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The Effects of Halo Assembly Bias on Self-Calibration in Galaxy Cluster Surveys

Self-calibration techniques for analyzing galaxy cluster counts utilize the abundance and the clustering amplitude of dark matter halos. These properties simultaneously constrain cosmological parameters and the cluster observable-mass relation. It was recently discovered that the clustering amplitude of halos depends not only on the halo mass, but also on various secondary variables, such as the halo formation time and the concentration; these dependences are collectively termed assembly bias. Applying modified Fisher matrix formalism, we explore whether these secondary variables have a significant impact on the study of dark energy properties using the self-calibration technique in current (SDSS) and the near future (DES, SPT, and LSST) cluster surveys. The impact of the secondary dependence is determined by (1) the scatter in the observable-mass relation and (2) the correlation between observable and secondary variables. We find that for optical surveys, the secondary dependence does not significantly influence an SDSS-like survey; however, it may affect a DES-like survey (given the high scatter currently expected from optical clusters) and an LSST-like survey (even for low scatter values and low correlations). For an SZ survey such as SPT, the impact of secondary dependence is insignificant if the scatter is 20% or lower but can be enhanced by the potential high scatter values introduced by a highly correlated background. Accurate modeling of the assembly bias is necessary for cluster self-calibration in the era of precision cosmology.


INTRODUCTION
The observed accelerating expansion of the Universe, which is often interpreted as evidence of dark energy, is one of the most surprising results of modern cosmology. In the ΛCDM paradigm, dark energy governs the late time expansion of the Universe, halting the growth of structures. Consequently, the evolution of the number of massive galaxy clusters provides one of the most powerful probes of dark energy (e.g. Wang & Steinhardt 1998;Haiman et al. 2001;Holder et al. 2001;Levine et al. 2002;Hu 2003;Majumdar & Mohr 2003;Rozo et al. 2007;Gladders et al. 2007;Mantz et al. 2007).
Several planned and ongoing surveys will identify massive clusters over substantial volumes using a variety of techniques, including optical galaxy counts (e.g. York et al. 2000;The Dark Energy Survey Collaboration 2005;Tyson 2002), the Sunyaev-Zel'dovich effect (e.g. Ruhl et al. 2004;Kosowsky 2003), and X-ray emissions (e.g. Ebeling et al. 2007;Burenin et al. 2007). These cluster surveys will complement a variety of future dark energy measurements using tools such as Type Ia supernovae, weak lensing, and baryon acoustic oscillations. Since each of these methods is subject to different systematics, combining them thus provides cross checks necessary to avoid biased inferences on the properties of dark energy (Albrecht et al. 2006).
While the abundance of clusters as a function of mass is well understood from a theoretical standpoint, measuring this abundance relies on observable mass tracers. This reliance is the single most significant obstacle confronting the use of clusters as cosmological probes. In particular, the statistical observable-mass relation needs to be understood to high accuracy in order to avoid systematic errors in the inference of cosmological parameters. Alternatively, additional observable quantities that depend on halo mass allow one to simultaneously constrain cosmology and the aforementioned observable-mass relation. One such observable quantity is the clustering amplitude of clusters, which depends sensitively on mass and can be determined through a counts-in-cells analysis. This general method is often referred to as "selfcalibration" (Majumdar & Mohr 2004;Lima & Hu 2004. In this work, we explore a possible systematic that arises in the self-calibration analysis, namely, the dependence of the clustering amplitude of halos on secondary variables. The clustering amplitude of halos is characterized by the halo bias, and recent studies have shown that halo bias depends not only on halo mass but also on additional halo properties, such as concentration, formation time, spin, substructure fraction, etc. (e.g. Gao et al. 2005;Harker et al. 2006;Wechsler et al. 2006;Wetzel et al. 2007;Bett et al. 2007;Jing et al. 2007). These dependences are often interpreted as arising from the different assembly histories of halos of the same mass, and we refer to these dependences collectively as "assembly bias" (e.g. Croton et al. 2007). If cluster selection is biased with respect to any of these variables, the observed clustering amplitude of clusters will deviate from the mean clustering amplitude of clusters with the same mass distribution. This deviation will lead to a biased inference of the observable-mass relation, and therefore to biased estimates for the cosmological parameters of interest.
We herein take the secondary parameter to be the halo concentration, which has been shown to play a role in halo bias for massive clusters by Wechsler et al. (2006; see also Wetzel et al. 2007;Jing et al. 2007). We then incorporate the concentration dependence of halo bias into the standard self-calibration formalism developed in Lima & Hu (2004. With modified Fisher matrix formalism, we investigate the impact of this additional dependence on cosmological parameter estimates from self-calibration. We specifically calculate the expected effects for four example galaxy cluster surveys, which represent the Sloan Digital Sky Survey (SDSS; assuming clusters selected from the photometric data), the Dark Energy Survey (DES), the South Pole Telescope (SPT), and the Large Synoptic Survey Telescope (LSST). We also explore various assumptions about the correlation between cluster observable and concentration. In detail, the significance of this systematic effect depends on the strength of this correlation as well as on the observablemass scatter. We find that the resulting bias in the inferred cosmological parameters is insignificant for the current SDSS photometric surveys, but it can be significant for upcoming photometric surveys such as DES and LSST. On the other hand, for SZ this systematic is less likely to be significant if the scatter is small and mainly intrinsic, but may still be significant if the correlated background dominates the scatter. This paper is organized as follows. In §2 we discuss why assembly bias may lead to biased cosmological parameter estimates in cluster counting experiments. In §3.1 we review the standard self-calibration formalism, and then proceed in §3.2 to include assembly bias into this formalism. Our statistical methodology for estimating the systematic errors due to assembly bias is described in §3.3. Details of our implementation can be found in §4. Section 5 presents our results and discussion. We summarize in §6.

