jsons_folder/2013MNRAS.434.2556W.json

{
  "ABSTRACT ": [
    "Dark matter N-body simulations provide a powerful tool to model the clustering of galaxies and help interpret the results of galaxy redshift surveys. However, the galaxy properties predicted from N-body simulations are not necessarily representative of the observed galaxy populations; for example, theoretical uncertainties arise from the absence of baryons in N-body simulations. In this work, we assess how the uncertainties in N-body simulations impact the cosmological parameters inferred from galaxy redshift surveys. Applying the halo model framework, we find that the velocity bias of galaxies in modelling the redshift-space distortions is likely to be the predominant source of systematic bias. For a deep, wide survey like BigBOSS, current 10 per cent uncertainties in the velocity bias limit kmax to 0.14 h Mpc−1 . In contrast, we find that the uncertainties related to the density profiles and the galaxy occupation statistics lead to relatively insignificant systematic biases. Therefore, the ability to calibrate the velocity bias accurately – from observations as well as simulations – will likely set the ultimate limit on the smallest length scale that can be used to infer cosmological information from galaxy clustering. ",
    "Key words: cosmological parameters – dark energy – dark matter – large-scale structure of Universe "
  ],
  "1 INTRODUCTION ": [
    "The large-scale distribution of galaxies has been used to probe the structure and composition of the universe for over three decades. From the pioneering analyses of the Lick cata-logue ((<>)Groth & Peebles (<>)1977) and the CfA Redshift Survey ((<>)Huchra et al. (<>)1983; (<>)Geller & Huchra (<>)1989) revealing the cosmic web, the APM Galaxy Survey hinting the departure from the standard cold dark matter model ((<>)Maddox et al. (<>)1990) to the subsequent 2dF Galaxy Redshift Survey ((<>)Col-(<>)less et al. (<>)2001), the Sloan Digital Sky Survey (SDSS; (<>)York (<>)et al. (<>)2000) and the VIMOS-VLT Deep Survey ((<>)Le F`evre (<>)et al. (<>)2005), galaxy redshift surveys have revolutionized the view of the large-scale structure of the universe. Recently, the WiggleZ Dark Energy Survey ((<>)Drinkwater et al. (<>)2010) and the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS; (<>)Schlegel et al. (<>)2009) have measured the galaxy clustering to unprecedented precision and provided stringent constraints on the cosmological parameters. ",
    "One of the most important features in the galaxy clustering is the baryon acoustic oscillations (BAO), originating ",
    "from the waves in the primordial electron–photon plasma before the recombination. The sound horizon at the end of recombination is manifested as a peak in the real-space two-point correlation function or as wiggles in the Fourier-space power spectrum. This characteristic scale of BAO is considered as a standard ruler of the different evolution stages of the universe, and as a dark energy probe with relatively well-controlled systematics ((<>)Blake & Glazebrook (<>)2003; (<>)Seo (<>)& Eisenstein (<>)2003). Indeed, since its discovery ((<>)Miller et al. (<>)2001; (<>)Cole et al. (<>)2005; (<>)Eisenstein et al. (<>)2005), BAO has been providing ever improving constraints on cosmological parameters (e.g., (<>)Percival et al. (<>)2010; (<>)Blake et al. (<>)2011; (<>)An-(<>)derson et al. (<>)2012). ",
    "Beyond the BAO feature, the full scale-dependence of the clustering of galaxies contains much more information and can be used to constrain cosmology (e.g., (<>)Tegmark et al. (<>)2006; (<>)Reid et al. (<>)2010; (<>)Tinker et al. (<>)2012; (<>)Cacciato et al. (<>)2013) and the halo occupation statistics (e.g., (<>)Abazajian (<>)et al. (<>)2005; (<>)Tinker et al. (<>)2005; (<>)van den Bosch et al. (<>)2007; (<>)Zheng & Weinberg (<>)2007; (<>)Zehavi et al. (<>)2011). From the perspective of power spectrum P (k), the number of modes increases as k3 , and the information content increases dramatically as one goes to smaller scales. However, when one tries to draw information from high k, especially at low redshift, ",
    "the density perturbations become non-linear and difficult to model (e.g., (<>)Smith et al. (<>)2003; (<>)Heitmann et al. (<>)2010; (<>)Jen-(<>)nings et al. (<>)2011), which can introduce significant systematic errors in the recovered cosmological parameters (e.g., (<>)de la (<>)Torre & Guzzo (<>)2012; (<>)Smith et al. (<>)2012). ",
    "The analysis of galaxy clustering often relies on N-body simulations and synthetic galaxy catalogues to model the non-linearity on small scales, as well as to estimate the cosmic and sample covariances. For example, the WiggleZ team has validated their model for the non-linear galaxy power spectrum using the GiggleZ Simulation 1 (<>)((<>)Parkinson et al. (<>)2012), while synthetic galaxy catalogues based on the Large Suite of Dark Matter Simulations (LasDamas 2(<>)) have been used in the galaxy clustering analysis of SDSS ((<>)Chuang & (<>)Wang (<>)2012; (<>)Xu et al. (<>)2013). ",
    "For upcoming surveys, synthetic catalogues generated from N-body simulations will likely be routinely used to calibrate galaxy surveys. However, N-body simulations are not free from systematics. In N-body simulations, galaxies are assigned to haloes or dark matter particles based on models such as halo occupation distribution (HOD; (<>)Pea-(<>)cock & Smith (<>)2000; (<>)Scoccimarro et al. (<>)2001; (<>)Berlind & (<>)Weinberg (<>)2002), abundance matching ((<>)Kravtsov et al. (<>)2004; (<>)Vale & Ostriker (<>)2004), or semi-analytic models ((<>)White & (<>)Frenk (<>)1991; (<>)Kauffmann et al. (<>)1993; (<>)Somerville & Primack (<>)1999; (<>)Cole et al. (<>)2000). The galaxy populations predicted by simulations can be affected by intensive stripping in dense environment (e.g., (<>)Wetzel & White (<>)2010) and the absence of baryons (e.g., (<>)Weinberg et al. (<>)2008; (<>)Simha et al. (<>)2012). On the other hand, when one uses dark matter particles to model the behaviour of galaxies, systematic errors may arise because the positions and velocities of galaxies do not necessarily follow those of dark matter particles ((<>)Wu et al. 2013a). Hydrodynamical simulations that include proper treatments of baryonic physics can be another avenue to predict the properties of galaxies more reliably; however, because these simulations are more computationally intensive, it is not yet practical to use them to achieve the statistics and high resolution required by upcoming large surveys. ",
    "In addition, it has been shown that galaxies predicted from N-body simulations cannot recover the spatial distribution of observed galaxies. For example, Wu et al. (in preparation) have shown that in high-resolution N-body simulations of galaxy clusters, subhaloes tend to be prematurely destroyed and fail to predict the location of galaxies (also see Appendix (<>)A). The need to include “orphan galaxies” (galaxies not associated with subhaloes in simulations) to improve the completeness of predicted galaxies has been frequently addressed in the community (e.g., (<>)Gao et al. (<>)2004; (<>)Wang et al. (<>)2006; (<>)Guo et al. (<>)2011); however, even including orphan galaxies does not lead to consistent galaxy clustering at all scales . For example, (<>)Guo et al. ((<>)2011) have shown that the galaxy population generated using the semi-analytic model applied to the Millennium Simulations overestimates the small scale clustering (also see (<>)Contreras et al. (<>)2013). ",
