THE RELATIVE ABUNDANCE OF ISOLATED CLUSTERS AS A PROBE OF DARK ENERGY

The observed rich clusters of galaxies are believed to have formed through the gravitational collapse of the highest local peaks of the initial density field (Bardeen et al. 1986). Although most of the rich clusters belong to the larger-scale superclusters (e.g., Wray et al. 2006; Einasto et al. 2007), there exist a few isolated massive clusters which reside in relatively low-density environments without any close neighbor clusters. The relative abundance of isolated clusters should depend on how strong gravitational clustering is on the largest scale and how frequently the clusters merge into superclusters in the universe. Therefore, it is expected that the relative abundance of isolated massive clusters may be useful to distinguish between different candidate dark energy models.

To predict the relative abundance of isolated clusters in different dark energy models, however, a theoretical framework is first required within which the number density of isolated clusters can be separately counted. The standard excursion set formalism has been conventionally employed to analytically calculate the number density of galaxy clusters as a function of mass, i.e., the mass function of galaxy clusters. Basically, the standard excursion set formalism counts the initial regions whose linearly extrapolated density contrast (δ) exceeds a fixed threshold value (δ_c) on a certain mass scale M, regarding them as the sites which would eventually collapse to form the bound objects of mass M (Press & Schechter 1974; Bond et al. 1991). The density threshold δ_c which is often called the collapse barrier has a constant value of δ_sc = 1.686 if the gravitational collapse process follows spherical dynamics (Gunn & Gott 1972).

Maggiore & Riotto (2010a, 2010b) have for the first time introduced the concept of stochastic collapse barrier to generalize the excursion set formalism. In light of the numerical result of Robertson et al. (2009), Maggiore & Riotto (2010a, 2010b) regarded the collapse barrier, δ_c, as a log-normal variable, and showed that in the high-mass limit the collapse barrier has a spherical average of 〈δ_c〉 = δ_sc = 1.686. Very recently, Corasaniti & Achitouv (2011a, 2011b) have further refined the mass function theory by incorporating ellipsoidal collapse dynamics into the generalized excursion set formalism. They extended the generalized excursion set formalism to the case where the mean collapse barrier, 〈δ_c〉, deviates from the spherical average, δ_sc = 1.686, in the low-mass limit. The collapse barrier higher than the spherical average, δ_c > 1.686, corresponds to the case where the formation of the low-mass halos experience a disturbing tidal effect from the surroundings.

We note here that the Corasaniti–Achitouv (CA) formalism should be applicable not only to the case where the gravitational collapse occurs non-spherically in the low-mass section but also to the case where the massive halos form in isolated low-density environments. Given that the massive cluster-size halos are likely to form in highly overdense regions but hardly in isolated low-density regions, the collapse barrier for the formation of the isolated clusters should be higher than the spherical average. Since the CA formalism allows the average collapse barrier to deviate from the spherical value of 1.686, their formalism may be capable of deriving the mass function of isolated clusters.

The organization of this paper is as follows. In Section 2, we briefly review the CA formalism of the halo mass function. In Section 3, we explain how the relative abundance of the isolated clusters can be evaluated by modifying the CA formalism and present the results of the numerical tests. In Section 4, we show how sensitively the relative abundance of the isolated clusters depends on the dark energy equation of state as well as the other key cosmological parameters. In Section 5, we summarize the results and conclude the work.

The mass function of bound halos, dN/dln M, is defined as the number density of bound objects whose masses belong to the differential mass interval [ln M, ln M + dln M] (Press & Schechter 1974). The classical mass function theory based on the standard excursion set formalism relates dN/dln M to the multiplicity function, f(σ), which gives the number density of the random walks crossing a specified collapse barrier, δ_c, when the pseudo time variable has the value of σ (Bond et al. 1991; Jedamzik 1995):

$$\begin{equation} \frac{dN(M, z)}{d\ln M} = \frac{\bar{\rho }}{M}\frac{d\ln \sigma ^{-1}}{d\ln M}f[\sigma (M,z)]. \end{equation} \tag{ 1 }$$