BIAS MATTERS
Halo bias characterizes the clustering amplitude of dark matter halos, and it is typically defined as the ratio between the density contrast of halos and that of the dark matter. In the hierarchical structure formation predicted by CDM, halo bias is a strong function of mass, increasing for more massive halos. This dependence on mass is now well calibrated from numerical simulations and can be approximated analytically with the excursion-set theory (e.g. Mo & White 1996;Sheth et al. 2001;Seljak & Warren 2004;Zentner 2007). Halo bias depends sensitively on dark energy in a way that is complementary to the dependence of the mass function on dark energy; thus, including the halo bias information in cluster counting experiments improves the dark energy constraints from using mean halo abundances alone.
Much work on halo bias has made the simplifying assumption that halo bias depends only on halo mass. However, recent studies based on N -body simulations have found evidence that secondary variables such as halo formation time and concentration do impact halo bias (e.g. Gao et al. 2005;Harker et al. 2006;Wechsler et al. 2006;Wetzel et al. 2007;Bett et al. 2007;Jing et al. 2007). In this work, we focus on the impact of halo concentration on halo bias, principally because among all secondary parameters, this dependence is the strongest at cluster scales and is the best understood statistically. The halo concentration describes the halo density profile and is defined as c = R vir /r s , where r s is the radius where the density profile has a log slope of −2. The halo concentration has been shown to correlate tightly with the halo formation epoch by e.g. Wechsler et al. (2002).
We specifically use the fitting formula given by Wechsler et al. (2006, eq. 6 (1) where b avg (M ) is the mean halo bias at fixed mass, b c (c|M/M * ) characterizes the concentration dependence of halo bias, and M * is the characteristic mass of gravitational collapse [quantitatively defined as σ(M * ) = 1.686, where σ(M ) is the r.m.s density fluctuation inside a sphere that encloses mass M ]. The superscript "ab" refers to "assembly bias," which we use as a generic term for the dependence of halo bias on secondary variables, based on the conjecture that these dependences arise through the different formation histories of halos of the same mass. We assume this formula holds for all clusters included in our fiducial surveys, although part of these clusters are outside the range where this formula has been calibrated with simulations. In addition, we note that Wechsler et al. (2006) calibrated this formula with M vir , while the mass function and the halo bias we use are not always well-calibrated with the same mass definition. We ignore the systematic errors that may be caused by these uncertainties.
The left panel of Figure 1 illustrates how concentration impacts halo bias in the fitting formula of Wechsler et al. (2006). As can be seen, for M 10 13.5 h −1 M ⊙ , low concentration halos are more clustered than high concentration ones of the same mass. This difference is potentially significant: if the cluster observable is correlated with concentration, one might measure cluster bias that differs from the mean halo bias for random halos of the same mass.
The right panel of Figure 1 shows how the effect of assembly bias can resemble that of a high dark energy density, with an extreme assumption of perfectly anticorrelated observable and concentration. Cumulative bias, which is relevant for halo samples above a certain observable threshold (see eq. [11]), is plotted here. As can be seen, if we tend to observe low concentration halos, the effect of assembly bias (dashed curve) makes the observed halo bias higher than the mean halo bias (averaged over random halos samples of the same mass) for the same fiducial cosmology (solid curve). This effect mimics a high dark energy density Ω DE = 0.9 (dotted curve), since high Ω DE will make structures rarer and more clustered. Thus, a wrong inference of Ω DE is possible if assembly bias is ignored in this case. In the following sections, we provide detailed formalism and analyses of such systematics under the framework of the self-calibration of observable-mass relation.  Wechsler et al. (2006). We assume a WMAP3 cosmology and log-normally distributed concentrations at a given halo mass. Halos are binned by concentration into four quartiles, and the halo bias of each quartile systematically deviates from the average halo bias (solid curve). Above 10 13.5 h −1 M ⊙ , low concentration halos (red and orange dotted curves) are more clustered than high concentration ones (green and blue dashed curves) of the same mass. The bottom panel shows the residual compared with the average halo bias. Right panel: Degeneracy between high dark energy density and assembly bias. The solid curve shows the cumulative bias (eq. [11], with the selection function nonzero above a threshold M th ) for the fiducial WMAP3 cosmology. The dashed curve shows the effect of assembly bias with the assumption of perfectly anti-correlated cluster observable and concentration (see §2 and 3.2 for details). This correlation can mimic the effect of high dark energy density (here assumed to be Ω DE = 0.9), shown as the dotted curve.
In a pixelated galaxy cluster survey, halo bias is related to the sample variance of cluster counts within the small sub-volumes of the survey (Hu & Kravtsov 2003). Including the sample variance in a counts-in-cells analysis allows one to "self-calibrate" the observable-mass distribution, which is one of the main uncertainties in modeling the surveys. This approach can thereby improve the dark energy constraints relative to "counts only" experiments (Majumdar & Mohr 2004;Lima & Hu 2004. In this section, we review the basic selfcalibration, closely following the formalism developed by Lima & Hu (2004).
Given a large-volume survey, consider a redshift slice which is sufficiently thin to make evolution ignorable. We then divide the area of this slice into equal-area cells and count the clusters in each cell. 3 The number of clusters in cell i, denoted by N i , is affected by the Poisson shot noise, which is modeled as N i ∼ Poisson(m i ). This Poisson mean m i varies from cell to cell due to the largescale clustering of matter and halos, and this fluctuation can be modeled as a normal distribution m i ∼ N(m, S), wherem is the mean halo abundance and S is the sample variance.