    "In this paper, we examine the impact of the systematics in N-body simulations on the predictions of galaxy clustering. We calculate the galaxy power spectrum based on the halo model, with inputs from the results of recent N-body simulations. We use the information of the full power spectrum of galaxies to forecast the cosmological parameter constraints and determine at which scale these systematics start to become relevant. We specifically explore how these uncertainties will limit our ability to utilize the cosmological information from small scale. ",
    "This paper is organized as follows. In Section (<>)2, we review the halo model prediction for galaxy power spectrum. In Section (<>)3, we present our fiducial assumptions and discuss the information content associated with P (k). Section (<>)4 explores the self-calibration of HOD parameters. Section (<>)5 addresses the impact of the uncertainties in the halo mass function on the cosmological constraints from galaxy clustering. Section (<>)6 focuses on various systematics associated with the properties of galaxies in dark matter haloes in N-body simulations and presents the required control of these sources of systematic error. We conclude in Section (<>)7. In Appendix (<>)A, we present the galaxy number density profile model used in this work. In Appendix (<>)B, we provide detailed derivation of the galaxy power spectrum based on the halo model. In Appendix (<>)C, we derive the power spectrum covariance. "
  ],
  "2 HALO MODEL AND GALAXY POWER SPECTRUM: A REVIEW ": [
    "Throughout this work, we use the power spectrum of galaxies P (k) as our clustering statistic. Possible alternatives include the three-dimensional correlation function ξ(r) and its two-dimensional analogue – the angular two-point function w(θ) or the projected two-point function wp(rp). While the Fourier-space power is more difficult to measure from the galaxy distribution, it is ‘closest to theory’ in the sense that the other aforementioned quantities are weighted integrals over P (k). Therefore, it is easiest to see the effect of the uncertainties in theoretical modelling by using the power spectrum. While these different functions measured in a given galaxy survey contain the same information in principle, in data analysis sometimes discrepancies occur (e.g., (<>)Anderson (<>)et al. (<>)2012). "
  ],
  "2.1 Basic model ": [
    "In this section, we provide the key equations of the galaxy power spectrum derived from the halo model, following (<>)Scherrer & Bertschinger ((<>)1991), (<>)Seljak ((<>)2000), and (<>)Cooray (<>)& Sheth ((<>)2002). The detailed derivation is provided in Appendix (<>)B. ",
    "The halo model assumes that all galaxies are inside dark matter haloes. To model the distribution of galaxies, we need the following distributions. ",
    "Statistics and spatial distribution of dark matter haloes: ",
    "• Halo bias, b2(M) = Phh(k)/Plin(k), where Phh is the power spectrum of haloes and Plin is the linear matter power spectrum. We limit our use of b(M) to large scales where b(M) is scale independent. ",
    "Statistics and spatial distribution of galaxies in a halo: ",
    "Galaxy number density profile, u(r|M), the radial dependence of the galaxy number densityinside a halo of a given mass. We normalize u such that u(r|M)d3 r = 1. We also use the density profile in Fourier space, u˜(k|M) = ",
    "d3 xu(x|M)e−ik·x , and u˜ → 1 for small k. ",
    ". ",
    "(1) ",
    "The power spectrum is contributed by two galaxies in two different haloes (the two-halo term, Pgg 2h) and two galaxies 1h in the same halo (the one-halo term, Pgg ): ",
    "Here X|M indicates the average value of quantity X at a given halo mass M. In the one-halo term, ",
    "which takes into account the contribution from central– satellite and satellite–satellite pairs ((<>)Berlind & Weinberg (<>)2002). "
  ],
  "2.2 Redshift-space distortions ": [
    "In observations, one cannot recover the exact three-dimensional spatial distribution of galaxies, because the redshifts of galaxies are impacted by their motions due to the local gravitational field and do not reflect their true distances. On larger scales, galaxies tend to move towards high-density regions along filaments, and these motions tend to squash the galaxy distribution along the line of sight and boost the clustering, a phenomenon known as the Kaiser effect ((<>)Kaiser (<>)1987). On small scales, the virial motions of galaxies inside a halo tend to make the galaxy distribution in the redshift space elongated along the line-of-sight, causing the so-called Fingers-of-God effect and reducing the small-scale power. In this section, we briefly describe the model we use ",
    "for the redshift-space distortions (RSD) for P (k), following (<>)Seljak ((<>)2001), (<>)White ((<>)2001), and (<>)Cooray & Sheth ((<>)2002). We adopt one of the simplified models – assuming the velocity distribution function to be Gaussian – and note that the improvement of the RSD model is currently an active research area. ",
    "Since the one-halo term involves the halo scale, we only consider the virial motions of galaxies inside a halo, which can be modelled as ((<>)Peacock (<>)1999) ",
    "(5) ",
    "where δ˜z and ˜ are the number density fluctuations ofgal δgal galaxies with and without the effect of RSD, σv(M) is the velocity dispersion of galaxies inside a halo of mass M and µ = kˆ · rˆ. We average over µ to obtain the angular averaged one-halo term ",
    "(6) ",
    "The factor ",
    "(7) ",
    "comes from averaging over µ. ",
    "For the two-halo term, we multiply the large-scale and small-scale effects together (see (<>)Peacock (<>)1999 and section 4 in (<>)Peacock & Dodds (<>)1994) ",
    "The first part is the familiar Kaiser result with f(ΩM) ≡ d ln D/d ln a, where D(a) is the linear growth function of density fluctuations and a is the scale factor. The density fluctuation of dark matter is denoted by δ˜m. The calculation thus includes not only the galaxy power spectrum, but also the matter power spectrum and the matter–galaxy cross power spectrum. After averaging over µ, we obtain ",
    "(9) ",
    "comes from the contribution of δ˜gal, and  ",
    "comes from the contribution of δ˜m. Here u˜m(k|M) denotes the dark matter density profile normalized the same way as u˜, and ρ¯ is the average matter density of the universe. We assume that u˜m(k|M) follows the Navarro–Frenk–White (NFW) profile ((<>)Navarro et al. (<>)1997) throughout the paper. The left-hand panel of Fig. (<>)1 shows an example of the ",
