Cosmological Constraints from a Combination of Galaxy Clustering and Lensing – III. Application to SDSS Data
We simultaneously constrain cosmology and galaxy bias using measurements of galaxy abundances, galaxy clustering and galaxy-galaxy lensing taken from the Sloan Digital Sky Survey. We use the conditional luminosity function (which describes the halo occupation statistics as function of galaxy luminosity) combined with the halo model (which describes the non-linear matter field in terms of its halo building blocks) to describe the galaxy-dark matter connection. We explicitly account for residual redshift space distortions in the projected galaxy-galaxy correlation functions, and marginalize over uncertainties in the scale dependence of the halo bias and the detailed structure of dark matter haloes. Under the assumption of a spatially flat, vanilla CDM cosmology, we focus on constraining the matter density, , and the normalization of the matter power spectrum, , and we adopt WMAP7 priors for the spectral index, , the Hubble parameter, , and the baryon density, . We obtain that and (95% CL). These results are robust to uncertainties in the radial number density distribution of satellite galaxies, while allowing for non-Poisson satellite occupation distributions results in a slightly lower value for (). These constraints are in excellent agreement (at the level) with the cosmic microwave background constraints from WMAP. This demonstrates that the use of a realistic and accurate model for galaxy bias, down to the smallest non-linear scales currently observed in galaxy surveys, leads to results perfectly consistent with the vanilla CDM cosmology.
keywords:galaxies: halos — large-scale structure of Universe — dark matter — cosmological parameters — gravitational lensing — methods: statistical
The last two decades have seen the emergence of a concordance cosmological model which describes the formation and evolution of cosmic structure in a scenario known as CDM. In these cosmological models, gravity is described by General Relativity, dark matter and dark energy are the major constituents of the Universe (with normal ‘baryonic’ matter only contributing percent), and density perturbations are seeded by quantum fluctuations in a scalar field, the inflaton, that dominated the energy density shortly after the Big Bang. In its most basic (‘vanilla’) form, the CDM model assumes a flat geometry, dark energy is modeled as Einstein’s cosmological constant, neutrino mass is assumed to be negligible, and the initial power spectrum of density perturbations is assumed to be a single power-law. Such CDM cosmologies are described by 5 parameters: the energy densities (in terms of the critical density) of baryons, , and cold dark matter, , the spectral index, , and normalization, , of the initial power spectrum, and the Hubble parameter, . The flat geometry implies that , and is strongly supported by the location of the first peak in the angular power spectrum of cosmic microwave background (CMB) temperature fluctuations (e.g., Balbi et al. 2000; Lange et al. 2001; Pryke et al. 2002; Netterfield et al. 2002; Ruhl et al. 2003) combined with the results on the Hubble constant from the Hubble Key Project (Freedman et al. 2001).
All these cosmological parameters have now been constrained at the few (-) percent level by a variety of probes, including, among others, temperature anisotropies in the CMB (e.g., Spergel et al. 2003, 2007; Reichardt et al. 2009; Dunkley et al. 2009; Komatsu et al. 2009, 2011), Cepheid distances (e.g., Freedman et al. 2001; Sandage et al. 2006; van Leeuwen et al. 2007), high redshift supernovae Ia (e.g., Riess et al. 1998; Perlmutter et al. 1999, Astier et al. 2006; Kowalski et al. 2008), measurements of the primordial deuterium abundance (e.g., Burles, Nollett & Turner 2001; O’Meara et al. 2006), cluster abundances (e.g., Vikhlinin et al. 2009; Rozo et al. 2010; Benson et al. 2011), cosmic shear (e.g., Benjamin et al. 2007; Fu et al. 2008; Lin et al. 2011), the integrated Sachs-Wolfe effect (e.g., Ho et al. 2008; Giannantonio et al. 2008), the Ly forest (e.g., Viel, Weller & Haehnelt 2004; McDonald et al. 2005; Desjacques & Nusser 2005), and strong gravitational lensing (e.g., Koopmans et al. 2003; Oguri et al. 2008). Despite some tension between a few subsets of all these independent constraints (see e.g., Dunkley et al. 2009 for a comprehensive overview), overall they are in good mutual agreement, giving rise to the notion of a true concordance cosmology.
Another potentially powerful probe for cosmology is the (large-scale) distribution of galaxies. Although stars make a negligible contribution to the total energy density of the Universe, the light from stars in galaxies can be observed directly and over cosmological scales, making galaxies useful tracers of the underlying dark matter density field. Unfortunately, the connection between galaxies and (dark) matter is muddled by the fact that galaxies are biased tracers of the mass distribution. The main problem is that this ‘galaxy bias’ is known to be extremely complicated: it is stochastic (e.g., Dekel & Lahav 1999; Tegmark & Bromley 1999), depends on galaxy properties such as luminosity, color and/or morphological type (e.g., Park et al. 1994; Guzzo et al. 2000; Norberg et al. 2001, 2002; Zehavi et al. 2005, 2011; Wang et al. 2007), and is scale dependent on small scales (e.g., Pervical et al. 2007; Reid, Spergel & Bode 2009; Cacciato et al. 2012). Based on these considerations, it is not surprising that galaxy bias is generally considered a nuisance when using galaxies to constrain cosmology. However, galaxy bias also contains a wealth of information regarding galaxy formation, especially on small scales (e.g., Cacciato et al. 2012). After all, it is the (poorly understood) physics of galaxy formation that determines where, how and with what efficiency galaxies form within the dark matter density field (see Mo, van den Bosch & White 2010). Therefore, ideally, one would like to simultaneously solve for cosmology and galaxy bias.