In a given mass range, the mean number counts of clusters in cell i depend onm, the bias integrated over the mass rangeb, and the mass overdensity δ i within this 3 Here we suppress all redshift dependence in our notation for simplicity. In practice, we consider the redshift dependence of the mass function, the halo bias, the observable-mass distribution, and the comoving survey volume. For readers of Lima & Hu (2004), note that our notation is slightly different. Since we consider a single redshift slice, our subscript i indicates the cell label of the same redshift, while in Lima & Hu (2004), their subscript i indicates a cell of redshift z i . cell with respect to the background: (2) The sample variance then has the form where Here P (k) is the matter power spectrum and W ( k) is the k -space window function of a cell of volume V , normalized such that V = d 3 xW ( x). Applying a counts-incells analysis, N i of each cell can be measured, andm and S can be obtained from a likelihood analysis. With additional knowledge of the matter power spectrum,b can be obtained.
Note that this sample variance should be more rigorously defined as the sample covariance In practice, our cell size is much larger than the correlation length of clusters; thus, the correlations between different cells are negligible. The off-diagonal elements are therefore much smaller then the diagonal ones, and the matrix S ij reduces to a diagonal matrix S ij = δ ij S, whose dimension equals n c , the number of cells in the redshift slice. We next relate these measurable quantities to theoretical models. Let M obs denote the observed mass proxy (the observable) of galaxy clusters. Given a differential mass function dn/dM and an observable-mass distribution P (M obs |M ), the differential observed cluster abundance is given as In terms of the binning function φ(M obs )-which is defined to be equal to unity if M obs falls in the bin corresponding to the observable range, and zero otherwiseand the cell volume V , the mean observed cluster abundance readsm which can be further simplified as if we define the selection function to be Given the halo bias b(M ), the bias integrated over the observable bin similarly reads From equations 9 and 11 we can see that if bothm andb are measured in the survey, the selection function φ|M can be self-calibrated.
In large-volume surveys, we often have several redshift bins and need to consider howm and S vary with redshift:m(z) and S(z). The sample variance is then generalized to the matrix S = diag(S ij (z 1 ), S ij (z 2 ), ...), where each S ij (z k ) has the dimension n c × n c . Similarly,m is generalized asm = (m(z 1 ),m(z 2 ), ...), with eachm(z k ) being a n c component vector. For future reference, we further define M = diag(m) and C = M + S; C is the covariance matrix in the limit of large cluster numbers in a cell (m i ≫ 1; see Lima & Hu 2004).
Constraints on dark energy parameters are extracted from the likelihood function that involves the countsin-cells data, the theoretical mean abundance, and the theoretical sample variance. For theoretical forecasts, the Fisher matrix-the expectation value of the second derivative of the minus log-likelihood function-is often applied. For a combination of the Poisson shot noise and the Gaussian sample variance, the Fisher matrix reads (Lima & Hu 2004) where the comma and subscript α indicates the partial derivative with respect to model parameter θ α . The Fisher matrix approach essentially approximates the likelihood function as a Gaussian distribution near its maximum likelihood point, and the curvature at this point is related to the constraints on the model parameters. The covariance matrix for model parameters is approximated by the inverse of the Fisher matrix. This basic picture will play a key role in §3.3, where we modify the Fisher matrix formalism for assessing the systematic errors.

Incorporating Assembly Bias into Self-Calibration
We now incorporate assembly bias into the selfcalibration formalism. The formalism we outline below is relevant for any secondary parameter which both affects the halo bias and correlates with the cluster mass proxy. We specifically consider the secondary parameter to be the halo concentration c and refer to this dependence throughout as "assembly bias." Note that although the halo concentration and assembly history are generally expected to be tightly correlated (Navarro et al. 1997;Wechsler et al. 2002), they may not have exactly the same effect on halo bias (see e.g. . Let b ab (M, c) be the halo assembly bias, which now depends on both mass and concentration, and let f (c|M ) be the distribution of concentrations for halos of mass M . In this case, the observable-mass distribution P (M obs |M ) needs to be generalized to an observablemass-concentration distribution P (M obs |M, c). With the secondary parameter c, the mean abundancem takes the form where This mean abundance remains the same as equation 9 since the concentration dependence only affects the halo bias but not the mass function. We thus require On the other hand, the bias integrated over the observable range is affected, and the analog of Equation 11 is (16) The corresponding sample variance in this case reads and we analogously define C ab = M + S ab . Replacing the corresponding matrices in Equation 12, we obtain the Fisher matrix incorporating assembly bias. The difference between P (M obs |M, c) and P (M obs |M ) depends on how M obs correlates with c. We leave these details to §4.1 and simply state here that our parametrization depends on the cross-correlation coefficient r relating M obs and c at fixed halo mass. When r = 0, assembly bias has no impact on self-calibration; when r = ±1, the impact of assembly bias is maximized. Figure 2 demonstrates the formalism described above (with an SPT survey assumption and a WMAP3 cosmology, see §4) and shows how the correlation between M obs and c changes the constraints on dark energy parameters, assuming that we have thorough knowledge of assembly bias and that r = ±1 (dotted and dashed curves). As can be seen, correlation between M obs and c actually improves the dark energy constraints if r is known a priori. This improvement is presumably due the dependence of bias on M * , which is also sensitive to dark energy, although we also note that the assumption of self-similarity in M/M * needs to be assessed in -Improvement of dark energy constraints assuming a thorough modeling of assembly bias and knowledge of the crosscorrelation relating M obs -the cluster's mass estimate based on a cluster observable-and c, the halo's concentration parameter. All error ellipses include the 68% confidence regions in the Ω DE -w plane. The solid ellipse shows the fiducial model of zero observableconcentration correlation (r = 0), in which case assembly bias has no effect. The dotted/dashed ellipse corresponds to an observable which is perfectly correlated/anti-correlated with concentration (r = 1/−1). If assembly bias is correctly modeled, the sensitivity of assembly bias to M * slightly improves dark energy constraints. the dark energy-dominated regime. In addition, with the knowledge of r, the scatter in M obs actually contains the information of halo concentration, which may also improve cosmological constraints. These extreme cases are mainly for demonstration, since we are unlikely to have sufficient astrophysical knowledge to specify both the assembly bias and this correlation. However, if individual concentrations can be measured for the most massive clusters (where the impact of assembly bias is most severe), they could provide observational evidence of assembly bias and increase the efficacy of self-calibration.