    "Figure 1. Galaxy power spectrum calculated based on the halo model. Left: the blue and red curves show the one-and two-halo terms, respectively. The solid curves include the RSD, while the dashed curves do not. RSD greatly reduce the power spectrum at small scale (the ‘Fingers-of-God’ effect), and only slightly shift the scale where one-and two-halo terms cross. Right: Our model fit to WiggleZ data from (<>)Parkinson et al. ((<>)2012). The blue solid curve shows the theoretical P (k) with the best-fitting HOD parameters and has been convolved with the observational window function. ",
    "contribution to the total galaxy power spectrum by the one-halo (blue) and two-halo (red) terms. The input of halo model will be detailed in Section (<>)3.1. The solid and dashed curves correspond to including and excluding the effect of RSD. As can be seen, including RSD significantly reduces the power at small scale. We also note that the scale where one- and two-halo terms cross shifts very slightly due to RSD. ",
    "The right-hand panel of Fig. (<>)1 presents the comparison between our model and one of the power spectra from the WiggleZ survey, provided by (<>)Parkinson et al. ((<>)2012). The green dashed/blue solid curve corresponds to the theoretical P (k) before/after convolving with the window function of WiggleZ. We assume that the HOD is described by the five parameters in equation ((<>)27); we fit for these five parameters and show the model corresponding to the best-fitting parameters. This figure is only for the purposes of illustration; details of the fitting procedure will be presented in a future paper. "
  ],
  "3 BASELINE MODEL AND FIDUCIAL DARK ENERGY CONSTRAINTS ": [],
  "3.1 Baseline assumptions ": [
    "We use the virial mass Mvir of dark matter haloes throughout this work and adopt the following functions in our halo model calculations: ",
    "Density profile: based on the universal NFW profile ((<>)Navarro et al. (<>)1997), which is described by one concentration parameter cvir ",
    "(12) ",
    "Concentration–mass relation: based on the relation in (<>)Bhattacharya et al. ((<>)2013), which will be further discussed in Section (<>)6.1. In the presence of significant scatter in the c–M relation, we perform the integration  ",
    "Throughout this paper, we assume that cvir has a Gaussian distribution for a given Mvir with a scatter of 0.33, based on the finding of (<>)Bhattacharya et al. ((<>)2013). ",
    "We convert the mass M200 to Mvir based on (<>)Hu & Kravtsov ((<>)2003). Since the scatter in the velocity dispersion is expected to be small (4 per cent), it is not included in our calculation. ",
    "• HOD: based on the parametrization from (<>)Zheng et al. ((<>)2005) and the fiducial parameters from (<>)Coupon et al. ((<>)2012), both of which will be discussed in detail in Section (<>)4. ",
    "We assume a fiducial galaxy survey covering fsky = 1/3 of the full sky (about 14 000 square degrees), similar to the BigBOSS experiment 3 (<>). We assume that the survey depth is comparable to the Canada–France–Hawaii Telescope Legacy Survey (CFHTLS) results presented in (<>)Coupon et al. ((<>)2012); specifically, we assume five redshift bins in the range 0.2 < z < 1.2, and the limiting magnitude in each bin is summarized in Table (<>)1. We assume no uncertainties in the redshift measurements of galaxies. Given that the assumption of such a deep, wide spectroscopic survey may be somewhat optimistic, our required control of systematic errors may be somewhat more stringent than what BigBOSS needs. ",
    "We include seven cosmological parameters, whose fidu-cial values are based on the Wilkinson Microwave Anisotropy Probe 7 constraints ((<>)Komatsu et al. (<>)2011): total matter density relative to critical ΩM = 0.275; dark energy equation of state today and its variation with scale factor w0 = −1 and wa = 0 respectively; physical baryon and matter densities Ωbh2 = 0.02255 and ΩMh2 = 0.1352; spectral index ns = 0.968; and the amplitude of primordial fluctuations A = Δ2ζ (k = 0.002h−1Mpc) = 2.43×10−9 . We assume a flat universe; thus, dark energy density ΩDE = 1 − ΩM. "
  ],
  "3.2 Likelihood function of P(k) and error forecasting ": [
    "Here we follow the derivations in (<>)Scoccimarro et al. ((<>)1999) and (<>)Cooray & Hu ((<>)2001) but use a different convention for the Fourier transform (see Appendix (<>)B). If we assume a thin shell in ln k space with width δ ln k around ln ki, the power spectrum estimator reads ",
    "(15) ",
    "where ",
    "(16) ",
    "and 1/n¯gal accounts for the effect of shot noise. The first term of Pˆ is calculated based on the halo model results described in Section (<>)2.2. ",
    "The covariance of is given byspectrumpower  ",
    "describing the non-Gaussian nature of the random field ",
    "We provide the detailed derivation in Appendix (<>)C. In equation ((<>)17) Vz is the volume of the redshift bin, Vz = Ωsurvey r 2(z)/H(z)dz, where the integral is performed  over the redshift extent of the bin. ",
    "The calculation of T¯ij involves four-point statistics, which is non-trivial to calculate. Fortunately, (<>)Cooray & Hu ((<>)2001) have shown that only the one-halo term dominates at the scale where the contribution of T¯ij to Cij is not negligible; therefore, we only need to calculate the one-halo contribution: ",
    "where ",
    "Analogous to the case of considered in Section (<>)2.1, the first term accounts for quadruplets composed of one central and three satellite galaxies, and the second term accounts for the quadruplets composed of four satellite galaxNsat|M4 /4!. ",
    "We employ the Fisher matrix formalism to forecast the statistical errors of the cosmological and nuisance parameters based on the fiducial survey. The Fisher matrix reads ",
    "where α and β are indices of model parameters, while i and j refer to bins in wavenumber which have a constant logarithmic width δ ln k and extend out to the maximum wavenumber kmax. We adopt δ ln k = 0.1, which has been tested to be small enough to ensure convergence. The best achievable error in the parameter θα is given by ",
    "(21) ",
    "Throughout this work, unless otherwise indicated, the full set of parameters considered is given by ",
    "Figure 2. Dark energy information content from P (k), based on our fiducial survey assumptions. The x-axis corresponds to the largest k (smallest scale) assumed to be reliably measured and interpreted. The blue curve corresponds to a Gaussian likelihood function and assumes no RSD; it leads to unrealistically tight constraints for large kmax. The green curve includes RSD, and the red curve further includes the trispectrum correction to the covariance matrix. As can be seen, including these two effects reduces the small-scale information. ",
    "The first seven are the cosmological parameters introduced in Section (<>)3.1, while the last five are the nuisance parameters describing the HOD and will be discussed in Section (<>)4.1. "
  ],
  "3.3 Fiducial constraints without systematics ": [
    "ht",
    " ",
    "200",
    "et al. ",