Early attempts to do so used galaxy power spectra measured from the two-Degree Field Galaxy Redshift Survey (2dFGRS; Colless et al. 2003) and/or Sloan Digital Sky Survey (SDSS; York et al. 2000) combined with a phenomenological fitting function for the non-linear, scale-dependent bias and marginalized over its free parameters (e.g., Cole et al. 2005; Tegmark et al. 2006; Padmanabhan et al. 2007). However, this approach has two problems. First, the fitting function used to describe the galaxy bias has no logical connection to galaxy formation. For this reason the free parameters used to parameterize the galaxy bias are merely considered as nuisance parameters. Second, as pointed out by Sánchez & Cole (2008) and Yoo et al. (2009), the particular fitting function used is often a poor description of the true scale dependence of galaxy bias, causing biased estimates of the cosmological parameters, especially for non-vanilla CDM cosmologies (e.g., Hamann et al. 2008). It is generally believed that the latter problem explains why there is still some tension between cosmological parameters (mainly ) inferred from different galaxy power spectra, or from using the same power spectrum, but using data covering different scales (e.g., Percival et al. 2007; Sánchez & Cole 2008; Dunkley et al. 2009)
This indicates that it is prudent to use a realistic, physically motivated model for galaxy bias. Under the assumption that all galaxies reside in dark matter haloes, a natural, realistic model for galaxy bias is provided by halo occupation models, which describe, in a statistical sense, how galaxies are distributed over dark matter haloes (e.g., Jing, Mo & Börner 1998; Peacock & Smith 2000; Scoccimarro et al. 2001; Berlind & Weinberg 2002; Yang, Mo & van den Bosch 2003). When combined with the halo model, which describes the (non-linear) matter distribution in terms of the dark matter halo building blocks (e.g., Neyman & Scott 1952; Seljak 2000; Ma & Fry 2000; Cooray & Sheth 2002), these halo occupation models provide a complete, accurate, and easy-to-interpret description of galaxy bias, all the way from the large, linear scales down to the small, non-linear scales of individual dark matter haloes.
The problem with this approach, though, is that the halo occupation models are also cosmology dependent, so that one typically needs constraints in addition to just large scale clustering. Several attempts have been made along these lines. Abazajian et al. (2005) simply used strong priors from the WMAP results. Yang et al. (2004) and Tinker et al. (2007) used peculiar velocities as inferred from the redshift space distortions in the two-point correlation function, and argued for a relatively low value of (for a vanilla CDM cosmology with ). Similar conclusions were reached by van den Bosch et al. (2003b) and Tinker et al. (2005), who used constraints on the (average) mass-to-light ratios of clusters, rather than peculiar velocities. Somewhat puzzling, a more recent analysis by Tinker et al. (2012) used the mass-to-number ratio of clusters, rather than the mass-to-light ratios, and found a relatively high value for of (again for ).
In this paper we use a combination of galaxy clustering and galaxy-galaxy lensing, as well as constraints on galaxy abundances, in order to simultaneously constrain cosmology and galaxy bias. Since galaxy-galaxy lensing probes the mass associated with the lensing galaxies, this is similar to using mass-to-light ratios as constraints. It has the advantage, though, that it probes mass-to-light ratios over a wide range in halo masses, and that the same halo occupation model used to compute the clustering of galaxies can also be used to compute the galaxy-galaxy lensing signal (e.g., Guzik & Seljak 2002; Yoo et al. 2006; Cacciato et al. 2009). A first application of this idea by Seljak et al. (2005), when combined with WMAP constraints, yielded a relative high value for of (for ). More recently, two different analyses based on the same galaxy-galaxy lensing data by Cacciato et al. (2009) and Li et al. (2009) both argued that a flat CDM cosmology with is in much better agreement with the data than a model, thus favoring again a relatively low value for . In this paper we improve on all these previous methods by (i) simultaneously constraining galaxy bias and cosmology, (ii) using a much more accurate analytical model, (iii) using the latest clustering data, (iv) modelling the latter accounting for residual redshift space distortions, (v) marginalizing over uncertainties related to the detailed structure of dark matter haloes and the scale dependence of the halo bias, and (vi) subjecting the analysis to a number of detailed tests that address how the results depend on certain assumptions inherent to the model.
This paper is the third in a series. In van den Bosch et al. (2012; hereafter Paper I), we presented the analytical model, which we calibrated and tested using detailed mock catalogs constructed from high resolution -body simulations. We demonstrated that our analytical model is accurate at the level of 10% or better over the entire range of scales covered by the data. In More et al. (2012b; hereafter Paper II), we presented a Fisher matrix analysis to identify parameter-degeneracies and to assess the accuracy with which various cosmological parameters can be constrained using our methodology. We demonstrated that the method can simultaneously constrain halo occupation statistics and cosmology, and we forecasted that, using existing data from the SDSS, we should be able to put constraints on and that are among the tightest ever achieved. In this paper we apply our method to existing data from the SDSS111A preliminary version of the main results presented in this paper were published in a conference proceedings by More et al. (2012a).. Although, as demonstrated in paper II, our method is also able to constrain extensions to the vanilla CDM cosmology, such as neutrino mass and the equation of state of dark energy, in this paper we focus solely on vanilla CDM cosmologies, and in particular on constraining the combination . We defer constraining neutrino mass, dark energy, and other modifications to the vanilla CDM cosmology to future papers.
This paper is organized as follows. In §2, we introduce the SDSS data used to constrain our models. In §3, we briefly review our analytical model to compute the galaxy luminosity function, the galaxy-galaxy correlation function, and the galaxy-galaxy lensing signal using the halo model combined with a model that describes halo occupation statistics as function of galaxy luminosity. The Bayesian analysis, used to infer posterior distributions for the cosmological parameters and for the parameters that describe the halo occupation statistics, is described in §4. Our main results are presented in §5, while §6 describes a number of tests that address the sensitivity of our results to several model assumptions. We summarize our findings in §7.
Throughout this paper, unless specifically stated otherwise, all radii and densities are in comoving units, and log is used to refer to the 10-based logarithm. Quantities that depend on the Hubble parameter are written in units of , defined above.
The data used to constrain our models consists of three components: galaxy abundances, in the form of the galaxy luminosity function, galaxy clustering, in the form of projected correlation functions for six different luminosity bins, and galaxy-galaxy lensing, in the form of excess surface densities (ESD), once again for six different luminosity bins. All these measurements are obtained from the Sloan Digital Sky Survey (SDSS; York et al. 2000).