In the following sections, we explore the question: if one were to perform the self-calibration analysis ignoring the effects of assembly bias (effectively, assuming r = 0), how would the estimated cosmological parameters be biased? As we shall see, the answer sensitively depends on r and on the scatter in the observable-mass distribution. We next include r as a free parameter in the Fisher matrix analysis and consider the effect of marginalization over r. However, a caveat for applying the Fisher matrix here is that since r is bound to the range [−1, 1], the likelihood function for r may not be well-approximated as Gaussian if r is close to ±1. Because the Fisher matrix is based on this Gaussian approximation, it may not apply to the case when r approaches ±1. On the other hand, our fiducial choices of this parameter, which are in the range |r| ≤ 0.5, may circumvent this problem.

Biased Parameter Estimation from Ignored
Systematics: A Modified Fisher Matrix Formalism In §2, we described how ignoring the impact of assembly bias can potentially lead to biased cosmological parameter estimates. In this section, we modify Fisher matrix formalism to quantitatively assess the significance of this systematic. We focus on how the parameter estimates are biased due to a wrong model assumption, and how significant this systematic error is when compared with statistical uncertainties. This formalism is motivated by the standard Fisher matrix formalism as presented in Tegmark et al. (1997).
We generally consider two models, denoted by model A and model B, each of which describes a data set x based on a parameter θ. Here θ can be generalized to a vector denoting a set of parameters (θ i values). We assume that the observed data set x is well described by model B but is mistakenly analyzed according to model A. If θ t denotes the true parameter in model B that corresponds to the observed data set x, we are interested in how the estimated parameterθ recovered based on model A differs from θ t . Our quantitative analysis can be summarized as follows: The data set x is assumed to be drawn from the probability distribution P B ( x|θ t ) for model B; in order to relate θ to θ t , we take average over x to compute ln L A (θ)|θ t .
2. We take the pointθ which maximizes ln L A (θ)|θ t as our estimator for the recovered cosmology. This step defines the functionθ(θ t ), the recovered model parameter varying with the input parameter θ t . We are particularly interested in δθ =θ(θ t ) − θ t , which is the systematic error in parameter inference due to assuming an incorrect model. 4 3. In order to assess the significance of the systematic error δθ, we compare it against the statistical uncertainty in θ. We calculate the modified Fisher matrixF ij (θ t ) = ∂ 2 (− ln L A )/∂θ i ∂θ j |θ t and obtain the corresponding error bar σ 2 θi = (F −1 ) ii . The systematic error is significant if δθ i σ θi .
A detailed derivation when both P A ( x|θ) and P B ( x|θ) are Gaussian can be found in the Appendix.
In this study, model A represents the standard selfcalibration analysis that ignores assembly bias, while model B is self-calibration analysis that includes the impact of assembly bias. The data set x is the number counts in each of the cells under consideration. The systematic errors of the recovered parameters are given by where C ab is the covariance matrices with assembly bias, and C and F are the same as those in equation 12. The modified Fisher matrix reads in which the modification comes from the change of covariance matrix due to assembly bias (see the Appendix).
We note that similar formalisms arising from different approaches can be found in e.g. Knox et al. (1998), Huterer & Turner (2001), Huterer & Linder (2007), and Amara & Refregier (2007). Figure 3 illustrates the results of our formalism as applied to the self-calibration analysis for an SPT-like survey in the specified WMAP3 cosmology (see §4 for details of implementation and assumptions). In each panel, the open circles indicate the assumed true values, while the filled circles show the recovered parameters from a self-calibration analysis that ignores assembly bias. The ellipses include the 68% confidence regions in the Ω DE -w plane; the dashed ellipses correspond to correctly-modeled assembly bias (assuming that we know the correlation coefficient r a priori; r will be mathematically defined in §4.1), while the solid ellipses correspond to the ignored assembly bias. Note that the shape of the confidence regions can also be changed by this systematic. The left panel shows the assumption of a small M obs -M scatter and low M obs -c correlation (σ ln M = 0.1 and r = −0.5), and the systematic errors are 0.22 and 0.23σ for Ω DE and w, respectively; the deviations of the parameter estimates are much less than the statistical uncertainties. The right panel shows the assumption of a larger scatter and perfectly anti-correlated M obs and c (σ ln M = 0.25 and r = −1), and the resulting systematic errors are 1.14 and 1.2σ for Ω DE and w, respectively; these deviations are significant and cannot be ignored. We thus expect the impact of assembly bias will be stronger if the observable-mass relation has a large scatter and if M obs is strongly correlated with c. The exact dependence of systematic error on these two quantities will be fully explored in §5.

Parameterizing the Observable-Concentration
Correlation In the absence of assembly bias, we follow Lima & Hu (2005) to parameterize the observable-mass relation P (ln M obs |M ). Given halo mass M , the corresponding log observables ln M obs are modeled as a Gaussian distribution with mean ln M + ln M bias -where M bias specifies the offset between the estimate mass and the true mass-and variance σ 2 ln M . This parameterization serves as the standard case as we generalize P (ln M obs |M ) to P (ln M obs |M, c) for analyzing the effect of assembly bias.
A priori, we do not know exactly how the estimated mass of a cluster M obs will depend on the cluster's concentration c, that is, the correct parameterization for P (ln M obs |M, c). In detail, this relation may depend on both physical and observational effects. However, we would like to demand a simple wish-list of properties of our parameterization: 1. When marginalized over concentration, P (ln M obs |M, c) should reduce to the Gaussian distribution P (ln M obs |M ) of the fiducial case (as required by eq. [15]), independent of any new parameters introduced (i.e. we should keep the total ln M obs -ln M scatter fixed).