    "To represent the statistical power of an upcoming galaxy redshift survey, in the limiting case of no nuisance parameters, we consider the inverse of the square root of the dark energy figure of merit, originally defined as the inverse of the forecasted 95 per cent area of the ellipse in the w0–wa plane ((<>)Huterer & Turner (<>)2001; (<>)Albrec6). In other words, our parameter of interest is σ(wa)σ(wp), where wp is the pivot that physically corresponds to w(a) evaluated at the scale factor where the constraint is the best. This quantity takes into account the temporal variation of dark energy, and the square root serves to compare it fairly to the constant w; the two quantities, σ(w) and σ(w0)σ(wp), tend to show very similar behaviour. For our fiducial survey, the in our parameter combination of interest is σ(wa)σ(wp) = 0.4 (or 0.003) for kmax = 0.1 (or 1) h Mpc−1 , without external priors. When we add the Planck Fisher matrix (Hu, private communication), σ(wa)σ(wp) becomes 0.002 (or 0.0002) for Mpc−1 . ",
    "Fig. (<>)2 presents the expected dark energy constraints as a function of kmax, without nuisance parameters or systematic errors for the moment, for three levels of sophistication in the theory. We proceed in steps: the blue curve corresponds to ",
    "no RSD (Section (<>)2.1) with a Gaussian likelihood function. In this case, the dark energy constraints increase sharply with kmax, indicating that these assumptions are unrealistic. The green curve includes the RSD (Section (<>)2.2), which reduce the dark energy information from small scales. The red curve further includes the effect of non-Gaussian likelihood [T¯ij from equation ((<>)18)], which reduces the information at high k even more. "
  ],
  "3.4 Systematic bias in model parameters ": [
    "In this work, we estimate the systematic shifts in parameter inference caused by using an inadequate model. In particular, if we assume a problematic model that produces a power spectrum Psys(k) that systematically deviates from the truth Pfid(k), we will obtain parameters that systematically deviate from their true values: θsys = θfid + Δθ. The systematic shifts in parameters can be obtained through a modified Fisher matrix formalism ((<>)Knox et al. (<>)1998): ",
    "(23) ",
    "(24) ",
    "To determine the significance of systematic errors, we calculate the systematic shifts Δχ2tot in the full high-dimensional parameter space, ",
    "(25) ",
    "where Δθ is the vector of the systematic shifts of parameters. Both Δθ and the Fisher matrix F include cosmological and nuisance parameters. The systematic bias is considered significant if the inferred θsys lies outside the 68.3 per cent confidence interval of the Gaussian likelihood function cen-tred on θfid; in other words, the bias is ‘greater than the 1 σ dispersion’. For example, in a full 12-dimensional parameter space considered here, the 68.3 per cent confidence interval corresponds to Δχ2tot = 13.7. "
  ],
  "4 SELF-CALIBRATION OF HOD PARAMETERS ": [
    "In this section, we focus on the efficacy of self-calibrating the HOD parameters, that is, determining these parameters from the survey concurrently with cosmological parameters. Since these HOD parameters are not known a priori, one usually marginalizes over them along with cosmological parameters (e.g., (<>)Tinker et al. (<>)2012), which inevitably increases the uncertainties in cosmological parameters. Here we focus on the statistical uncertainties and assume no systematic error; in the next section, we will compare these statistical errors with systematic shifts of parameters. ",
    "We focus on two parametrizations of HOD: one is based on (<>)Zheng et al. ((<>)2005) and the other is based on a piecewise continuous parametrization. ",
    "Table 1. Fiducial values for the HOD parameters, adopted from (<>)Coupon et al. ((<>)2012) based on CFHTLS. ",
    "Figure 3. Self-calibration of HOD parameters. We show the dark energy constraints as a function of the highest k used in the survey. The red curve corresponds to no nuisance parameters. The dark blue curve corresponds to five nuisance parameters based on the parametrization in (<>)Zheng et al. ((<>)2005), while the cyan curve corresponds to a piecewise continuous parametrization for satellite galaxies, with one parameter in each of the five mass bins. Including nuisance parameters in either parametrization systematically increases the dark energy uncertainties by one or two orders of magnitude. The dashed curves include the Planck prior and assume the same nuisance parameters as their solid-curve counterparts. "
  ],
  "4.1 Zheng et al. parametrization ": [
    "The HOD describes the probability distribution of having N galaxies in a halo of mass M. In principle, the HOD is spec-ified by the full distribution P (N|M); in practice, modelling of the two-point statistics only requires Ncen|M, Nsat|M, and Nsat(Nsat − 1)|M. We follow the HOD parametrization from (<>)Zheng et al. ((<>)2005), which separates the contribu-",
    " ",
    "(26) ",
    "(27) ",
    "The first equation describes the contribution from the central galaxy; Mmin corresponds to the threshold mass where a halo can start to host a galaxy that is observable to the survey, and σlog10 M describes the transition width of this threshold. The second equation describes the contribution from satellite galaxies, whose number is assumed to follow a power law, and M0 is the cutoff mass. In addition, we make the widely-adopted assumption that P (Nsat|M) follows a Poisson distribution, i.e., ",
    "(28) ",
    "We adopt the fiducial values from (<>)Coupon et al. ((<>)2012), which are constrained using the projected angular two-point correlation function w(θ) from the CFHTLS out to z = 1.2. We use the same binning and limiting magnitude as in (<>)Coupon et al. ((<>)2012); the values are summarized in Table (<>)1. We do not use the error bars quoted there as our priors because we would like all parameters to be self-calibrated consistently. ",
    "Under these assumptions, we have five nuisance parameters (log10 Mmin, σlog10 M , log10 M0, log10 M1, αsat) for each of the five redshift bins, i.e., 25 parameters in total. We assume that each of the five distinct nuisance parameters varies coherently across the five redshift bins, and is therefore described by a single parameter. Under this assumption, instead of 25 nuisance parameters, we only use five nuisance parameters to describe the uncertainties of all HOD parameters. We parametrize the variations around the fiducial values: ",
    "(29) ",
    "where hi are the dimensionless parameters describing the uncertainties of the aforementioned 5 HOD parameters. We note that this choice of five HOD parameters only represents one possible model; depending on the data available and the astrophysical motivation, in principle one can use a more general model to describe the evolution of HOD. Increasing the number of degrees of freedom describing the evolution of HOD will inevitably lead to degradation in the dark energy constraints, and it will be very important to establish the total number of degrees of freedom necessary to model the HOD and its uncertainties. ",