We use the -band galaxy luminosity function, , (hereafter LF) of Blanton et al. (2003a)222available at http://cosmo.nyu.edu/mb144/lf.html, sampled at 32 magnitudes in the range , where indicates the -band magnitude of galaxies KE corrected to following the procedure of Blanton et al. (2003b). For each of these magnitude bins we use the (statistical) errors on quoted by Blanton et al. (2003a). Unfortunately, we do not have a full covariance matrix for this data set. There are two main sources of covariance for the luminosity function: magnitude errors, which cause covariance between neighboring magnitude bins, and large scale structure (‘sample variance’)333Following Scott, Srednicki & White (1994) we use the term ‘sample variance’, rather than the more common ‘cosmic variance’. The LF of Blanton et al. is sampled at magnitude intervals of mag, which is much smaller than the typical error on individual magnitudes ( mag). In order to suppress the covariance due to these magnitude errors, we only sample the LF of Blanton et al. at 32 magnitude intervals of , roughly three times as large as the typical magnitude error. This should eliminate virtually all covariance due to errors in the magnitudes of individual galaxies. This leaves the covariance due to sample variance, which can be ‘modeled’ to reasonable accuracy as an up or down shift of the entire LF (i.e.,. the error bars are fully covariant, e.g. Blanton et al. 2003a). Throughout this study, we ignore this sample variance, which effectively implies that we assume that the volume probed by the SDSS is a fair representation of the average Universe (see discussion in §6.4).
The galaxy clustering data used in this paper are taken from Zehavi et al. (2011), and based on the SDSS DR7 (Abazajian et al. 2009). Using all galaxies in the main galaxy sample with apparent magnitudes , Zehavi et al. measured the projected correlation functions, , over the radial range for six volume limited samples (see Table 1). These have been obtained according to
with or , depending on the luminosity sample used (see Table 1). Here is the projected separation between two galaxies, is the redshift-space separation along the line-of-sight, and is the two-dimensional correlation function, which is anisotropic due to the presence of peculiar velocities. As discussed in Paper I, the fact that is finite results in residual redshift space distortions that need to be corrected for in the modeling (see also § 6.3). The errors on are characterized by their full covariance matrices, kindly provided to us in electronic format by I. Zehavi, and obtained from 144 spatially contiguous subsamples using the jackknife technique.
Finally, for the galaxy-galaxy lensing data we use the excess surface densities, , covering the radial range , obtained by Seljak et al. (2005) and Mandelbaum et al. (2006), and kindly provided to us in electronic format by R. Mandelbaum. These measurements have been obtained using a catalogue of lens galaxies with apparent magnitude in the redshift range taken from the main galaxy catalogue of the SDSS Data Release 4 (Adelman-McCarthy et al. 2006). This sample is split in 8 (flux-limited) luminosity bins. However, since both the faintest and brightest bins have extremely poor signal-to-noise, we only use the 6 intermediate luminosity bins listed in Table 2. We refer the reader to Mandelbaum et al. (2006) for a detailed description of the data and of the method used to determine the excess surface density profiles. Since the covariance in is only very small over the radial scales covered by the data (R. Mandelbaum, private communication), we only use the diagonal elements in what follows.
Throughout this paper we refer to these three data sets as the LF, WP and ESD data. Note that each of these data sets adopted a flat CDM cosmology with (LF and WP) or (ESD) when computing distances and/or absolute magnitudes. Changing the assumed cosmology, as we do in our analysis, in principle therefore also has a small impact on the observational measurements by changing the distance-redshift relation and thus shifting galaxies among luminosity bins and galaxy pairs among radial separation bins. However, in this paper we restrict ourselves to cosmologies that only differ mildly from a CDM cosmology with . Even at our outer redshift limit of z = 0.2, the effect of lowering from 0.3 to 0.25 is only 1 percent in distance, so the measurements used here are effectively independent of cosmological parameters within their observational uncertainties. Therefore, we ignore this small effect in what follows, and always use the observational data as described above.
3 Model Description
As described in detail in Paper I, the observables444The projected correlation function and the excess surface density are modeled per luminosity bin . However, to keep the notation concise, we do not explicitly write down the dependence on and . , and can be computed for a given cosmology, which determines the properties of the dark matter distribution (e.g, halo mass function, halo bias function, halo density profiles), and a given description of the galaxy-dark matter connection (i.e., halo occupation statistics). The dependence on redshift is included to emphasize that we model each observable at its mean redshift. For the luminosity function we adopt , while the mean redshifts for the different luminosity bins of the and measurements are listed in Tables 1 and 2, respectively.
In this section we give a concise overview of our model. Readers interested in a more thorough description are referred to Paper I, while those readers that are already familiar with our model, or that are mainly interested in the results, may want to skip this section and proceed immediately to § 4.
3.1 Cosmological Parameters
Throughout this paper we consider ‘vanilla’ CDM cosmologies in which gravity is described by standard General Relativity, neutrino mass is negligible, the initial power spectrum is a single power-law, and dark energy is modeled as Einstein’s cosmological constant with . These cosmologies are completely specified by five parameters; the matter density in units of the critical density, , the normalization of the matter power spectrum, , the Hubble parameter , the initial spectral index of the matter power spectrum, , and the baryon density in units of the critical density, . Hence, our cosmological model parameters are described by the vector
Note that the baryon density only enters in our analysis in the transfer function, and always in the combination .
The main goal of this paper is to use the observational data on , and discussed in §2 to constrain and . Throughout this paper we will therefore not use any priors on these two parameters. For , and , on the other hand, we include prior information from the seven year analysis of the cosmic microwave background data from WMAP (hereafter WMAP7; Komatsu et al. 2011), as described in §4 below. For this reason we refer to , and as our secondary cosmological parameters in what follows.