2. In order to study how self-calibration is affected as the dependence of M obs on c is "turned on," the parameterization should have a tunable parameter. Note. -All surveys use cells of area 10 deg 2 and ∆z = 0.1 When this tunable parameter is set to zero, our analysis should reduce to the standard case.
In the interest of simplicity, we take P (ln M obs |M, c) to be Gaussian in ln M obs , and assume that the halo concentration slightly shifts ln M obs relative to ln M , so that the mean and the variance of ln M obs are given by In the above expressions, r is the correlation coefficient between ln M obs and c ′ at fixed ln M , σ ln M is the scatter in ln M obs at fixed M , and c ′ is defined via Note that when r = 0, all of the observed scatter in ln M obs at fixed ln M is intrinsic, and our model reduces to the standard case. Conversely, for r = 1, the scatter in ln M obs at fixed ln M is entirely due to the scatter in halo concentration at fixed mass. As a consistency check, we find that if we marginalize P (ln M obs |M, c) over concentration (assuming a log-normal distribution for c at fixed mass, see e.g. Jing 2000, Bullock et al. 2001, and Neto et al. 2007), the resulting distribution P (ln M obs |M ) is exactly that of the standard case; that is, our parameterization preserves the total scatter in ln M obs at a given ln M .

Survey Assumptions, Cosmological Models, and Nuisance Parameters
With the Fisher matrix analysis, we statistically forecast the systematic effects for four galaxy cluster surveys: the Sloan Digital Sky Survey (SDSS, York et al. 2000; assuming the volume using photometric data), the Dark Energy Survey (DES 5 ), the South Pole Telescope (SPT 6 ), and the Large Synoptic Survey Telescope (LSST 7 ). The survey areas are assumed to be 7500 deg 2 for SDSS, 5000 deg 2 for DES, 4000 deg 2 for SPT, and 20000 deg 2 for LSST, with survey depths of for Ω DE and w, respectively; in this case the effects of assembly bias are ignorable. On the other hand, the right panel shows that for an extreme assumption of σ ln M = 0.25 and r = −1, the systematic errors are 1.14 and 1.2σ for Ω DE and w, respectively; in this case the effects of assembly bias are significant. and LSST, the observational threshold is assumed to be M obs ≥ 10 13.5 h −1 M ⊙ , and the counts in each of these surveys are binned in three observable bins. The survey parameters for all four surveys are detailed in Table 1.
While the mass threshold of SZ observations has little redshift dependence (e.g. Carlstrom et al. 2002), the mass threshold of optical surveys has more uncertainties. Clusters with mass 10 13.5 h −1 M ⊙ have been shown to be detectable, with high purity and completeness, with more than 10 bright red galaxies (∼ 0.4L * ) in the SDSS photometric survey out to z ∼ 0.3 Johnston et al. 2007). We note that our choice of the minimum mass for the optical surveys assumes that such clusters can still be detected with high purity and completeness out to the maximum redshift z max . This assumption may be reasonable out to z = 1, where clusters have been shown to have a robust red sequence, but the efficacy of this method will eventually break down at higher redshifts. In any case, it will need to be tested in detail with both realistic simulations and the data itself. We note that for LSST, one may wish to detect clusters using peaks in the lensing shear instead of from assumptions about the galaxy distribution (e.g. Kaiser 1995;Hennawi & Spergel 2005), in which case self-calibration could serve as a consistency check for the predictions for the observed shear signal made directly from simulations. In §5, we consider one example case for LSST, which has similar assumptions to the lower z max optical surveys, for reference.
In this work, we consider two sets of cosmological parameters, namely the best fit cosmologies to WMAP1 (Spergel et al. 2003) and WMAP3 (Spergel et al. 2007), whose parameter values are listed in Table 2. Both of them are flat ΛCDM cosmologies but differ mainly in the relative contribution of dark energy to the global energy density, in the normalization of fluctuations (δ ζ or σ 8 ), and in the spectral index (n). The impact of these differences on our analysis will be presented in §5. In our statistical forecast, we do not put any priors on dark energy parameters, but we assume Plancklike priors on the rest of the cosmological parameters (see Table 2). Finally, in our forecast models we use the halo mass function by Jenkins et al. (2001), the bias function by Sheth et al. (2001), and the assembly bias b ab (M, c) found by Wechsler et al. (2006, this assumption was shown in Fig. 1).
With regard to the observable-mass relation, our model involves three nuisance parameters: the bias in the estimated mass (ln M bias ), the scatter of ln M obs given ln M (σ ln M ), and the cross-correlation coefficient between ln M obs and the normalized halo concentration c ′ (r). Throughout, we take ln M bias = 0 as our fiducial model. Our choice for the fiducial values for the scatter and the cross-correlation coefficient in each of the surveys requires further discussion.
Let us first focus on the scatter. For an SPT-like survey, the observational mass proxy is the SZ decrement of the cosmic microwave background due to the hot, ionized gas permeating the inter-cluster medium. At present, this scatter has only been predicted from numerical simulations but has not been determined from observations. White et al. (2002) argued that the three main sources of scatter are the evolution of the M -T relation, asphericity in the matter distribution, and line-of-sight projection. Motl et al. (2005) and Nagai (2006) showed that the scatter is 10%-15%, and the scaling relation is insensitive to the detailed physical processes involved in galaxy formation, with a good agreement with self-similar models. However, Shaw et al. (2007) showed that at least 20% intrinsic scatter exists due to the internal properties of Note.
galaxy clusters. They also demonstrated that this scatter could be reduced by choosing different aperture radius for defining M and Y , or by removing cluster samples with many substructures. Moreover, it may be possible to reduce the scatter even further using cluster structural properties. For example, Afshordi (2007) proposed a "fundamental plane" among the cluster mass, the total SZ flux, and the SZ half-light radius R SZ,2 ; in simulations, this relation reduced the scatter in mass estimates to ∼ 14%. Further, Haugboelle et al. (2007) found that by constructing an empirical model for the SZ profile, which includes a scaling parameter r 0 , they could reduce the scatter down to 4%. In this work, we take the largest of these range of values, namely 20%, as our fiducial scatter for SPT. If SPT is insensitive to halo assembly bias for this largest possible scatter, then it will also be insensitive for smaller values of scatter.