    "We explore how well these parameters can be self-calibrated by P (k) without the aid of priors. Fig. (<>)3 shows the dark energy constraints as a function of kmax, with fixed nuisance parameters (red) and with these 5 marginalized nuisance parameters (dark blue). The RSD and the full covariances of P (k) are included in this calculation. Clearly, the dark energy constraints are weakened by approximately about one or two orders of magnitude when we marginalize over HOD parameters. "
  ],
  "4.2 Piecewise continuous parametrization of HOD parameters ": [
    "One potential worry with the parametrization in equation ((<>)27) is whether Nsat|M is accurately described by a power law. To address this, we propose a less model-dependent, piecewise continuous parametrization for Nsat|M. We divide the halo mass range into nbins bins and assign a parameter describing the uncertainties of HOD in each bin. That is, ",
    "(30) ",
    "where Θi(M) defines the binning and equals 1 in [Mi, Mi+1] and 0 elsewhere, while fi is the free parameter in bin i and describes the uncertainty of Nsat|M in this bin. ",
    "(31) ",
    "We start with one parameter per decade in mass, using nbins = 5 parameters between 1011 and 1016 h−1M, equally spaced in log10 M. We assume these parameters to be independent of redshift. The cyan curve in Fig. (<>)3 corresponds to marginalizing over these five piecewise continuous parameters for Nsat|M and two parameters (log10 Mmin, σlog10 M ) for Ncen|M, with no prior on them. ",
    "Fig. (<>)4 shows the dependence of dark energy constraints on the number of parameters describing Nsat|M per decade of mass, Nper decade. The three panels correspond to kmax = 0.1, 0.4, and 1 h Mpc−1 . The Planck prior is included in this calculation. The black curve corresponds to no prior on fi and shows strong degradation with increasing Nper decade as one would expect. When kmax is small, the prior knowledge of HOD is important to improve the dark energy constraints. On the other hand, when kmax is large, HOD can be well self-calibrated, and the prior is not as important. ",
    "To enable a fair comparison of priors, however, we would like to increase the freedom in the HOD model while fixing the overall uncertainty per decade. To do this, we impose a fixed prior per decade of mass:  ",
    "(32) ",
    "so that the total prior per unit log10 M, when we add the Fisher information from all fi, is σ0 regardless of the value of Nper decade. ",
    "Figure 5. Impact of the uncertainty in the halo mass function on P (k). The main panel shows the systematic shifts of P (k) when the mass function is shifted by a constant 5 per cent (independent of mass and redshift). The inset shows Δχtot 2 as a function of kmax when the mass function is shifted by 1 or 5 per cent. The horizontal dashed line marks Δχtot 2 = 13.7, the 1-σ deviation in the 12-dimensional parameter space. As can be seen, 5 per cent (1 per cent) allows kmax up to 0.15 (0.25) h Mpc−1 . ",
    "The red/green/blue curves in Fig. (<>)4 correspond to imposing σ0 = 1/0.1/0.01. For kmax = 0.1 h Mpc−1 , the dark energy constraints converge when we use one parameter per decade of mass regardless of the prior on nuisance parameters. When kmax > 0.1 h Mpc−1 , a few more parameters per decade in mass are required for the results to converge. For example, for kmax = 0.4 (1.0) h Mpc−1 , we need two (three) parameters per decade to ensure convergence. The required number of parameters also somewhat depends on the prior. ",
    "We note that the HOD parameters are progressively better self-calibrated when we go to higher kmax; when kmax = 1 h Mpc−1 , self-calibrating the five HOD parameters only moderately degrades the dark energy constraints. This finding encourages future surveys to further push towards high kmax for rich cosmological and astrophysical information. "
  ],
  "5 SYSTEMATIC ERRORS DUE TO THE UNCERTAINTIES IN HALO MASS FUNCTION ": [
    "In this section, we focus on the effect of the uncertainties in the halo mass function on the cosmological constraints from galaxy clustering. The mass function has been widely explored analytically (e.g., (<>)Press & Schechter (<>)1974) as well as numerically using dark matter N-body simulations (e.g., (<>)Sheth & Tormen (<>)1999; (<>)Sheth et al. (<>)2001; (<>)Jenkins et al. (<>)2001; (<>)Evrard et al. (<>)2002; (<>)Reed et al. (<>)2003; (<>)Warren et al. (<>)2006; ",
    "Figure 4. Dark energy constraints with self-calibrated piecewise continuous HODs. The three panels correspond to kmax = 0.1, 0.4 and 1 h Mpc−1 . The x-axis corresponds to the number of parameters used to describe Nsat|M per decade of mass, and the y-axis corresponds to the dark energy constraints. The black curve corresponds to no prior, and the constraints are degraded with larger number of parameters. The other curves correspond to consistently adding a fixed total prior per decade of mass; that is, σfi = σ0 Nper decade , where σ0 = 1, 0.1, or 0.01. We note that one parameter per decade is sufficient for kmax = 0.1 h Mpc−1 , while two or three parameters are needed for higher kmax. Note that for higher kmax, the HOD parameters are better self-calibrated, and the dark energy constraints are less dependent on the prior on HOD nuisance parameters. ",
    "(<>)Luki´c et al. (<>)2007; (<>)Cohn & White (<>)2008; (<>)Tinker et al. (<>)2008; (<>)Luki´c et al. (<>)2009; (<>)Crocce et al. (<>)2010; (<>)Bhattacharya et al. (<>)2013; (<>)Reed et al. (<>)2013; (<>)Watson et al. (<>)2013) and hydrodynamical simulations (e.g., (<>)Rudd et al. (<>)2008; (<>)Stanek et al. (<>)2010; (<>)Cui et al. (<>)2012). The different fitting formulae for the mass function are often based on different halo identification methods and mass definitions; therefore, instead of drawing a direct comparison between different fitting formulae, we choose one specific fiducial model and explore the uncertainties relative to this model. 4 (<>)",
    "We use the fitting function from (<>)Tinker et al. ((<>)2008, described in Section (<>)3.1), which has been calibrated based on a large suite of simulations implementing different N-body algorithms and different versions of ΛCDM cosmology; therefore, it is likely to fairly represent the uncertainties in the mass function calibration. (<>)Tinker et al. ((<>)2008) quoted a statistical uncertainty of  5 per cent at z = 0 (∼ 1 per cent around M∗). However, the uncertainties are presumably larger at higher redshift and can further increase if the effects of baryons are taken into account. ",
    "We explore the effect of a small constant shift of the halo mass function, parametrized as ",
    "(33) ",
    "The main panel of Fig. (<>)5 shows the impact of  = 0.05 on P (k). For the two-halo term (small k), P (k) changes by ",
    "less than 1 per cent, because a constant shift in the mass function only affects Fv describing the large-scale RSD (see equation (<>)9). For the one-halo term (large k), P (k) changes by −5 per cent, which can be easily seen from equation (<>)6; the numerator includes one integration of dn/dM (galaxy pairs in one halo) while the denominator includes the square of such an integration. ",
