in Eisenstein et al. (2007b), and in Appendix A.3 we show that using a spline fit to instead of the Eisenstein & Hu (1998) formula for does not affect the likelihood surface in the region of interest.
3.2 Non-linear structure growth
As the small perturbations in the early universe evolve, gravitational instability drives the density field non-linear, and power on small scales is enhanced as structures form. halofit (Smith et al., 2003) provides an analytic formalism to estimate the real space non-linear matter power as a function of the underlying linear matter power spectrum. While Eqn. 10 accounts for the effects of non-linear growth of structure on the BAO features in , halofit provides a more accurate fit to the smooth component of the non-linear growth in the quasi-linear regime () when evaluated with an input spectrum rather than the linear matter power spectrum containing BAO wiggles:
Eqn. 12 is our modified halofit model real space power spectrum, using Eqn. 10 to account for BAO damping and halofit for the smooth component. The bottom left panel of Fig. 3 shows that and agree at the level for in our fiducial cosmology. Since we normalize the final model using our mock catalogues at the fiducial cosmology , in practice halofit only provides the cosmological dependence of the non-linear correction to the matter power spectrum:
is our model for the ratio of the non-linear matter power spectrum to the damped linear power spectrum. The normalization of accounts for the small offset between the -body and halofit results in Fig. 3 at the fiducial cosmology. In the space of cosmologies consistent with the data, the small cosmology-dependence of this correction is primarily through . In Section 5.2 we find that the LRG-only likelihood surface is independent of the assumed value of over the range 0.7 to 0.9.
3.3 Halo bias
In our likelihood calculation we marginalize over the overall amplitude of , so in this Section we are concerned only with the scale dependence of the relation between the reconstructed halo and matter power spectra. Smith et al. (2007) show that the scale dependence of halo bias in real space is large for the most massive haloes, but should be rather weak for the halo mass range which host the majority of the LRGs; Matsubara (2008) demonstrates this analytically in redshift space in the quasi-linear regime. Indeed, Reid et al. (2008) find that the power spectrum of the (redshift space) reconstructed halo density field is nearly linearly biased with respect to the underlying real space matter power spectrum for and our fiducial CDM model, and we assume this should remain approximately true in the narrow range of cosmologies consistent with the data. For the fiducial cosmology, we can use our simulations to calibrate the relation between the halo and matter spectra:
This is our model for the smooth component of the bias between the halo and dark matter power spectra. To account for any dependence of on the cosmological model and other remaining modeling uncertainties, we introduce a smooth multiplicative correction to the final model containing three nuisance parameters , and :
where we set . The parameter is the effective bias of the LRGs at the effective sample redshift, , relative to galaxies (Eqn. 18 of Percival et al. 2004). In Section 4 we will use consistency checks between the observed and mock catalogue galaxy density fields as well as the halo model framework to establish the allowed region of parameter space. An allowed trapezoidal region in space is completely specified through two parameters, and . These two parameters specify the maximum absolute deviation allowed by away from 1 for () and (). When evaluating the likelihood of a particular cosmological model we marginalize analytically over using a flat prior on , and we marginalize numerically over the allowed region with a flat prior in this region. We discuss the impact of these priors on the cosmological constraints in Appendix C.
3.4 Model fits and evolution with redshift
Our final model halo power spectrum at fixed redshift treats each of the three non-linear effects independently: Eqn. 10 converts the linear power spectrum to the damped linear power spectrum, converts the damped linear power spectrum to the real space non-linear matter power spectrum, converts the real space non-linear matter power spectrum to the redshift space reconstructed halo density field power spectrum (assuming this relation is cosmology independent), and allows for smooth deviations from our model due to modeling errors, uncertainties, and unaccounted cosmological parameter dependencies:
The model in Eqn. 16 is strictly only valid at a single redshift. In order to match our model to the observed redshift distribution of the LRGs and their associated haloes, we use the mock halo catalogues constructed in Reid et al. (2008) at three redshift snapshots. These are centered on the NEAR (), MID (), and FAR () LRG subsamples of Tegmark et al. (2006). Fig. 4 shows our fits to for each redshift snapshot. We first fit for in Eqn. 10 using our LRG mock catalogue results . We include modes between and in the fit and marginalize over an arbitrary fourth order polynomial to account for the smooth deviations from with . We find Mpc, Mpc, and Mpc. These numbers are roughly consistent with the results presented in Eisenstein et al. (2007b), and are somewhat degenerate with the smooth polynomial correction.
After fixing these values for , we calibrate the smooth component of the model, . For we fit to a second order polynomial, and a fourth order polynomial for . This component of the fit is shown in the first three panels of Fig. 4 by the dotted curves, while the solid lines show the full fit to . Both the BAO-damping and smooth increase in power with are well described by our fits out to .
where specifies the weight of each redshift subsample. The lower right panel of Fig. 3 shows that the smooth correction for the non-linear matter power spectrum varies by over the redshift range of the LRGs. Moreover, the lower right panel of Fig. 4 shows that the relative shape of the power spectrum of the reconstructed halo density field varies by between the redshift subsamples, so moderate biases in the determination of these weights will induce negligible changes in the predicted shape .