In optical surveys, the usual observational mass proxy is the optical richness, namely the galaxy number in a galaxy cluster. Other choices are also possible, including the total optical luminosity or combinations of parameters (e.g. Rykoff et al. 2008;Becker et al. 2007;Reyes et al. 2008). Determining a reasonable choice for the scatter for a DES-like survey is somewhat less straightforward, as predictions from simulations are less robust and the scatter can highly depend on both the richness measure and the cluster finder. Gladders et al. (2007) applied a self-calibration analysis to a catalog from the Red-Sequence Cluster Surveys (RCS), finding the fractional scatter f sc to be 0.69 ± 0.20 and 0.71 +0.19 −0.17 (based on different priors). Using the velocity information of galaxies in maxBCG clusters, Becker et al. (2007) estimated that the optical richness had a mass dependent scatter which varied from about 0.75 for massive clusters to 1.2 for group scale objects. Cross-correlation with the X-ray data on these same clusters suggests a considerably smaller scatter of about 0.5 (E. Rozo et al. in preparation). Here, we choose 0.5 as our fiducial scatter value for two reasons: First, as we shall see, even with this amount of scatter, halo assembly bias has a significant impact on self-calibration studies; this small scatter thus provides a baseline value for the impact of assembly bias. Second, we note that current optical richness estimates have all used fairly crude measures of richness. We think it is highly probable that in the near future we will start seeing richness measures that are considerably more strongly correlated with mass than those used at present. Thus, we have opted to select a scatter value that is closer to the lowest scatter estimated in current samples.
We now turn to the correlation relating observable and concentration, r. Currently, there are no observational constraints on r for either optical or SZ mass proxies. We also find no quoted values for the correlation between SZ and halo concentration in the literature, although we note that Reid & Spergel (2006) and Shaw et al. (2007) investigated the impact of halo concentration on the scatter in Y SZ and found that a considerable fraction of the scatter in SZ is due to variations in halo concentration at fixed mass. In this work, we choose the fiducial value to be r = 0.4, which is the value observed in simulations of clusters using the hydrodynamical ART code (D. Rudd 2008, private communication).
The M obs -c correlation r for optical clusters is somewhat better understood. We are not aware of current observational constraints on the correlation between optical cluster richness and halo concentrations, although with a sufficiently large sample, this value could in principle be measured from lensing data. Zentner et al. (2005) and Wechsler et al. (2006) have shown that the amount of substructures in a cluster-size halo is negatively correlated with halo concentration. On the other hand, selection effects could modify this correlation. For instance, high concentration halos might, on average, be assigned higher richness than low concentration halos of the same mass due to the larger galaxy density near the cluster core. In this work, we choose r = −0.5 as our fiducial value, which is roughly consistent with the numerical results of Zentner et al. (2005) and Wechsler et al. (2006). We note that the results presented here assume that all of these parameters are constant with both redshift and mass.

RESULTS AND DISCUSSION
We now present the effect of assembly bias for a set of specific assumptions about galaxy cluster surveys. We focus on the systematic errors in the two dark energy parameters Ω DE and w when compared with the statistical errors expected from each survey, |δΩ DE |/σ ΩDE and |δw|/σ w . Figure 4 shows how these two ratios vary with the scatter in ln M obs -ln M (σ ln M ) and the crosscorrelations coefficient of ln M obs -c ′ (r) for our fiducial SDSS, DES, and SPT surveys. All plots assume a WMAP3 cosmology. As can be seen, a high degree of correlation and/or large scatter can result in significantly biased cosmological estimates for both DES and SPT, while for the current SDSS the statistical uncertainty is sufficiently large that halo assembly bias is insignificant.
How halo assembly bias differently affects DES and SPT is worth discussing. For a fixed scatter and correlation coefficient, the cosmological constraints from DES are considerably less biased than those of SPT. The reason for this difference is two-fold. First, DES clusters  Fig. 4.-Systematic errors for Ω DE (l eft) and w (r ight) estimators, as a function of scatter in the observable given mass (σ ln M ) and the correlation between observable and halo concentration (r). The ratios of the systematic error and the statistical uncertainty (|δθ|/σ θ ) are shown for three of our main survey assumptions: SDSS, DES, and SPT, from top to bottom. High scatter value and strong correlation/anti-correlation correspond to high deviation of estimators. We also mark the fiducial values of σ ln M and r in each panel according to our current knowledge from observations and numerical simulations. We note that these plots are also applicable to other surveys (e.g. X-ray) for which the survey volume and mass threshold are the same as those assumed here. See §5 for discussion. probe a lower mass scale than SPT clusters do. This difference is important because the effect of concentration on halo bias is important for high mass halos, but non-existent for halos of mass near M * ∼ 10 13 h −1 M ⊙ (Wechsler et al. 2006). Consequently, the cosmological constraints coming from low mass clusters (groups) should be unbiased. 8 The second important difference between SPT and DES is that in our fiducial surveys we have assumed binned counts for DES clusters but only thresholded counts for SPT clusters. Consequently, all the cosmological information provided by the shape of the halo mass function (which is unaffected by assembly bias) does not contribute to the SPT constraints. Thus, SPT constraints are considerably more sensitive to the effects of assembly bias than the DES constraints given the same scatter and correlation coefficient.