Here $\bar{\rho }$ is the mean mass density of the universe and the pseudo time variable σ(M, z) is equivalent to the rms density fluctuation of the linear density field smoothed on the mass scale M at redshift z. It scales with the linear growth factor, D(z), as σ(M, z) ≡ D(z)σ(M) with σ(M) ≡ σ(M, z = 0). Throughout Section 2 and Section 3, we assume a flat ΛCDM cosmology (with Ω_m = 0.25, Ω_Λ = 0.75, Ω_b = 0.045, h = 0.7, n_s = 0.95, and σ₈ = 0.8), to be consistent with the WMAP7 cosmology (Komatsu et al. 2011), for which the linear growth factor has the following analytical expression (Lahav et al. 1991):

$$\begin{equation} D(z)\!\propto\! \frac{5}{2}\Omega _{m} [\Omega _{m}(1\,+\,z)^{3}\,+\,\Omega _{\Lambda }]^{1/2}\!\!\int _{z}^{\infty }\!\!dz^{\prime }\! \frac{1+z^{\prime }}{[\Omega _{m}(1+z^{\prime })^{3} + \Omega _{\Lambda }]^{3/2}}. \end{equation} \tag{ 2 }$$

Here D(z) is normalized to satisfy D(z = 0) = 1.

In the original Press–Schechter theory where gravitational collapse is assumed to occur spherically, the collapse barrier has a constant value of δ_sc = 1.686. In the Press–Schechter variants developed afterward to improve the accuracy of the halo mass function by accounting for the ellipsoidal collapse process (e.g., Bond & Myers 1996; Monaco 1997; Audit et al. 1997; Lee & Shandarin 1998; Sheth & Tormen 1999; Jenkins et al. 2001; Warren et al. 2006; Tinker et al. 2008; Robertson et al. 2009), the collapse barrier is described not as a constant but as a function of the mass scale M (or equivalently, a function of the rms density fluctuation, σ).

Recently, Maggiore & Riotto (2010a, 2010b, hereafter MR10) pointed out that not only the constant spherical collapse barrier but also the scale-dependent ellipsoidal collapse barrier is incapable of describing the true nature of the complicated halo formation process and suggested that the collapse barrier δ_c should be treated as a stochastic variable. MR10 derived an analytic expression for the halo mass function by generalizing the excursion set formalism for the case where the collapse barrier is stochastic and the random walk process is non-Markovian. It was shown by MR10 that in the high-mass limit (M ⩾ 10¹⁴ h⁻¹ M_☉) the average of the stochastic collapse barrier equals the constant spherical value, 〈δ_c〉 = δ_sc = 1.686, and its variance is scale-dependent as 〈Δ²D_B〉 = σ²(M)D_B, where D_B is called the diffusion coefficient. The stochasticity of the collapse barrier is caused by the disturbance from the surroundings and the ambiguity in the halo-identification procedure (see Maggiore & Riotto 2010b for a detailed explanation).

In light of the MR10 work, Corasaniti & Achitouv (2011a, 2011b, hereafter CA11) have extended the generalized excursion set formalism to the ellipsoidal collapse case in which the average of δ_c deviates from the spherical value, 1.686, in the low-mass section. The resulting CA11 multiplicity function has two characteristic coefficients: the diffusion coefficient D_B defined as in the MR10 formalism and the drifting average coefficient, β, defined as 〈δ_c〉 ≡ δ_sc + β. The non-zero value of this drifting average coefficient, β, quantifies the degree of the deviation of the average of the collapse barrier from the spherical average 1.686, while the non-zero value of the diffusion coefficient D_B quantifies the degree of the stochasticity of the collapse barrier δ_c.