3.2 The Conditional Luminosity Function
Under the assumption that each galaxy resides in a dark matter halo, , and can be computed using a statistical description of how galaxies are distributed over dark matter haloes of different mass. To that extent we use the conditional luminosity function (hereafter CLF) introduced by Yang et al. (2003). The CLF, , specifies the average number of galaxies with luminosities in the range that reside in a halo of mass . We split the CLF in two components,
where describes the contribution due to central galaxies (defined as those galaxies that reside at the center of their host halo), while characterizes satellite galaxies (those that orbit around a central). Throughout we ignore a potential redshift dependence of the CLF. Since the data that we use to constrain the CLF only covers a narrow range in redshift, this assumption will not have a strong impact on our results.
Our parameterization of the CLF model is motivated by the results obtained by Yang, Mo & van den Bosch (2008) from a large galaxy group catalogue (Yang et al. 2007) extracted from the SDSS Data Release 4, and by Tal et al. (2012) from a study of the luminosity function of satellite galaxies of luminous red galaxies. In particular, the CLF of central galaxies is modeled as a log-normal function:
and the satellite term as a modified Schechter function:
which decreases faster than a Schechter function at the bright end. Note that , , , and are all functions of the halo mass .
Following Cacciato et al. (2009), and motivated by the results of Yang et al. (2008) and More et al. (2009, 2011), we assume that , which expresses the scatter in of central galaxies at fixed halo mass, is a constant (i.e. is independent of halo mass and redshift). In addition, for , we adopt the following parameterization;
Hence, for and for . Here is a characteristic mass scale, and is a normalization.
For the satellite galaxies we adopt
(i.e., the faint-end slope of is independent of mass and redshift), and
with . Note that neither of these functional forms has a physical motivation; they merely were found to adequately describe the results obtained by Yang et al. (2008) from the SDSS galaxy group catalog.
To summarize, our parameterization of the CLF thus has a total of nine free parameters, characterized by the vector
3.3 Galaxy Luminosity Function
Once the CLF is specified, the galaxy luminosity function at redshift , , simply follows from integrating over the halo mass function, ;
In what follows, we will always be concerned with galaxies in a specific luminosity interval . The average number density of such galaxies follows from the CLF according to
is the average number of galaxies with that reside in a halo of mass .
The first step towards computing the projected correlation functions, , and ESD profiles, , is to compute the galaxy-galaxy power spectrum, , and the galaxy-matter cross power spectrum, . These power spectra are the Fourier space analogs of the galaxy-galaxy and galaxy-matter correlation functions.
The galaxy-galaxy power spectrum can be expressed as a sum of the one-halo (1h) and the two-halo (2h) terms, each of which can be further subdivided based upon the type of galaxies (central or satellite) that contribute to the power spectrum, i.e.,
Similarly, the galaxy-matter power spectrum can be written as
As shown in paper I, these terms can be written in compact form as
where ‘x’ and ‘y’ are either ‘c’ (for central), ‘s’ (for satellite), or ‘m’ (for matter), describes the power-spectrum of haloes of mass and (see Appendix A), and we have defined
Here and are the average number of central and satellite galaxies in a halo of mass , which follow from Eq. (13) upon replacing by and , respectively. Furthermore, is the Fourier transform of the normalized number density distribution of satellite galaxies that reside in a halo of mass , and is the Fourier transform of the normalized density distribution of dark matter within a halo of mass .
3.5 Computing and
Once and have been determined, it is fairly straightforward to compute the projected galaxy-galaxy correlation function, , and the excess surface density (ESD) profile, . We start by Fourier transforming the power-spectra to obtain the two-point correlation functions:
where ‘x’ is either ‘g’ (for galaxies) or ‘m’ (for matter).
The projected galaxy-galaxy correlation function, , is related to the real-space galaxy-galaxy correlation function, , according to
Here is the maximum integration range used for the data (see Table 1), is the separation perpendicular to the line-of-sight, is the separation between the galaxies in redshift space, is the Legendre polynomial, and , , and are given by
with the scale factor, the linear growth rate, and
the mean bias of the galaxies in consideration. Note that Eq. (22) accounts for the large scale redshift space distortions due to infall (the ‘Kaiser’-effect), which is necessary because the measurements for have been obtained using a finite . Note that the in Eqs. (23)-(25) is the non-linear galaxy-galaxy correlation function. Although the Kaiser formalism (Kaiser 1987) is only strictly valid in the linear regime, this simple modification results in a more accurate correction for the residual redshift space distortions (see Paper I for details). As shown in Paper I, not taking these residual redshift space distortions into account results in systematic errors that can easily exceed 20 percent on large scales (), causing systematic errors in the inferred galaxy bias (see also Padmanabhan, White & Eisenstein 2007; Norberg et al. 2009; Baldauf et al. 2010; More 2011).
Finally, the excess surface density profile, , is defined as
Here is the projected surface mass density, which is related to the galaxy-dark matter cross correlation, , according to
3.6 Model Ingredients
In this subsection we briefly describe the ingredients of our model. A more detailed description can be found in Paper I.
In our fiducial model we require both the linear and the non-linear power spectra of the matter distribution, and , respectively. The latter enters in the computation of the halo-halo correlation function, as described in detail in Appendix A. Throughout we compute using the fitting formula of Smith et al. (2003)555We use the small modification suggested on John Peacock’s website http://www.roe.ac.uk/jap/haloes/, although we have verified that this has no significant impact on any of our results., while for we use the linear transfer function of Eisenstein & Hu (1998), neglecting any contribution from neutrinos and assuming a CMB temperature of 2.725K (Mather et al. 1999).