In the limit that most pairs of galaxies contributing power to mode come from the same redshift, the fractional contribution to the power spectrum from a large redshift subsample is
where , , and specify the average number density, bias, and weight of the sample at redshift as defined in Percival et al. (2004). Since the integrand is slowly varying with redshift, this approximation should be fairly accurate. We derive weights , , and .
3.5 Comparison with fiducial model
Our fiducial model is calibrated on simulations with the WMAP5 recommended parameters (Komatsu et al., 2009): ( = (0.2792, 0.0462, 0.7208, 0.960, 0.817, 0.701). For the 45 observed bandpowers satisfying , if we hold nuisance parameters and choose to minimize ; our fiducial model is therefore sufficiently close to the measured to be used to calibrate the cosmology-dependent model. The best-fitting nuisance parameters within the allowed range that we determine in Section 4.3, and , lower the to 40.9 for 42 DOF. The best-fitting model to the LRG-only likelihood presented in Section 5.1 is lower by only for the same treatment of the three nuisance parameters.
4 Quantifying Model Uncertainties and Checks for Systematics
While the non-linear evolution of a collisionless dark matter density field can be accurately studied using -body simulations, there remain many uncertainties in the mapping between the galaxy and matter density field. We first review the generic halo model predictions for a galaxy power spectrum, which provide the context for exploring the uncertainties in the relation between the galaxy and matter density fields. We summarize the results of Appendix B, which presents our modeling assumptions and consistency checks between the mock catalogue and SDSS DR7 LRG density fields that constrain the level of deviation from our modeling assumptions. The ultimate goal of this Section is to establish physically-motivated constraints on the nuisance parameters and in Eqn. 15 by determining and defined in Section 3.3. These nuisance parameter constraints will then be used to compute cosmological parameter constraints in Section 5.
4.1 Galaxy power spectra in the halo model
In the simplest picture for a galaxy power spectrum in the halo model, one considers a separation of the pairs into galaxies occupying the same dark matter halo, which contribute to , and those occupying different dark matter haloes, which contribute to (Cooray & Sheth, 2002):
On large scales, treating the haloes as linear tracers of the underlying matter density field (Eqn. 21) and ignoring the spatial extent of haloes in Eqn. 20 are good approximations (Reid et al., 2008). Therefore, in real space, the dominant effect of the inclusion of satellite galaxies is an excess shot noise given by Eqn. 20, though they also upweight highly biased halo pairs and slightly increase as well. However, in redshift space, satellite galaxies are significantly displaced along the line of sight from their host haloes by the FOGs, and power is shuffled between scales, and even the largest scale modes along the line of sight are damped by the FOG smearing. There will be residual non-linear redshift space distortions in the reconstructed halo density field from imperfect reconstruction, and potentially from peculiar motion of isolated LRGs in their host haloes as well.
4.2 Summary of tests for systematics and remaining uncertainties
In the context of the halo model, both uncertainty in the distribution of galaxies in groups as it enters Eqn. 20 and uncertainty in the structure of the FOG features will introduce uncertainty in the relation between the reconstructed halo and matter density fields, and thus their power spectra. Appendices B.1 and B.2 discuss the modeling assumptions we have used to derive the Reid et al. (2008) mock LRG catalogues from -body simulation halo catalogues, and state the expected impact on the relation between the reconstructed halo and matter power spectra.
Appendix B.3 introduces several distinct consistency checks of the uncertainties in Appendices B.1 and B.2. In Section 2.2 we define the CiC group finder by which we identify haloes. We demonstrate that this group finder produces group multiplicity functions that are in good agreement between the mock and observed LRG density fields, once fiber collisions are accounted for. While this agreement demonstrates that our mock catalogues reproduce small scale higher-order clustering statistics and FOG features of the observed density field, this is not a consistency check since the mocks were designed to match these statistics. We find consistency when we compute a second CiC group multiplicity function allowing a wider separation between pairs perpendicular to the line of sight ( Mpc). If the observed satellite galaxies were significantly less concentrated than in our mock catalogues, we would detect these galaxies when increases from 0.8 Mpc to 1.2 Mpc. From this comparison we conclude that residual shot noise errors from inaccurate halo density field reconstruction are of the total shot noise correction and do not dominate our systematic uncertainty. The second consistency check between the mock and observed LRG catalogues is the distribution of line of sight separations between pairs of galaxies in the same CiC group (Fig 15). This check probes the accuracy of our model of the FOG features coming from galaxies occupying the same halo, and the agreement we find indicates that the residual FOG features in the reconstructed observed and mock halo density fields will be in satisfactory agreement. Appendix B.4 presents the difference between the power spectra with and without the halo density field reconstruction preprocessing step ( and , respectively). This difference agrees with the mock catalogues, provided one carefully accounts for the impact of fiber collisions. In other words, while the treatment of fiber collisions can substantially impact , is unaffected. In Appendix B.5 we demonstrate that the luminosity weighting used to compute but not accounted for in the mock catalogues does not alter the effective shot noise level of . Appendix B.6 presents evidence that the cosmology dependence of the model is sufficiently accurate. Finally, we note that Lunnan et al. (prep) have compared the Reid et al. (2008) mock catalogue genus curve with the observed genus curves (Gott et al., 2009), and find good agreement with no free parameters.