The CA11 multiplicity function f(σ) was written at second order as

$$\begin{eqnarray} f(\sigma; D_{B},\beta) &\approx& f^{(0)}(\sigma; D_{B}) + f^{(1)}_{\beta =0}(\sigma; D_{B})\nonumber\\ && + f^{(1)}_{\beta }(\sigma; D_{B}) + f^{(1)}_{\beta ^2}(\sigma; D_{B}), \end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray} f^{(0)}(\sigma;D_{B})&=&\frac{\delta _{sc}}{\sigma \sqrt{1+D_B}} \sqrt{\frac{2}{\pi }}\,e^{-\frac{(\delta _{sc}+\beta \sigma ^2)^2}{2\sigma ^2(1+D_B)}}, \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} f^{(1)}_{\beta =0}(\sigma;D_{B})&=&{-}\tilde{\kappa }\frac{\delta _{sc}}{\sigma } \sqrt{\frac{2a}{\pi }}\left[e^{-\frac{a\delta _{sc}^2}{2\sigma ^2}} -\frac{1}{2}\Gamma \left(0,x\right)\right], \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} f^{(1)}_{\beta }(\sigma;D_{B})&=& {-}\beta \,a\,\delta _{sc}\big[f^{(1)}_{\beta =0}(\sigma)+\tilde{\kappa }\, \hbox{Erfc}\left(x\right)\big], \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&f^{(1)}_{\beta ^2}(\sigma;D_{B})=\beta ^{2}a^{2}\delta ^{2}_{sc}\tilde{\kappa } \biggl \lbrace \hbox{Erfc}\left(x\right)\nonumber\\ &&\quad+ \frac{\sigma }{a\delta _{sc}}\sqrt{\frac{a}{2\pi }}\biggl [e^{-\frac{a\delta _{sc}^2}{2\sigma ^2}}\left(\frac{1}{2}-2x\right)+\frac{3}{4}\frac{a\delta _{sc}^2}{\sigma ^2}\Gamma \left(0,x\right)\biggr ]\biggr \rbrace, \end{eqnarray} \tag{ 7 }$$

with $x\equiv (\delta _{sc}/\sigma)\sqrt{a/2}$, a = 1/(1 + D_B), $\tilde{\kappa }=\kappa \,a$, k = 0.475, and incomplete Gamma function Γ(0, x). Here, f⁽⁰⁾(σ; D_B) coincides with the MR10 multiplicity function which has only one parameter D_B. Computing the halo mass functions through Equations (1)–(7) and comparing them with the fitting formula given by Tinker et al. (2008), CA11 have determined the best-fit values of the two coefficients as β = 0.057 and D_B = 0.294. CA11 have also shown how the shape of the halo mass function changes with the values of D_B and β (see Figure 3 in Corasaniti & Achitouv 2011b). According to their results, the variation of the diffusion coefficient D_B alters the high-end slope of f(σ) while the variation of the drifting average coefficient β affects the overall amplitude of f(σ).

Now, we would like to confirm the validity of the CA11 mass function by testing it against the high-resolution MICE simulations (Crocce et al. 2010) which traced the evolution of 2048³ dark matter particles (each having mass of 23.42 × 10¹⁰ h⁻¹ M_☉) on a periodic box with linear size of 3 h⁻¹ Gpc for a ΛCDM cosmology. We utilize the publicly available catalog of the cluster halos that were identified via the standard friends-of-friends (FoF) algorithm (Davis et al. 1985). The catalog provides information on each halo's (comoving) position, peculiar velocity, and FoF mass (calculated as the sum of the masses of all the dark matter particles belonging to each halo) at three different redshifts: z = 0, 0.5, 1. For a detailed description of the MICE simulations and the cluster catalogs, we refer the readers to Crocce et al. (2010).

Binning the logarithmic masses of the cluster halos from the MICE catalog and counting the number densities of the cluster halos belonging to each logarithmic mass bin, we obtain the numerical mass function of all cluster halos, dN_T/dln M, at each redshift. Fitting the CA11 formula (Equations (1)–(7)) to the numerical results from the MICE simulations at each redshift, we have determined the best-fit values of β and D_B with the help of the χ² statistics, the results of which are listed in Table 1.

Note that the diffusion coefficient, D_B, has the same best-fit value of 0.38 at all three redshifts while the best-fit value of the drifting average coefficient, β, increases as the redshift decreases. The result indicates that at earlier epochs the gravitational collapse process is closer to spherical dynamics than at present epoch. Since a bound halo is harder to form at earlier epochs, those high-z halos must correspond to higher peaks in the initial density field. As shown analytically by Bernardeau (1994), the higher a local density peak is, the more spherically its gravitational collapse proceeds.