Throughout, we define dark matter haloes as spheres with an average density of 200 times the background density. We assume that their density profiles follow the NFW profile (Navarro, Frenk & White 1997), with a concentration-mass relation given by
Here is the average concentration-mass relation of Macciò et al. (2007), properly converted to our definition of halo mass. We treat as a free nuisance parameter which accounts for (i) the fact that there is an uncertainty of percent in the average concentration mass relation as obtained by different authors (e.g., Navarro, Frenk & White 1997; Eke, Navarro & Steinmetz 2001; Bullock et al. 2001; Macció et al. 2007; Zhao et al. 2009), and (ii) realistic dark matter haloes are triaxial, rather than spherical, have substructure, and have scatter in the concentration-mass relation. As shown in Paper I, setting modifies the one-halo term of by more than 20 percent on small scales (). As discussed at length in Paper I, an uncertainty in of this amount is more than adequate to capture the inaccuracies in our model that arise from the various oversimplifications and uncertainties regarding the structure of dark matter haloes. Hence, in what follows, we adopt a Gaussian prior on , centered on and with a standard deviation .
For the halo mass function, , and the halo bias function, , we use the fitting functions of Tinker et al. (2010). Note that these functions obey the normalization condition
as required by the fact that, on average, on large scales matter is unbiased with respect to itself. As described in Appendix A, the scale dependence of the halo bias in the quasi-linear regime is described by a modified version of the empirical fitting function of Tinker et al. (2005). This modification is needed to account for the fact that we use a different definition of dark matter haloes, and is characterized by one free ‘nuisance’ parameter, . In Paper I we calibrated using numerical -body simulations of structure formation in a CDM cosmology, and found that can accurately fit the simulation results. In order to account for the fact that we cannot rule out that is cosmology dependent, we include an uncertainty of percent on in our cosmological analysis. In particular, we treat as a free parameter, but adopt a Gaussian prior centered on and with .
For our fiducial model, we assume that the radial number density distribution of satellite galaxies follows that of the dark matter particles, i.e., . In other words, we assume that satellite galaxies follow a NFW profile with the same concentration-mass relation (Eq. ) as dark matter haloes. In §6.1 we relax this assumption and examine how changes in impact on our results.
Finally, we emphasize that the expression for the 1-halo term of the galaxy-galaxy correlation function (Eq. ) has made the implicit assumption that the halo occupation number of satellite galaxies obeys Poisson statistics, i.e., that (see Paper I for details). In §6.2 we will relax this assumption, and explore how deviations of from a Poisson distribution impact on our results.
The main goal of this paper is to obtain constraints on the cosmological parameters and , and the halo occupation distribution, as characterized by the CLF, using the SDSS data described in §2 and the analytical model described in §3. We use Bayesian inference techniques to determine the posterior probability distribution of the model parameters , given the data . According to Bayes’ theorem,
where is the likelihood of the data given the model parameters, is the prior probability of these parameters, and
is the marginal probability of the data, also called evidence for the model. Since, we do not intend to perform model selection, the evidence just acts as a normalization constant which need not be calculated. Therefore the posterior distribution is given by
where is a sum of the following terms
The first three terms quantify the goodness of the fit to the data and correspond to the likelihood of the data given the parameters, while the last term corresponds to the prior information we adopt. The likelihood terms are given by
Here denotes the model prediction for the observable , is the corresponding error, is the vector of the projected clustering measurements in the –th luminosity bin, and is the covariance matrix of these measurements.
As detailed in Paper I, our analytical model is accurate at the level of 10 (in most cases 5) percent, in reproducing the 3-dimensional galaxy-galaxy correlation and the galaxy-matter cross-correlation from mock galaxy catalogs. Since the differences are not systematic, the accuracy is expected to be much better for the projected galaxy-galaxy correlation function, , and the galaxy-galaxy lensing signal, . Therefore, we do not account for any systematic uncertainty from our modeling in our likelihood estimate.
For our fiducial model, the set of model parameters, , includes our primary cosmological parameters of interest, and , the set of secondary cosmological parameters , the CLF parameters and a set of nuisance parameters . Throughout we adopt uniform, non-informative priors on our primary cosmological parameters as well as on all CLF parameters. For the secondary cosmological parameters, , we include priors from the WMAP7 analysis. In order to obtain the covariance matrix of these parameters (), we have used the Monte Carlo Markov chains from the WMAP7 analysis (kindly provided to us by E. Komatsu) and marginalized over all other parameters present in their analysis. Finally, for the two nuisance parameters we adopt Gaussian priors, as discussed in §3.6. Hence, we have that
where the summation is over the two nuisance parameters and .
We sample the posterior distribution of our model parameters given the data using a Monte-Carlo Markov chain (MCMC). In particular, we implement the Metropolis-Hastings algorithm to construct the MCMC (Metropolis et al. 1953; Hastings 1970). At any point in the chain, a trial model is generated using a method specified below. The chi-squared statistic for the trial model, , is calculated using Equations (36)–(40). This trial model is accepted to be a member of the chain with a probability given by
where denotes the for the current model in the chain. We initialize the chain from a random position in our multi-dimensional parameter space and obtain a chain of models. We discard the first models (the burn-in period) allowing the chain to sample from a more probable part of the distribution. We use this chain of models to estimate the confidence levels on the parameters and different observables of interest.
A proper choice of the proposal distribution is very important in order to achieve fast convergence and a reasonable acceptance rate for the trial models. The posterior distribution in a multi-dimensional parameter space, such as the one we are dealing with, will have numerous degeneracies and in general can be very difficult to sample from. We have adopted the following strategy to overcome these difficulties. During the first half of the burn-in stage, we chose an independent Gaussian proposal distribution for every model parameter, as is common for the Metropolis-Hastings algorithm. Half-way through the burn-in stage, we perform a Fisher information matrix analysis at the best fit model found thus far. The Fisher information matrix, given by
is a symmetric matrix, where denotes the number of parameters in our model, and is the likelihood. The inverse of the Fisher matrix gives the covariance matrix, , of the posterior constraints on the model parameters (see Paper II)666The subscript ‘prop’ indicates that this matrix is used to describe the proposal distribution.. More importantly, the eigenvectors of the covariance matrix are an excellent guide to the numerous degeneracies in the posterior distribution, and the corresponding eigenvalues set a scale for how wide the posterior ought to be in a given direction. Therefore, for the second half of the burn-in period, we utilize this information and use a proposal distribution which is a multi-variate Gaussian centered at the current value of the parameters and with a covariance equal to . In practice, the trial model () can be generated from the current model () using
where is a vector consisting of standard normal deviates, the matrix is such that , and is a parameter that we have chosen to achieve an average acceptance rate of %. We repeat the Fisher matrix analysis once again at the end of the burn-in period (using the best fit model found thus far) and use the covariance matrix to define our proposal distribution to be used for the MCMC. We have found this strategy to be extremely efficient in sampling our posterior distributions777A general-purpose python implementation of the MCMC sampler we have used in our work is available from the authors upon request..