    "We next see how this systematic shift in P (k) impacts cosmological parameters. We use Δχ2tot = 13.7 [1 σ errors in a 12-dimensional parameter space; see equation ((<>)25)] as our criterion of significant impact from the systematic error. We calculate Δχ2tot using the Fisher matrix for seven cosmological parameters (Section (<>)3.1) and five HOD parameters (Section (<>)4.1). Throughout this and the next section, we use the Planck prior but no priors on HOD parameters. We believe that these two assumptions reflect reality in the next 5–10 years, when Planck data will firmly pin down certain combinations of cosmological parameters, while the determination of the nuisance HOD quantities will still be in flux. We note that unbiased priors always decrease the resulting systematic bias (for a proof, see appendix A of (<>)Bernstein (<>)& Huterer (<>)2010) and make the theoretical requirements less stringent. Thus, any prior on HOD parameters will alleviate the systematic biases and make the required accuracy of theory less stringent. ",
    "The inset of Fig. (<>)5 shows how Δχ2 depends on kmax, for 5 per cent (blue) and 1 per cent (green) systematic shifts in the mass function. As can be seen, a 5 per cent (1 per cent) shift in the mass function can cause a significant systematic error at kmax = 0.15 (0.25) h Mpc−1 . We note that at these scales, P (k) is still dominated by the two-halo term; therefore, the systematic shifts caused by the mass function are mainly related to the large-scale redshift distortion (the Kaiser effect). However, this large-scale effect can ",
    "be mitigated by prior knowledge of σ8 (e.g., σ8 constraints from galaxy cluster counts; (<>)Rozo et al. (<>)2010), with which the large-scale galaxy bias can be calibrated. Therefore, by calibrating the large-scale clustering amplitude, one can in principle reduce the impact of the uncertainties in the mass function. ",
    "Finally, we note that the halo bias b(M) is also currently being actively studied (e.g., (<>)Tinker et al. (<>)2010; (<>)Ma (<>)et al. (<>)2011; (<>)Manera & Gazta˜naga (<>)2011; (<>)Paranjape et al. (<>)2013). The uncertainty in the halo bias is related to the uncertainty in the mass function; for example, (<>)Tinker et al. ((<>)2010) have indicated that their fitting function for the halo bias has an ∼ 6 per cent uncertainty, which is related to the uncertainty of their mass function. In addition, the uncertainties and systematics in b(M) will lead to a constant shift in the two-halo term (see equation (<>)9). In this case, holding the galaxy bias fixed will cause a huge systematic shift in cosmological parameters (for example, σ8); therefore, it is necessary to fit the overall galaxy bias to the large-scale clustering data. In this work, we do not specifically explore the impact of uncertainties of the halo bias because the halo bias determines the large-scale clustering amplitude, which can be observationally calibrated when combined with independent knowledge of σ8. On the other hand, we note that a scale-dependent bias can arise from the primordial non-Gaussianity (e.g., (<>)Dalal et al. (<>)2008) or small-scale nonlinearity (e.g., (<>)Smith et al. (<>)2007). In this case, one could resort to multiple tracers of large-scale structure (e.g., (<>)Seljak (<>)2009; (<>)Cacciato et al. (<>)2013), knowledge of primordial non-Gaussianity from the cosmic microwave background (e.g., (<>)Planck Collaboration (<>)2013) or higher order statistics (e.g., (<>)Mar´ın et al. (<>)2013) to better calibrate the scale dependence of the galaxy bias. "
  ],
  "6 SYSTEMATIC ERRORS DUE TO THE UNCERTAINTIES IN HALO PROPERTIES ": [
    "In this section, we explore the impact of four sources of theoretical uncertainties related to the properties of dark matter haloes coming from N-body simulations on the constraining power of P (k). These sources of systematics are as follows: ",
    "In particular, we address the following points. ",
    "We again use Δχ2tot to assess the impact of systematic errors on the cosmological parameters, as described in the previous section. The summary of the impact of these systematics is presented in Fig. (<>)6 and Table (<>)2. "
  ],
  "6.1 Concentration–mass relation ": [
    "In the halo model, the one-halo term depends on the number density profile of galaxies, u˜(k|M). We assume that the galaxy distribution follows the dark matter distribution, which is well described by an NFW profile. We then use the concentration–mass relation of dark matter haloes from the literature to compute u˜(k|M). ",
    "The concentration–mass relation has been calibrated with dark matter N-body simulations (e.g., (<>)Bullock et al. (<>)2001; (<>)Neto et al. (<>)2007; (<>)Duffy et al. (<>)2008; (<>)Macci`o et al. (<>)2008; (<>)Kwan et al. (<>)2013; (<>)Prada et al. (<>)2012; (<>)Bhattacharya et al. (<>)2013) and hydrodynamical simulations (e.g., (<>)Lau et al. (<>)2009; (<>)Duffy et al. (<>)2010; (<>)Rasia et al. (<>)2013). Several observational programmes are also working towards pinning down this relation (e.g., (<>)Coe et al. (<>)2012; (<>)Oguri et al. (<>)2012). However, 10–20 per cent of uncertainties in the concentration–mass relation remain, and the concentration–mass relation also varies with cosmology and the implementation of baryonic physics (see, e.g., the review in (<>)Bhattacharya et al. (<>)2013). ",
    "We investigate the impact of uncertainties in the concentration–mass relation by comparing the models from (<>)Bullock et al. ((<>)2001, B01 hereafter) and the recent calibration from (<>)Bhattacharya et al. ((<>)2013, B13 hereafter). These two models represent two extreme cases of the concentration–mass relation; therefore, using these two extreme cases sets the upper limit of the systematic bias caused by the c–M relation. We assume a scatter of 0.33 for the c–M relation in both cases. Ignoring this scatter will lead to an approximately 0.5 per cent difference in P (k) at k ≈ 1 h Mpc−1 . ",
    "Our baseline model is from the recent formula given by B13 (based on virial overdensity): ",
    "We compare it with the model from B01: ",
    "(35) ",
    "These two calibrations agree near M∗ at z = 0. ",
    "The top-left panel of Fig. (<>)6 shows the relative change in the power spectrum P (k), evaluated at five redshifts, due to the difference between B01 and B13. We find that P (k) based on B01 is in general lower than that based on B13, because B01 predict lower concentrations at the high-mass end. Although B01 predict higher concentrations at the low-mass end, these haloes rarely contribute to the one-halo term and thus do not significantly boost clustering. ",
    "The inset in this panel shows the systematic shifts in the parameter space caused by different models, which are characterized by Δχ2tot. It can be seen that the systematic error starts to be comparable to the statistical error (Δχ2tot = 13.7, marked by a horizontal dashed line) at kmax = 1.2 h Mpc−1 , which makes it a relatively unimportant source of systematic error. ",
    "We would now like to study the effects of improved calibration in the c–M relation. A natural way to do this is to assume that the difference between the two extreme predic-",