It is also worth mentioning that our result on the best-fit value of D_B = 0.38 is different from the original value D_B = 0.27 used in the CA11 formalism. This discrepancy must be due to the fact that we use the FoF masses available in the MICE catalog while the spherical overdensity (SO) masses were considered in the original CA11 work.

Figure 1 plots the CA11 mass functions with the best-fit values of β and D_B (solid line) and compares them with the numerical results from the MICE simulations (square dots) at z = 0, 0.5, and 1 in the top left, top middle, and top right panels, respectively. The Jackknife method is employed to calculate the errors: dividing the halos into eight subsamples (each having the same number of the halos) and determining dN_T/dln M separately from each subsample, the errors are calculated as 1σ scatter of dN_T/dln M among the eight Jackknife resamples. Figure 1 also plots the ratio of the analytic mass function to the numerical result as a function of mass at z = 0, 0.5, and 1 in the bottom left, bottom middle, and bottom right panels, respectively. As can be seen, the analytical and numerical mass functions agree with each other quite well at all three redshifts, except in the high-mass section (M > 10¹⁵ h⁻¹ M_☉) where the Jackknife errors are very large.

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Applying the FoF algorithm with the linkage length parameter of b to the MICE cluster catalogs at each redshift, we first find the clusters of clusters (i.e., superclusters) that have more than one neighbor cluster within the FoF linkage length, $b\bar{l}$, where $\bar{l}$ is the mean cluster separation. The isolated clusters are then identified as those clusters that do not belong to any superclusters. Note that the degree of their isolation depends on the value of b: the larger the value of b, the more isolated they are. Here we consider the extreme case in which the isolated clusters experience no disturbance from the surrounding large-scale structures and the difference between their FoF and SO masses is negligible. In this extreme case, the value of the diffusion coefficient, D_B, must vanish since the non-zero value of D_B indicates the presence of disturbance from the surroundings and the existence of the difference between the FoF and SO masses.

The ad hoc value of the linkage length parameter for this extreme case is determined to be b = 0.4 using the following procedures. We first investigate how the best-fit value of D_B changes as the value of b varies from 0.25 to 0.45 in the FoF algorithm applied to the MICE cluster catalogs. For each case of b, we select the isolated clusters from the MICE cluster catalog and obtain the numerical mass function of the isolated clusters. Then, we fit the numerical result to the analytic CA11 formula (Equations (1)–(7)) to determine the best-fit values of D_B and β with the help of the χ² statistics. Figure 2 plots the best-fit value of D_B versus the linkage length parameter b. As can be seen, the best-fit value of D_B drops to zero when the linkage length parameter b reaches 0.4. Table 2 lists the best-fit values of D_B and β for the mass function of the isolated clusters when the ad hoc value of the linkage length parameter is set at b = 0.4.

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Now, we write the total mass function of all cluster halos, dN_T/dln M, as the sum of the mass function of the isolated cluster halos, dN_I/dln M and that of the non-isolated cluster halos, dN_NI/dln M:

$$\begin{equation} \frac{dN_{T}(M, z)}{d\ln M}=\frac{dN_{\rm I}(M, z)}{d\ln M} + \frac{dN_{\rm NI}(M, z)}{d\ln M}. \end{equation} \tag{ 8 }$$

Putting D_B = 0 in the CA11 formula, we express the mass function of the isolated clusters as

$$\begin{equation} \frac{dN_{I}(M, z)}{d\ln M} = \frac{\bar{\rho }}{M}\frac{d\ln \sigma ^{-1}}{d\ln M}f[\sigma;\beta, D_{B}=0]. \end{equation} \tag{ 9 }$$