Having described the data, the model, and the methodology, we now turn to our results. In this section we describe the cosmological constraints obtained for our Fiducial model, whereas §6 discusses the robustness of these results to model variations. The constraints on galaxy bias, as characterized via the CLF, are discussed in §5.3.
As discussed above, our Fiducial model consists of 16 free parameters; the two primary cosmological parameters of interest, and , for which we use uniform, non-informative priors, the secondary cosmological parameters , and , for which we use priors from WMAP7 (including their covariance), the 9 CLF parameters that describe the halo occupation statistics, also with uniform, non-informative priors, and finally the 2 nuisance parameters, and , for which we adopt Gaussian priors as described in §3.6. With a grand total of 182 constraints (32 data points for the LF, six bins of 13 data points each for the projected correlation function888Although the galaxy-galaxy clustering data points have covariance, we have verified that the covariance matrix for each luminosity bin has rank equal to , and therefore does not reduce the number of constraints. and six bins of 12 ESD data points), this implies degrees of freedom, which is the number we have used to compute the reduced values listed in the final column of Table 3.
Figs. 1 and 2 compare the predictions of the Fiducial model (shaded regions, indicating the 95% confidence levels) to the data used to constrain the model (solid dots with error bars, indicating the 68 % confidence levels). Fig. 1 shows that the model accurately fits the -band galaxy luminosity function. Although most data points agree with the model predictions at the level, the data reveals a few small ‘wiggles’ at the faint end that are not reproduced by the model, and which contribute dominantly to , the value of which is listed in Table 3.
The left-hand side of Fig. 2 shows the projected correlation functions, , for six different magnitude bins. We caution that, because of the covariance in the data, which is accounted for in the modeling (see §4), the quality of the fit cannot be judged by eye. However, it is evident from the values of the best-fit Fiducial model (see Table 3), that the total is clearly dominated by . In particular, , even though the projected correlation functions only have times as many data points. It turns out is dominated by the contribution from the data in the magnitude bin. Interestingly, this bin covers the volume that encloses the Sloan Great Wall (SGW), a huge supercluster at and the largest coherent structure detected in the SDSS (Gott et al. 2005). As discussed in Zehavi et al. (2011), pruning the data sample so as to exclude the SGW region results in a significantly reduced clustering strength for galaxies in the magnitude range (i.e., the correlation length is reduced from to ). We return to this issue, and its potential impact on our cosmological constraints, in §6.4 when we discuss the potential impact of sample variance.
Finally, the right-hand side of Fig. 2 shows the excess surface densities, , again for six different magnitude bins as indicated. The model nicely reproduces the overall trends in the data, with only a few data points that fall outside the 95% confidence region of the model. Overall, we conclude that our Fiducial model is consistent with the data at a satisfactory level. In particular, the most important features in the data are nicely reproduced by the model and find a natural explanation within the framework of the halo model. For example, the fact that brighter galaxies reveal stronger clustering and higher excess surface densities is consistent with the common notion that brighter galaxies reside in more massive haloes. The lensing signal is directly sensitive to this aspect because it probes the matter distribution around galaxies, whereas the clustering signal is affected by it only indirectly due to the fact that more massive haloes are more strongly clustered than less massive ones (e.g., Mo & White 1996). Also, the relatively weak deviations of and from pure power-laws typically reflect transitions from scales where the signal is dominated by different components of the power spectra. Examples are the 1-halo to 2-halo transition (e.g., Zehavi et al. 2004) and the 1-halo central to 1-halo satellite transition for the excess surface densities (e.g., Cacciato et al. 2009).
5.1 Cosmological Parameters
Fig. 3 shows the constraints on our two primary cosmological parameters of interest; and . The blue contours show the 68% and 95% CLs of the joint two-dimensional, marginalized posterior distribution obtained from our simultaneous analysis of the abundance, clustering and lensing of galaxies in the SDSS. The green contours show the corresponding CLs for the WMAP7 analysis of the CMB (Komatsu et al. 2011), and are shown for comparison. Note that our results are in excellent agreement with those from WMAP7, strengthening the case for a true concordance cosmology. In particular, our analysis yields and (both 95% CL), while the WMAP7 analysis has and (both 95% CL). Note also that the degeneracy between and inherent in our analysis runs perpendicular to that inherent in the CMB data. This indicates that a combined analysis will be able to significantly tighten the constraints on and (see also Paper II). Finally, Fig. 3 suggests that our constraints are even tighter than those from the WMAP7 analysis. However, we emphasize that this is not a fair comparison since we have used priors from WMAP7 on the secondary cosmological parameters , and , but not on or (see Paper II for the case with no priors on , and ).
Fig. 4 shows the one-dimensional (histograms) and joint two-dimensional (contour plots) marginalized posterior distributions on all five cosmological parameters. Solid contours indicate the 68% and 95% CLs obtained from the analysis presented here, while the dotted contours are the 68% and 95% CLs from the WMAP7 analysis, shown for comparison. The strongest parameter degeneracies are between and (cross-correlation coefficient ), between and (), and between and (). All other combinations are only weakly correlated with .