    "Figure 6. Systematic differences in P (k) caused by the four sources of errors discussed in Section (<>)6. In each panel, the main figure shows the fractional difference in P (k) in the five redshift bins, while the inset shows the systematic error Δχtot 2 as a function of kmax. The 1 σ deviation in the 12-dimensional parameter space, Δχtot 2 = 13.7, is marked by the horizontal dashed line in each inset. ",
    "tions has been reduced by some constant factor, and that the new value interpolates between the two original extremes. We define the interpolated value as ",
    "where cfid(M) and calt(M) are respectively the fiducial (say, B13) and the alternate (say, B01) models for the concentration–mass relation. Here fsys is a tunable parameter that allows us to assess the effect of a fraction of the full ",
    "systematics. The limiting cases are: ",
    "fsys = 0 ⇐⇒ no systematics ",
    "fsys = 1 ⇐⇒ fiducial systematics . ",
    "For a higher kmax, the tolerance of systematics is smaller, and fsys provides a measure for required reduction of systematics. For a given kmax, we search for the appropriate fsys value that makes the systematic negligible 5 (<>). ",
    "Figure 7. Required reduction of systematic errors, shown as the fraction of the current errors, for the four sources of systematic errors discussed in this paper, as a function of the maximum wavenumber considered in the survey. Note that the velocity bias requires the greatest improvement relative to the current knowledge. ",
    "The blue curve in Fig. (<>)7 shows the requirement on fsys from the c-M relation as a function of kmax. For all practical kmax values, c–M does not require more precise calibrations from N-body simulations. The results are summarized in the ‘c–M relation’ row of Table (<>)2. "
  ],
  "6.2 Galaxy number density profile: deviation from NFW ": [
    "Our fiducial model assumes that the galaxy distribution inside a halo is described by the NFW profile. However, N-",
    "ially lead to the same fractional shift in P (k) because the c–M relation (and most other systematics) enters non-linearly into P (k). We therefore need to perform a separate calculation of P (k) for each fsys. ",
    "body simulations have shown that the distribution of sub-haloes in cluster-size haloes tends to be shallower than the NFW profile, and also shallower than the observed galaxy number density profile (e.g., (<>)Diemand et al. (<>)2004; (<>)Nagai & (<>)Kravtsov (<>)2005). These deviations could be related to insuf-ficient resolution or the absence of baryons in N-body simulations – the so-called overmerging issue. Several authors have proposed models for ‘orphan galaxies’ to compensate the overmerging issue; however, these models do not always recover the observed galaxy clustering (e.g., (<>)Guo et al. (<>)2011). The exact cause for these issues is still uncertain; nevertheless, the uncertainties associated with the distribution of subhaloes will likely impact the modelling of galaxy clustering. Based on the comparisons between dark matter and hydrodynamical simulations (e.g., (<>)Macci`o et al. (<>)2006; (<>)Wein-(<>)berg et al. (<>)2008), the observed galaxy density profile is likely to be bracketed by the density profiles of the subhalo number and dark matter. ",
    "In this section, we investigate whether the uncertainties in the galaxy number density profile lead to a significant systematic bias. To model the possibility that the galaxy distribution is shallower than dark matter in the inner region of clusters, we adopt the subhalo number density profile measured from Wu et al. (in preparation), which is also illustrated in Appendix (<>)A. Fig. (<>)A1 presents one example of the galaxy number density profile measured from an N-body simulation. Based on this result, we model the subhalo number density profile as ",
    "where uNFW(r|M) is the NFW profile , and fsurv is the “surviving fraction” of galaxies given by ",
    "(38) ",
    "We note that fsurv is smaller for higher host halo mass and smaller radius, where the effect of overmerging is stronger. ",
    "The top-right panel of Fig. (<>)6 shows the difference in P (k) caused by this cored profile. As expected, the deficit of the galaxy number at small scales leads to lower power at high k. In addition, the suppression is stronger at low redshift because massive clusters are more abundant at low z. The inset shows the corresponding Δχ 2tot as a function of kmax; the systematic shifts dominate at kmax = 1 h Mpc−1 . ",
    "We model the interpolated systematic error in the density profile as ",
    "(39) ",
    "Here our fiducial model is the NFW profile, and the alternative profile is given by equation (<>)37. We again search for the required fsys as a function of kmax. The result is shown by the green curve in Fig. (<>)7. Like the c–M relation, the density profile of galaxies does not require more precise calibrations for all practical kmax. The results are summarized in the ‘Profile’ row of Table (<>)2. "
  ],
  "6.3 Deviation from the Poisson distribution ": [
    "In our fiducial model, P(Nsat|M) is assumed to be Poisson distributed; that is, the second mo(<>)Kolchin et al. ((<>)2010) have shown that the number of subhaloes for a given halo mass deviates from the Poisson distribution (their fig. 8). In addition, (<>)Wu et al. (2013a) have shown that the extra-Poisson scatter depends on how subhaloes are chosen and depends on the resolution. Therefore, it is still unclear whether P(Nsat|M) follows a Poisson distribution. To assess the impact of the possible extra-Poisson scatter, we adopt α = 1.02 in our one-halo term (following (<>)Boylan-Kolchin et al. (<>)2010), noting that this choice of α brackets the various possibilities explored in (<>)Wu et al. (2013a, fig. 3 therein). ",
    "The bottom-left panel in Fig. (<>)6 shows the impact of α = 1.02 on P(k), relative to the fiducial Poisson case with α = 1. The extra-Poisson scatter only impacts the one-halo term; therefore, the large-scale P(k) is unaffected. At small scales, P(k) is boosted by less than 3 per cent. For different redshifts, ΔP/P takes off at different k, reflect-ing the varying scale where one-halo and two-halo terms cross. We also note that at high k, ΔP/P bends downwards, reflecting the fact that the one-halo term includes When P(Nsat|M) is super-Poisson, more galaxy pairs are expected, and the one-halo term gets more weighting of thus, P(k) becomes lower at high k. ",
    "The inset in the bottom-left panel of Fig. (<>)6 shows that the systematic shifts dominate statistical errors at kmax = 0.3 h Mpc−1 . We model the partial uncertainties in α as ",
    "(40) ",
    "The red curve in Fig. (<>)7 shows the required fsys as a function of kmax; for kmax = 1 h Mpc−1 , the required fsys = 0.2. The results are summarized in the P(Nsat) row of Table (<>)2. "
  ],
  "6.4 Velocity bias ": [
    "The small-scale RSD (also known as the ‘Fingers-of-God’ effect) are usually modelled as an exponential suppression of power with the term e −(kσv µ)2 . Here σv is the velocity dispersion of galaxies inside a cluster σvgal(Mvir). Assuming that the motions of galaxies trace those of dark matter particles, we use the velocity dispersion of dark matter particles ",