Here the multiplicity function of the isolated clusters which is characterized by one coefficient β is written at second order as

$$\begin{equation} f(\sigma; D_{B}=0,\beta) \approx f^{(0)}(\sigma)\,+\, f^{(1)}_{\beta =0}(\sigma)\,+\, f^{(1)}_{\beta }(\sigma) + f^{(1)}_{\beta ^2}(\sigma),\nonumber\\ \end{equation} \tag{ 10 }$$

where

$$\begin{eqnarray} f^{(0)}(\sigma)&=&\frac{\delta _{sc}}{\sigma } \sqrt{\frac{2}{\pi }}\,e^{-\frac{(\delta _{sc}+\beta \sigma ^2)^2}{2\sigma ^2}}, \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} f^{(1)}_{\beta =0}(\sigma)&=&{-}\kappa \frac{\delta _{sc}}{\sigma } \sqrt{\frac{2}{\pi }}\left[e^{-\frac{\delta _{sc}^2}{2\sigma ^2}} -\frac{1}{2}\Gamma \left(0,x\right)\right], \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} f^{(1)}_{\beta }(\sigma)&=& {-}\beta \,\delta _{sc}\big[f^{(1)}_{\beta =0}(\sigma)+\kappa \,\hbox{Erfc} \left(x\right)\big], \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} f^{(1)}_{\beta ^2}(\sigma)&=&\beta ^{2}\delta ^{2}_{sc}\kappa \biggl \lbrace \hbox{Erfc}\left(x\right)+ \frac{\sigma }{\delta _{sc}}\sqrt{\frac{1}{2\pi }}\biggl [e^{-\frac{\delta _{sc}^2}{2\sigma ^2}}\left(\frac{1}{2}-2x\right)\nonumber\\ &&+\frac{3}{4}\frac{\delta _{sc}^2}{\sigma ^2}\Gamma \left(0,x\right)\biggr ]\biggr \rbrace. \end{eqnarray} \tag{ 14 }$$

Figure 3 plots the analytical mass functions of the isolated clusters (solid lines) with the best-fit values of β and compared them with the numerical results from the MICE simulations (square dots) at z = 0, 0.5, and 1 in the top left, top middle, and top right panels, respectively. As can be seen, the analytical mass functions of the isolated clusters agree quite well with the numerical results at all three redshifts. The ratios of the analytic CA11 formula to the numerical results are plotted in the bottom panels. Although the agreements between the analytical and the numerical results for the case of the isolated clusters are not so excellent as for the case of all cluster halos (see Figure 1), the ratio at each redshift is still quite close to unity: the discrepancy is less than 20% except in the high-mass limit where the Jackknife errors are very large.

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This result indicates that the CA11 formula with D_B = 0 indeed works fairly well for the determination of the mass function of the isolated clusters. It is worth emphasizing here that this analytical mass function of the isolated clusters (Equation (9)) has only one coefficient β and there is no fitting normalization constant. The amplitude of dN_I/dln M is found to automatically match the numerical result when the best-fit value of β is put into the CA formula.

Figure 4 plots the ratio of the mass function of the isolated clusters to that of the non-isolated clusters at z = 0, 0.5, and 1 in the left, middle, and right panels, respectively. The errors are calculated as 1σ scatter of the ratio among the eight Jackknife resamples. As can be seen, the numerical result (square dots) agrees fairly well with the analytic prediction (solid line) based on the CA11 formula at each redshift. We believe that it would reduce the existing discrepancy between the analytical and numerical results even more to include the higher order terms in the calculation of the multiplicity function, f(σ; D_B, β). Throughout this paper, however, we consider only the second-order approximation of f(σ; D_B, β). Note that the ratio drops with mass more rapidly at higher redshifts, indicating that at high redshifts most of the very massive clusters have stronger tendency to form in the highly overdense regions than at present epoch.

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In a ΛCDM cosmology the dark energy equation of state (defined as the ratio of the pressure density to the energy density) is a perfect constant given as w ≡ P_Λ/ρ_Λ = −1, whereas in dynamic dark energy models the value of w may vary with time and can deviate from −1 at z = 0 (Wang & Steinhardt 1998). This difference in the dark energy equation of state results in the different functional shape of D(z) (Chevallier & Polarski 2001; Linder 2003; Basilakos 2003; Percival 2005).