Overall, there is good agreement between our constraints and those inferred from the WMAP7 data (see also Table 4). However, there is some tension regarding the secondary cosmological parameters, which is evident from the fact that the posterior and prior distributions (indicated by red, solid curves) reveal an offset. This is most pronounced for the Hubble parameter ; whereas the WMAP7 prior used has our posterior distribution has (both 95% CL). For comparison, the revised parallaxes for Cepheid stars by van Leeuwen et al. (2007) raises the value for from the HST Key Project from (Freedman et al. 2001) to (68% CL), and the Cepheid-based determination of Sandage et al. (2006) from to (68% CL). Hence, despite some tension with the WMAP7 based constraints, our posterior distribution for the Hubble constant is well within the range of values inferred from Cepheids. Interestingly, our constraint on is correlated with the constraints on both and : according to our analysis, a relatively low Hubble parameter of implies and , while for the results presented here suggest that and . Clearly, an improved constraint for the Hubble parameter could help to significantly tighten the constraints on both and .
5.2 Nuisance parameters
As discussed in §3.6, our model contains two nuisance parameters: , which enters in the description of the scale dependence of the halo bias, and , which sets the normalization of the halo concentration-mass relation. As discussed in detail in Paper I, the freedom in also characterizes model uncertainties arising from our oversimplifications regarding the structure of dark matter haloes.
The upper panels of Fig. 5 show the posterior distributions for (left-hand) and (right-hand) of our Fiducial model. The contours in the other panels show 68% and 95% confidence levels of the joint two-dimensional marginalized posteriors with and . Clearly, neither nor shows significant correlation with or (in all cases the cross-correlation coefficient ). As already discussed in Paper II, this is an important result, as it indicates that the uncertainties in the scale dependence of the halo bias and the oversimplifications regarding the structure of dark matter haloes do not have a significant impact on the cosmological constraints presented here.
The solid, red curves in the upper panels of Fig. 5 reflect the Gaussian priors that we imposed on our model. In the case of , the posterior distribution (, 95% CL) is significantly narrower than the prior distribution (, 95% CL), indicating that the prior did not have a significant impact on our results. In the case of , however, the posterior distribution (, 95% CL) is clearly offset from the prior (, 95% CL) to larger values. This might raise concern that a less restrictive prior might have resulted in significantly different cosmological constraints. However, this is not the case for the following two reasons. First, as already mentioned above, and discussed in more detailed in Paper II, is only very poorly correlated with the cosmological parameters. Second, as discussed in the Appendix, when increases the radial bias function asymptotes to the empirical fitting function of Tinker et al. (2005). In other words, once increases beyond a certain value, any further increase has zero impact. For the best-fit cosmology of our Fiducial model, this critical value of is .
5.3 The Galaxy-Dark Matter Connection
One of the powerful aspects of the method used here, is that the data is used to simultaneusly constrain cosmology and halo occupation statistics. Fig. 6 shows the one-dimensional (histograms) and joint two-dimensional (contour plots) marginalized posterior distributions of model Fiducial for the nine CLF parameters that describe the relation between galaxy luminosity and halo mass. The medians and 95% CLs are also listed in Table 5. All nine parameters are tightly constrained, with tight degeneracies between the parameters , and , that describe the normalization of the satellite CLF (see Eq. ), and between the parameters , and , that describe the relation between halo mass and the luminosity of its central galaxy (see Eq. ).
The constraint on the faint-end slope of the satellite CLF is (95% CL), in good agreement with results obtained from galaxy group catalogues (e.g., Eke et al. 2004; Yang et al. 2008). The constraints on the scatter in the CLF of central galaxies is (95% CL), which is in excellent agreement with a variety of other constraints, from satellite kinematics (More et al. 2009), from clustering and lensing (Cacciato et al. 2009; Moster et al. 2010), and from galaxy group catalogues (Yang et al. 2008). Interestingly, such an amount of scatter is also in excellent agreement with predictions from semi-analytical models for galaxy formation (e.g., Wang et al. 2006; see also More et al. 2009).
In the left-hand panel of Fig. 7 we compare the constraints on the - relation of our Fiducial model (shaded area, indicating the 68% confidence region) to the results obtained by Yang et al. (2008) from the SDSS galaxy group catalogue (symbols with errorbars indicating the 68% confidence level). Here we have converted the group masses listed in their Table 1 to our definition of halo mass and the cosmology of the best-fit Fiducial model using the abundance matching technique described in Eq. (15) of Yang et al. (2007). Finally, the right-hand panel of Fig. 7 compares the satellite fractions as function of galaxy luminosity obtained from our Fiducial model (blue shaded area, indicating the 68% CL) to constraints obtained by Cooray (2006; black, open squares), Yang et al. (2008; red, open circles), van den Bosch et al. (2007; green shaded area, indicating the 68% CLs), and Tinker et al. (2007; orange shaded area, indicating the 68% CL). All these constraints are obtained comparing halo occupation models to data from the SDSS or 2dFGRS, and are in excellent agreement with each other and with the constraints from our Fiducial model.
Based on all these comparisons, we conclude that the constraints on the galaxy-dark matter connection for our Fiducial model are in excellent agreement with a wide variety of independent constraints. We emphasize that, contrary to many other studies, our combined analysis of abundance, clustering, and lensing of galaxies also accounts for uncertainties in cosmological parameters. This strongly supports that our method of simultaneously constraining cosmology and halo occupation statistics yields reliable results.
The Fiducial model, discussed in the previous section, relies on two assumptions regarding satellite galaxies that may not be entirely accurate. These concern the radial number density distribution of satellites and the Poisson nature of the satellite occupation numbers. In this section we gauge the impact of these assumptions on our results. In addition, we also address the importance of properly accounting for the residual redshift space distortions present in the projected correlation functions used to constrain the models. We do so by running a series of MCMCs in which we include small variations with respect to our Fiducial model. These models, and their respective -values for the best-fit model in the chain, are listed in Table 3, while the cosmological constraints are summarized in Table 4.