    "inside a halo, σvDM(Mvir), which has been well established using simulations ((<>)Evrard et al. (<>)2008). However, the velocity dispersion of galaxies inside a cluster σvgal is not necessarily the same as σvDM . The ratio between the two is defined as the velocity bias ",
    "(41) ",
    "The exact value of bv and its redshift dependence are still under debate. Subhaloes from N-body simulations have shown bv > 1 (e.g., (<>)Col´ın et al. (<>)2000). In addition, (<>)Wu et al. (2013b) have shown that the exact value of bv depends on the selection criteria applied to subhaloes, on the resolution of simulations, and on the location of subhaloes. On the other hand, a simulated galaxy population based on assigning sub-haloes to dark matter particles (e.g., (<>)Faltenbacher & Die-(<>)mand (<>)2006) or based on hydrodynamical simulations with cooling and star formation (e.g., (<>)Lau et al. (<>)2010; (<>)Munari (<>)et al. (<>)2013) tends to have unbiased velocities. ",
    "Since this paper focuses on the possible systematics from N-body simulations, we adopt bv > 1 observed in N-body simulations. Based on the recent calibration from (<>)Munari et al. ((<>)2013), we adopt the value of velocity bias to be ",
    "(42) ",
    "(estimated from the dotted curve in their fig. 7A, which corresponds to subhaloes in their N-body simulations.) The bottom-right panel of Fig. (<>)6 shows the systematic error in P(k) caused by this velocity bias. Introducing higher velocity dispersion of galaxies clearly leads to larger suppression on small scales. Note that each curve showcases a dip near k ≈ 1 h Mpc−1 , which roughly corresponds to the scale where one-halo and two-halo terms cross. As shown in Section (<>)2.2, the exponential suppression of RSD enters the one-halo and two-halo terms differently; modifying the RSD will therefore slightly change the scale of one-halo to two-halo transition. Also note that the shift in P(k) does not vanish even for very small k, because the exponential suppression enters the two-halo term as well. ",
    "The inset in the bottom-right panel of Fig. (<>)6 shows that the systematic shifts associated with velocity bias (difference between no velocity bias and positive velocity bias) dominate the statistical error even for kmax = 0.14 h Mpc−1 . Because the deviation of P(k) starts at large scales and increases towards small scales, the velocity bias is dominant among the four sources of systematic errors studied in this paper. ",
    "As before, we consider values of the velocity bias that interpolate between the two extreme values considered: ",
    "(43) ",
    "where fsys = 0 corresponds to bv = bfid = 1.0 while fsys = 1.0 corresponds to bv = 1.1 − z/15. The cyan curve= bv, alt in Fig. (<>)7 shows the required reduction of fsys for a given kmax. For example, to extend the survey just out to the usually conservative wavenumber kmax = 0.3 h Mpc−1 , better-",
    "than-current knowledge of the velocity bias (fsys = 0.11 < 1) is required.6 (<>)",
    "Given that a biased bv value can lead to a significant systematic error, it is necessary to marginalize over bv to mitigate the systematic bias. We find that marginalizing over an additional parameter bv in the Fisher matrix calculation does not significantlydegrade the dark energy constraints; the statistical error σ(wa)σ(wp) is increased by a factor of 2 at most. Since P (k) is sensitive to the change in bv (as shown in the last panel of Fig. (<>)6), it is not surprising that bv can be well constrained by data when set free. In addition, the effect of bv does not seem to be degenerate with the effects of other nuisance parameters and is likely to be well constrained. ",
    "While the preparation of this paper was near completion, we learned about the related work from (<>)Linder & Sam-(<>)sing ((<>)2013). These authors have focused on a particular RSD model from (<>)Kwan et al. ((<>)2012) and assess the impact of uncertainties in this model on cosmological constraints. These authors have found that, if the model parameters are fixed, they often require sub-per cent accuracy; on the other hand, if these model parameters are self-calibrated using the data, they do not significantly degrade the cosmological constraints. This trend is consistent with our findings regarding fixing versus marginalizing over the velocity bias. ",
    "We emphasize that the main goal of this paper is to see to what extent the theoretical uncertainties associated with calibrating galaxy clustering using N-body simulations lead to errors in the cosmological parameters. Given the diffi-culty of predicting clustering beyond k  0.5 h Mpc−1 using purely theoretical methods (e.g., the perturbation theory), resorting to calibration with N-body simulations is required, and this will remain to be the case for years to come. Our findings suggest that the velocity information of galaxies predicted from N-body simulations is likely to generate biases. "
  ],
  "7 SUMMARY ": [
    "As the interpretation of the galaxy clustering measurements from deep, wide redshift surveys often relies on synthetic galaxy catalogues from N-body simulations, the systematic uncertainties in N-body simulations are likely to lead to systematic errors in the cosmological results. In this paper, we have studied several theoretical uncertainties in the predictions of N-body simulations, including the statistics, the spatial distribution and the velocity dispersion of subhaloes. In particular, we have applied the halo model to calculate the galaxy power spectrum P (k), with inputs from recent N-body simulations. We have investigated how the uncertainties from these inputs impact the cosmological interpretation of P (k), and how well these systematics need to be ",
    "controlled for future surveys. Our main findings can be summarized as follows: ",
    "Velocity bias is likely to be the most important source of systematic error for upcoming surveys. The current uncertainty of 10 per cent at z = 0 is likely to introduce 3 ",
    "The sensitivity of P (k) to velocity bias leads to the question of what can be done to alleviate the potential systematic bias. Calibration through both observations and simulations is certainly one obvious solution. Another trick that is increasingly being used for large-scale structure surveys is to self-calibrate the systematic error(s); in the velocity-bias case, this would mean marginalizing over bv. With this marginalization, we expect to be left with vastly diminished biases and only a modest degradation in the cosmological parameters. We do not expect the bias to vanish completely, however, since second-order effects (e.g., redshift-and scale-dependence of bv) will remain and will cause systematic shifts. Given that we currently do not have a good model of bv(z, k), we have not attempted the full self-calibration exercise, but we definitely expect this to be modus operandi of galaxy clustering analyses in the future. "
  ],
  "ACKNOWLEDGEMENTS ": [
    "We thank Andrew Hearin, Eric Linder, Chris Miller, and Zheng Zheng for many helpful suggestions. We also thank the anonymous referee for helpful comments. This work was supported by the U.S. Department of Energy under contract number DE-FG02-95ER40899. "
  ]
}