The evolution of the abundance of galaxy clusters has been regarded as one of the most powerful probes of w(z) (e.g., Wang & Steinhardt 1998; Haiman et al. 2001; Weller et al. 2002; Basilakos et al. 2010). The most serious systematics in constraining w(z) with the evolution of the abundance of the galaxy clusters may come from the mass assignment of the high-z clusters. Several statistical methods have been suggested so far to overcome the systematics but none of them have yet to be fully satisfactory (e.g., Rykoff et al. 2008; Cunha & Evrard 2010; Stanek et al. 2010, and references therein. See also Allen et al. (2011) for the latest review). We suggest here that the relative abundance of the galaxy clusters at a given epoch provides a complimentary probe of the dark energy equation of state.

The relative abundance of the isolated clusters is defined as the ratio of the cumulative mass function of the isolated clusters to that of the non-isolated clusters:

$$\begin{eqnarray} \xi _{\rm I}(M_{c};z)&\equiv&\frac{\left[\int _{M_{c}}^{\infty }d\ln M \frac{dN_{\rm I}(M, z)}{d\ln M}\right]}{\left[\int _{M_{c}}^{\infty }d\ln M \frac{dN_{\rm NI}(M, z)}{d\ln M}\right]}, \end{eqnarray} \tag{ 15 }$$

where the mass function of the non-isolated cluster dN_NI/dln M is obtained by dN_T/dln M − dN_I/dln M. To demonstrate how the relative abundance of the isolated clusters ξ_I changes with the dark energy equation state w, we consider the specific case where the dark energy equation of state is redshift-dependent as w(z) = w₀ + w₁z/(1 + z) (Chevallier & Polarski 2001; Linder 2003). Basilakos (2003) and Percival (2005) showed that for these dynamic dark energy models the linear growth factor can be approximated as

$$\begin{equation} D(z)=\frac{5\Omega _{mz}}{2(z+1)} \left[\Omega ^{\alpha }_{mz}-\Omega _{Qz} + \left(1+\frac{\Omega _{mz}}{2}\right) \left(1+{\cal A}\Omega _{Qz}\right)\right]^{-1}, \end{equation} \tag{ 16 }$$

where

$$\begin{eqnarray} \alpha &=&\frac{3}{5-2/(1-w)}+\frac{3}{125} \frac{(1-w)(1-3w/2)}{(1-6w/5)^{3}}[1-\Omega _{mz}],\nonumber\\ \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} {\cal A}&=&{-}\frac{0.28}{w+0.08}-0.3. \end{eqnarray} \tag{ 18 }$$

Here Ω_mz and Ω_Qz represent the matter density and dark energy density parameters at z, respectively, related to their present values, Ω_m and Ω_Q, as

$$\begin{equation} \Omega _{mz}=\frac{\Omega _{m}(1+z)^{3}}{E^{2}(z)}, \quad \Omega _{Qz}=\frac{\Omega _{Q}}{E^{2}(z)(1+z)^{f(z)}}. \end{equation} \tag{ 19 }$$

where

$$\begin{eqnarray} E^{2}(z)&=&\Omega _{m}(1+z)^{3}+\Omega _{Q}(1+z)^{-f(z)}, \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} f(z) &=& {-}3(1+w_{0}){-}\frac{3w_{1}}{2\ln (1+z)}. \end{eqnarray} \tag{ 21 }$$

We first calculate ξ_I for the ΛCDM case and then repeat the calculation of ξ_I for the four different dynamic dark energy models at z = 0.5. Figure 5 plots the relative abundances of the isolated clusters for the dynamic dark energy models and compares them with the ΛCDM case. The errors for the ΛCDM case are calculated as 1σ scatter of ξ_I among the eight Jackknife resamples from the MICE cluster catalog. As can be seen, the five different dark energy models predict different relative abundances of the isolated clusters. For the case of w₁ > 0 (w₁ < 0), the relative abundance of the isolated clusters has lower (higher) amplitude than for the ΛCDM case. This can be explained by the following logic. In a dark energy model with w₁ > 0 (w₁ < 0) the largest-scale powers are higher (lower) than in the ΛCDM case and thus boost the merging of the isolated clusters into the superclusters, reducing the relative abundance of the isolated clusters.

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Furthermore, the differences in ξ_I between each dynamic dark energy case and the ΛCDM case are larger than the statistical errors calculated through the Jackknife resampling. If a future cluster survey can find as many clusters as the MICE simulations, the sensitivity shown in Figure 5 suggests that the relative abundance of the isolated clusters will be a powerful probe of the dark energy equation of state in practice.