6.1 Radial Number Density Profile of Satellites
In our Fiducial model we have assumed that the radial number density distribution of satellite galaxies in a halo of mass follows an NFW profile with a concentration parameter that is identical to that of the dark matter density profile, i.e., we assumed that (see §3.4). This has observational support from a number of studies of the radial distribution of satellite galaxies in groups and clusters (e.g., Beers & Tonry 1986; Carlberg, Yee & Ellingson 1997a; van der Marel et al. 2000; Biviano & Girardi 2003; Lin, Mohr & Stanford 2004; van den Bosch et al. 2005b; Sheldon et al. 2009).
However, a number of recent studies have argued that the spatial distribution of satellite galaxies is less centrally concentrated than the dark matter (e.g., Yang et al. 2005; Chen 2008; More et al. 2009). Interestingly, an opposite result has been obtained for luminous red galaxies (e.g., Masjedi et al. 2006; Watson et al. 2010, 2012; Tal, Wake & van Dokkum 2012). In fact, there are indications that the radial distribution of satellite galaxies appears to have some dependence on the color and/or morphologies of the central (e.g., Lorrimer et al. 1994; Sales & Lambas 2005; Lares, Lambas & Dominguez 2011; Guo et al. 2012). From a theoretical point of view, one expects the radial distribution of satellite galaxies to reflect that of dark matter subhaloes. Numerical simulations have shown that subhaloes do not follow the same spatial distribution as the dark matter; subhaloes tend to populate preferentially the outskirts of their host haloes (e.g., Diemand, Moore & Stadel 2004; Gao et al. 2004; Springel et al. 2008). However, the stellar components of satellite galaxies may be more resilient against tidal disruption than their dark matter components, such that the radial profile of satellite galaxies is not necessarily well represented by that of dark matter subhaloes in pure -body simulations.
In conclusion, the simple assumption that is likely oversimplified. To address the potential impact of this assumption on our results, we now consider a model in which we adopt
Here is the scale radius of the dark matter density profile for a halo of mass , and is a free parameter. For this profile reduces to that of our Fiducial model, i.e., . Hence, is a parameter that controls how satellites are concentrated with respect to the dark matter. In order to gauge how our results depend on , and whether the data prefers values of that differ from unity, we run a MCMC in which is a free parameter for which we adopt Gaussian prior with mean equal to unity and standard deviation, . In what follows we refer to this MCMC as our Radial model.
As is apparent from the values of the best-fit model in the Radial chain (see Table 3), the extra freedom in the model does not result in a significantly better fit to the data. More importantly, the cosmological constraints are also unaffected (see upper left-hand panel of Fig. 8 and Table 4). This is also evident from the contour plots in the left-hand column of Fig. 9, which show the 68 and 95% confidence intervals of the joint two-dimensional posterior distributions for and (middle row) and for and (bottom row). With cross-correlation coefficients , it is clear that neither nor is significantly influenced by potential uncertainty in . Finally, the histogram in the upper left-hand panel of Fig. 9 shows the posterior distribution for . With (95% CL) it is clear that the posterior distribution is narrower than the prior distribution (solid, red curve), and that the data prefers values of that are close to our fiducial value of unity. We conclude that our results are robust to uncertainties in the radial number density profile of satellite galaxies, and that the data used here is consistent with satellite galaxies being an unbiased tracer of the mass distribution within their host haloes (i.e., ). Interestingly, as mentioned before, this result is supported by studies of satellite galaxies in groups and clusters (e.g., Beers & Tonry 1986; Carlberg, Yee & Ellingson 1997a; van der Marel et al. 2000; Biviano & Girardi 2003; Lin, Mohr & Stanford 2004; van den Bosch et al. 2005b; Sheldon et al. 2009) but it is somewhat in disagreement with a number of studies which indicate that the radial distribution of satellite is less centrally concentrated than dark matter (e.g., Yang et al. 2005; Chen 2008; More et al. 2009). It is worth mentioning that results based on groups might be potentially affected by the offset of central galaxies (see e.g. Skibba et al. 2011). Clearly, more dedicated studies are required to further constrain the radial distribution of satellite galaxies.
6.2 Poisson Statistics of Satellite Galaxies
In our Fiducial model we have assumed that the occupation numbers of satellite galaxies obey Poisson statistics, i.e.,
As shown in Yang et al. (2008), this assumption has strong support from galaxy group catalogs. Additional support comes from numerical simulations which show that dark matter subhaloes (which are believed to host satellite galaxies) also follow Poisson statistics (e.g., Kravtsov et al. 2004). However, recently there have been several claims that the occupation statistics of subhaloes and/or satellite galaxies may deviate slightly, but significantly, from Poisson, i.e., (e.g., Porciani, Magliocchetti & Norberg 2004; van den Bosch et al. 2005a; Giocoli et al. 2010; Boylan-Kolchin et al. 2010; Busha et al. 2011). As shown in Paper I, the satellite-satellite term of the 1-halo power spectrum scales linearly with , i.e., . Hence, any deviation of from unity has a direct impact on the projected correlation functions on small scales, at least if the satellite-satellite term dominates over the central-satellite term (which is independent of ).
In order to quantify how our results depend on uncertainties related to the exact form of , we run a MCMC in which we include as a free parameter. Motivated by the empirical results of Yang et al. (2008), which suggest that the occupation statistics of satellite galaxies in the SDSS are not too different from Poisson, we adopt a Gaussian prior for with mean equal to unity and standard deviation, . In what follows we refer to this MCMC as our Poisson model.
The upper right-hand panel of Fig. 8 shows the 68 and 95% confidence levels of the joint, two-dimensional marginalized posterior distribution of and for model Poisson (yellow contours), compared to that for our Fiducial model (blue contours). As is evident, the extra freedom in results in a best-fit value for that is slightly lower than for the Fiducial model (see Table 4), but the change is only marginally significant (i.e., ).
The marginalized posterior distribution for , shown as the shaded histogram in the upper right-hand panel of Fig. 9, has (95% CL), indicating that the data prefers a sub-Poisson probability distribution . The middle and bottom right-hand panels of Fig.