That is not, however, the end of the story. In order to fairly compare SPT to DES, one also needs to consider the regions of parameter space relevant to each of these surveys. We noted earlier that numerical simulations predict that the intrinsic scatter in the SZ signal is approximately 20% or even less (e.g. Motl et al. 2005;Nagai 2006;Shaw et al. 2007;Haugboelle et al. 2007). As can be seen from the bottom panels of Figure 4, these scatter values do not result in significant biasing of the recovered cosmological parameters for any value of r. Although the expected intrinsic scatter is small, the projection effect may raise or even dominate the total scatter (see e.g. White et al. 2002;Hallman et al. 2007). Holder et al. (2007) found that the SZ background can generate errors larger than 20% in recovered flux if σ 8 is near 0.7. If the extra scatter is due to the randomly aligned structures along the line of sight, then we do not expect assembly bias to have a significant impact. On the other hand, if the projection effect is dominated by nearby structures, the extra scatter due to projection will be strongly correlated with the environment, resulting in higher correlation between concentration and observable. In fact, if the scatter due to projection is as high as Holder et al. (2007) predicted and is also dominated by nearby correlated structure, the effect of assembly bias may be strengthened.
Photometric surveys like DES, in contrast, are very likely to be sensitive to the impact of assembly bias. In this case, we know that the optical richness-mass relation has a scatter 50% (e.g. Gladders et al. 2007;Becker et al. 2007;E.Rozo et al. in preparation). As can be seen in the middle panels of Figure 4, even moderate correlations between M obs and c, e.g. |r| 0.5, can result in significant biasing of the recovered cosmological parameters. It is likely, therefore, that cosmological analysis of the DES optical cluster sample will need to include halo assembly bias in order to avoid systematic errors in dark energy inference, unless the analysis can be done with an observable that is more tightly correlated with mass. On the other hand, DES will have additional mass measurements, including its weak lensing and SZ signals from SPT. If these measurements are included in the analysis, the effect of assembly bias may be dimin-ished. Observational inference of assembly bias may even be possible with mass profile measurements.
We note that in these figures, the only differences are between the survey volumes and the threshold in M obs , without assuming any other information specific to the optical or SZ surveys. Therefore, these results are applicable to other surveys (e.g. X-ray surveys) with the same survey conditions.
In Figure 5 we (1) explore the systematic effects due to assembly bias under different cosmological parameters (WMAP1 and WMAP3) and (2) extend the calculation to include an assumption for an LSST-like optical cluster survey. (For SDSS, DES, LSST: r = −0.5 and σ ln M = 0.5. For SPT: r = 0.4 and σ ln M = 0.2.) First, the most relevant difference between WMAP1 and WMAP3 is that WMAP3 has higher Ω DE and lower normalization δ ζ or σ 8 values, as listed in Table 2. As a result, the WMAP3 cosmology has fewer clusters, and the sample variance of the clusters is smaller. These differences increase the statistical errors of the surveys (see also Lima & Hu 2007), thus making the impact of assembly bias less significant in the WMAP3 cosmology than in the WMAP1 cosmology. Overall, our main conclusions remain unchanged. Second, for LSST, we find the systematic due to assembly bias is very significant for our fiducial values; this systematic is likely to be significant for even small values of σ ln M and r.
We especially note that the systematic of assembly bias impacts Ω DE and w differently. From the survey point of view, increasing z max above 1 largely improves the constraints on w, but barely improves the constraints on Ω DE . For w, the systematic error due to assembly bias increases monotonically with z max , while for Ω DE , this systematic error somewhat cancels and then changes its sign as z max increases. This difference is due to the fact that Ω DE and w affect the observed large-scale structure differently in different regimes. Dark energy affects the observed large-scale structure through two mechanisms: the growth function and the comoving volume. High Ω DE and high w both result in stronger suppression of structure. On the other hand, the volume dependence works differently: high Ω DE and low w correspond to larger volumes. The effects work in opposite directions for Ω DE ; higher Ω DE leads to less structure but more volume. Before the onset of dark energy domination, the comoving volume effect dominates; after dark energy take-over, the growth function effect dominates. Thus near the onset of dark energy domination, the observed structure is insensitive to Ω DE , leading to no extra information from this regime. The effects of assembly bias on Ω DE before and after the dark energy domination have opposite signs and thus cancel each other. On the other hand, for w, both effects work in the same direction; thus, including more survey volume will always increase the amount of information on w, and the systematic effects do not cancel. That is why the systematic effect of assembly bias on w increases monotonically with z max .
Another interesting question is how the constraints on cosmological parameters are degraded if we include r as an additional nuisance parameter that needs to be marginalized over. However, as we mentioned earlier, the fact that the likelihood function is non-Gaussian in r if r is close to ±1 implies that the Fisher matrix estimates may not apply. Therefore, the following constraints with -Impact of assembly bias for two different cosmologies and four survey conditions. The ratio |δθ|/σ θ for Ω DE and w are plotted as circles and squares respectively. Solid and open symbols are for WMAP1 and WMAP3 cosmologies respectively. DES and LSST are clearly sensitive to assembly bias, while SPT is marginally sensitive to it, with the effect being stronger for WMAP1 than WMAP3. A current SDSS-like survey is not sensitive to assembly bias. (Fiducial values assumed for other parameters include: r = −0.5 and σ ln M = 0.5 for SDSS, DES, and LSST, and r = 0.4 and σ ln M = 0.2 for SPT) marginalization over r are only to be taken as rough indicators.
We use moderate values for r (0.4 for SZ and −0.5 for optical) to compare the cosmological constraints assuming (1) fixed r values, and (2) r to be a free parameter in the Fisher matrix. Table 3 contains our results for three of the survey assumptions. As can be seen, while the error bars for DES are only slightly affected by marginalization over r, those for SPT increase by a factor of 2 to 3. The reason is again related to the mass binning; since our fiducial SPT survey does not include mass binning, there is no information about the shape of the halo mass function, which, if present, can improve the constraints on the scatter in the observable-mass relation. In the absence of this shape information, the constraints on the scatter is modest, which means that marginalization of Ω DE and w over the acceptable region of the parameter space will reach areas with very large scatter. Since those areas are highly sensitive to the effects of halo assembly bias, the marginalized errors will be significantly larger. In the last row of Table 3 (SPT5), we assume five narrow observable bins for SPT with bin size ∆log 0 M obs = 0.2. In this case, the dark energy constraints are barely degraded after marginalizing over r. Thus, mass binning is a crucial component of the data analysis for both DES and SPT to maximize their potential as cosmological probes. Note that, in all cases, r itself cannot be well-constrained like other nuisance parameters. Since the dependence on r only affects the sample variance but not the abundance, the information for constraining r is insufficient.