As discussed in detail in Appendix B.2, the vast majority of LRGs () are expected to reside at the centre of their host dark matter haloes (Zheng et al., 2008; Reid et al., 2008). The principal modeling uncertainty we identify in Appendix B is the velocity of these central LRGs within their host haloes; substantial intrahalo velocities for these galaxies will suppress power in a scale-dependent manner (Fig. 14). Note that none of the tests from Appendix B can directly constrain the level of central LRG velocity dispersion.
4.3 Constraints on
In Section 3.3 we introduced a quadratic function to account both for errors in our modeling at the fiducial cosmology and for any errors in the cosmology dependence of our model. We parametrized the amplitude of the total modeling uncertainty through and . These parameters, which we determine in this subsection, specify the maximum fractional deviation from the model power spectrum at and , respectively. We choose these values of because is usually considered safely in the linear regime, while is the maximum wavenumber we attempt to model.
The dominant uncertainty in our model is in the relation between the power spectrum of the reconstructed halo density field and the underlying matter power spectrum, which we describe by Eqn. 14. At in the mock catalogues, the reconstructed halo density field and the redshift space central galaxy power spectra agree well below the percent level. The total one-halo correction in real space is 7-10%. If we conservatively assume that the halo reconstruction algorithm incorrectly subtracts the real space one-halo term by 20%, then the systematic error at , , is allowed to be 2%. At , the same error would translate to 5% in real space, though in redshift space this term is mitigated. In Appendix B.4 we find that the shape difference of and is only 18% at , and only 8% after accounting for the shot noise introduced by the fiber collision corrections. If we assume that our modeling and treatment of the one-halo contribution to the FOGs are accurate at the level, we can estimate a conservative error at of 5%. Therefore, for all the modeling uncertainties besides central velocity dispersion that we have discussed, and encompass the estimated uncertainties. These are our ‘fiducial’ nuisance function constraints.
In Appendix B.2 we find that a large amount of central galaxy misidentification or central–halo velocity bias can reduce the amplitude of by a smoothly varying function of at a level that exceeds these fiducial bounds on and . Our approach to mitigating the impact of uncertain central LRG peculiar velocities is twofold. First, for all of the analysis in Section 5 we adopt more conservative bounds for the nuisance function: and , which nearly encompass the change in power spectrum shape in Fig. 14 for the extreme velocity dispersion model. Furthermore, we calibrate a second model from the mocks with extreme velocity dispersion, and in Appendix C we determine the cosmological parameter constraints with this model to establish the level of remaining systematic uncertainty in our final results.
5 Cosmological Constraints
In this Section we explore the cosmological constraints derived from the power spectrum of the reconstructed halo density field, . We first consider constraints obtained from alone, and then combine the LRG likelihood with WMAP5 and the Union Supernova Sample (Kowalski et al., 2008) to explore joint constraints in several cosmological models. Throughout, we make use of the COSMOMC package (Lewis & Bridle, 2002) to compute cosmological constraints using the Markov Chain Monte-Carlo method. A stand-alone module to compute the likelihood is made publicly available.222http://lambda.gsfc.nasa.gov/toolbox/lrgdr/
5.1 Constraints from the halo power spectrum
|weak + prior|
|VD + prior|
|NW + prior|
|NW + prior|
In this subsection we examine the cosmological constraints derived from the alone and in combination with a prior on from WMAP5. In the model , the scale factor in Eqn. 8 is evaluated at . For comparison with other works, we scale our constraint on using the fiducial distance-redshift relation, for which ; the variation of this ratio with cosmological parameters is negligible. Following Eisenstein et al. (2005), we consider two free parameters and . In this subsection we hold , , and fixed at their values in the fiducial cosmological model, and assume a flat CDM model; in §5.2 we relax these assumptions.
For the 45 bandpowers satisfying , is minimized when and with best-fitting nuisance parameters and : for 40 degrees of freedom. Thus the assumed model power spectrum and covariance matrix provide a reasonable fit to the observed spectrum. In a CDM model, this point corresponds to and . Fig 5 shows contours in the parameter space, while Table 2 reports marginalized one-dimensional constraints for several combinations of these parameters.
The information in can be roughly divided into broad-shape information and information from the BAO scale. Since in this subsection is fixed, the shape information is the location of the turnover in the power spectrum set by matter-radiation equality, which constrains ; information from the BAO scale constrains . Here, is the sound horizon at the baryon-drag epoch, which we evaluate using Eqn. 6 of Eisenstein & Hu (1998). These two scales correspond to constraints on and respectively, in a CDM cosmology (Tegmark et al., 2006).
To isolate information from the power spectrum turnover and exclude that of the BAO scale, we alter our model so that in Eqn. 10. The dashed lines in Fig. 5 show the constraints when using this ‘no wiggles’ model. Most of the available shape information comes from large scales with ; we demonstrate this in Table 2 by fitting the