We examine the degeneracy between w₀ and w₁. Varying the value of w₀ from −1 to −0.9 and the value of w₁ from −0.6 to 0.6, we repeatedly calculate ξ_I at z = 0.5, setting the cutoff mass scale at M_c = 3.35 × 10¹³ h⁻¹ M_☉ (the lowest cluster mass in the MICE catalog). The contour curves of ξ_I in the w₀–w₁ plane are plotted in Figure 6. When w₀ is fixed, the relative abundance of the isolated clusters decreases as the value of w₁ increases, which is consistent with Figure 5.

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We also examine the degeneracy between Ω_m and w₁ and between σ₈ and w₁ at z = 0.5, which are plotted in the left and right panels of Figure 7, respectively. For these plots the value of w₀ is set at −1. As can be seen, when w₁ is fixed, the relative abundance of the isolated clusters, ξ_I, increases as the value of Ω_m increases. This implies that if there is stronger gravitational effect due to the larger amount of dark matter, the formation of a cluster can occur more easily even in isolated low-density environments. For a given value of ξ_I, w₁ increases with Ω_m. A similar trend is found in the σ₈–w₁ degeneracy curves. When w₁ is fixed, ξ_I increases as the value of σ₈ increases. The overall high density powers make it less hard for a cluster to form in isolated low-density environments.

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It is worth mentioning here that the degeneracy trends shown in Figures 6 and 7 are different from those obtained from the total cluster abundance (see Wang & Steinhardt 1998; Weller et al. 2002). The increase of w₁ has the same effect on the cluster abundance as the increase of Ω_m and σ₈: the cluster abundance increases as the three parameters increase. In contrast, when the relative abundance of isolated clusters is used, the increase of w₁ has the opposite effect: the relative abundance of the isolated clusters increases as w₁ decreases and as Ω_m and σ₈ increase. This result implies that the relative abundance of the isolated clusters may be helpful to break the degeneracy between w₁ and the other key cosmological parameters.

We have derived the mass function of isolated clusters in the frame of the recently developed CA formalism, assuming that for the case of isolated clusters there is no disturbance from the surroundings and no ambiguity in the mass determination. The numerical results from the MICE simulations have been used to determine empirically the value of the drifting average coefficient, β, which quantifies the degree of the deviation of the collapse barrier for the formation of isolated clusters from the standard average value.

Extrapolating the validity of the CA formalism to the dynamic dark energy models and using our analytic result on the mass function of the isolated clusters, we have shown that the relative abundance of isolated clusters, ξ_I, defined as the ratio of the cumulative mass function of the isolated clusters to that of the non-isolated clusters at a given epoch, depends sensitively on the dark energy equation of state. This result proves our theoretical concept that the relative abundance of isolated clusters is in principle a powerful probe of dark energy.

Yet, several additional works have to be done before constraining the dark energy equation of state with ξ_I. First, it will be necessary to derive the functional form of the drifting average coefficient, β(z), for the isolated and non-isolated cases, separately. Second, it has to be confirmed if the CA formalism indeed works not only in a ΛCDM cosmology but also in dynamic dark energy models and how the drifting average coefficient changes with the background cosmology. Third, it has to be examined whether or not the higher order perturbation terms in the CA formalism have any non-negligible effect on the relative abundance of the isolated clusters. We plan to conduct these works and wish to report the results elsewhere in the future.

We would also like to mention that our analytic model for the mass function of isolated clusters will be useful not only as a probe of dark energy equation of state but also for predicting more accurately cluster-related statistics since isolated massive clusters may differ from non-isolated ones in their physical properties such as average shape, dynamical state, gas entropy profileetc (E. Komatsu 2011, private communication).

I thank a referee for helpful suggestions and E. Komatsu for stimulating discussion. I acknowledge the use of data from the MICE simulations that are publicly available at http://www.ice.cat/mice. I also acknowledge the financial support from the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST, No. 2011-0017895) and from the National Research Foundation of Korea to the Center for Galaxy Evolution Research.