SUMMARY
Self-calibration analysis in galaxy cluster surveys relies on the dependence of the halo bias on mass to simulta-neously constrain cosmology and the cluster observablemass distribution. Recent work has shown that halo bias is sensitive not only to halo mass, but also to secondary parameters related to the assembly history. Here we consider the effect of halo concentration on the bias as a specific case of the secondary parameters (generally termed assembly bias), and show how it might affect self-calibration analyses. In particular, if halo selection depends on halo concentration, the observed clustering amplitude of the corresponding cluster sample will deviate from that of a random selection of clusters with the same mass distribution. This deviation in the observed clustering amplitude can result in biased inferences of cosmological parameters, depending on (1) the amount of scatter between halo mass and the observational mass proxy, and (2) the correlation between the mass proxy and halo concentration. For current surveys like SDSS, the statistical uncertainty is still sufficiently large that the systematic error due to assembly bias is negligible. On the other hand, for an SPT-like survey, the expected small amount of intrinsic scatter between the SZ decrement and halo mass suggests that the impact of assembly bias on parameter estimation is negligible; however, if the projection effect results in higher scatter in high density regions, assembly bias may have significant impact. For a DES-like survey, where the mass proxy is likely to have considerably larger scatter, we estimate that assembly bias can displace the recovered dark energy parameters from their true values by about 1σ. For an LSST-like survey, this systematic error can exceed 2σ in w. In the last two cases, halo assembly bias may need to be explicitly included in the cosmological analysis to avoid biasing of the recovered dark energy parameters. We emphasize, however, that our analysis has assumed the specific dependence of halo bias on halo concentration found by Wechsler et al. (2006). If this dependence is shown to be smaller at high masses, if the correlation relating the observable mass proxy and halo concentration can be shown to be small, or if observables that are more tightly correlated with mass can be found, the effect will be mitigated. We have shown that binning in mass is crucial for both optical and SZ surveys, as marginalization over this correlation coefficient can increase the expected errors of dark energy parameters by a factor of a few if we only use thresholded counts. Note. -Cosmological constraints with fixed cross-correlation coefficient r, and with marginalized r. We assume a WMAP3 cosmology, and the nuisance parameters are the same as those in Fig. 5. After marginalization over r, the constraints from binned cluster samples (SDSS, DES, and SPT5) are barely degraded, while the constraints from thresholded samples (SPT) are degrade by a factor of 2 to 3. This result demonstrates the importance of mass binning. In all cases, r cannot be well-constrained like other nuisance parameters since it only affects the sample variance but not the abundance. We emphasize, however, that the second part of this table are to be interpreted as rough indicators, since the likelihood function may not be Gaussian in r.

BIASED PARAMETER ESTIMATION FROM INCORRECT MODELS
In this section, we explicitly implement the modified Fisher matrix formalism developed in §3.3 for the case in which both P A ( x|θ) and P B ( x|θ) are Gaussian. Let µ(θ) and C(θ) be the mean and covariance matrix defining P A ( x|θ) in model A, which is related to the likelihood function; let µ B (θ) and C B (θ) be the corresponding quantities in model B, which represent the observed data. Note that µ(θ) and C(θ) contain the model parameter θ that we are trying to fit, while µ B and C B contain the true parameter value θ t . The log-likelihood function of model A reads (up to a constant) 2L = −2 ln L( x|θ) = ln det C + ( x − µ) T C −1 ( x − µ) .
To proceed further, we focus on two simple examples of interest. The first example is the effect of assembly bias; model A corresponds the standard self-calibration, while model B corresponds to self-calibration with assembly bias. In this case, model B changes the sample variance but not the mean; thus µ(θ) = µ B (θ) for all θ values, but C(θ) = C B (θ). After linearizing with respect to δθ, the linear equations for δθ read Tr{C −1 C ,i (1 − C −1 C B + j C −1 C ,j C −1 C B δθ j )} + 2 j µ T ,j C −1 µ ,i δθ j = 0 .
After solving the linear equations, we obtain the parameter deviation δθ where is the Fisher matrix of the Gaussian likelihood function. Note that the bias in the recovered parameters is proportional to the difference between models A and B. Note that in analyzing the data generated by model B using model A changes not only the recovered parameters but also their error bars. By performing a similar calculation, the modified Fisher matrix with systematics now reads The error bar for all parameters estimated in model A using the data generated by model B can be recovered by invertingF. However, in the case of counts-in-cells, the likelihood function is not perfectly Gaussian; it is convolution of Poisson and Gaussian (see e.g. Lima & Hu 2004;Hu & Cohn 2006). The modified Fisher matrix thus reads As a second example, we consider the case in which model B changes the mean but not the variance of the data. One example is the effect of modified gravity on the weak lensing shear cross power spectrum (e.g. Huterer & Linder 2007). Here model A is the General Relativity prediction, while model B is the modified gravitational prediction. In this case, µ(θ) = µ B (θ) while C(θ) = C B (θ). The linear equation for δθ j reads Tr{ j C −1 C ,i C −1 C ,j δθ j } − 2( µ B − µ) T C −1 µ ,i + 2 j µ T ,j C −1 µ ,i δθ j = 0 , which is equivalent to Our formalism thus provides a different and generalizable route of obtaining the systematic error.