GALAXY CLUSTERS AS A PROBE OF EARLY DARK ENERGY

Ujjaini Alam; Zarija Lukić; Suman Bhattacharya

doi:10.1088/0004-637X/727/2/87

1. INTRODUCTION

Over the last decade, observational evidence has mounted in favor of dark energy, the mysterious component which dominates the energy content of the universe at present, and causes the expansion of the universe to accelerate (Kowalski et al. 2008; Komatsu et al. 2010). The nature of dark energy is one of the most tantalizing mysteries of present-day cosmology. The simplest model for dark energy, the cosmological constant, fits the current data well (Kowalski et al. 2008), however, there are no strong constraints on the time evolution of dark energy at present. Thus, evolving models of dark energy remain as alternative candidates for dark energy. Many non-cosmological constant models have been suggested for dark energy, including scalar field quintessence models, modifications of the Einstein framework of gravity, etc. (see reviews Sahni & Starobinsky 2000; Carroll 2001; Peebles & Ratra 2003; Copeland et al. 2006; Nojiri & Odintsov 2007; Sahni & Starobinsky 2006; Frieman et al. 2008; and references therein), but as of now there is no clear consensus on the nature on dark energy. An interesting class of models which have been suggested in the literature is early dark energy (EDE) models, in which the early universe contained a substantial amount of dark energy. Geometric probes of dark energy put strong constraints on the present-day nature of dark energy, and these constraints are expected to improve with future surveys. However, very little is known as to the nature of dark energy at early times due to the paucity of data beyond redshifts of few. It has thus been hypothesized that even if dark energy at present behaves like the cosmological constant, in the past it could have had completely different behavior, leading to the idea of the EDE models. Different facets of these models have been studied in recent works (Dodelson et al. 2000; Skordis & Albrecht 2002; Doran & Robbers 2006; and references therein) and have been analyzed with respect to observations extensively in recent times in Linder & Robbers (2008), Francis et al. (2008), Grossi & Springel (2009), Fedeli et al. (2009), Xia & Viel (2009), Jennings et al. (2010), Alam (2010), & de Putter et al. (2010). Current data place some constraints on these models but do not rule them out. In this work, we use a parameterization of the equation of state of dark energy to study the possibility of constraining EDE models using future large-scale structure surveys.

Ground and space-based telescopes targeting clusters in microwave (Atacama Cosmology Telescope (ACT; Hincks et al. 2010), the South Pole Telescope (SPT; Staniszewski et al. 2009), and Planck³) have begun operation or will return data shortly. Also current (Ebeling et al. 2010) and future missions have been planned for detecting clusters in X-ray waveband (Predehl et al. 2007). It has been recognized for some time now that galaxy cluster surveys in X-ray or microwave via Sunyaev–Zel'dovich (SZ) effect could be precision probes of cosmological parameters, in particular the dark energy density and its equation-of-state parameter (Haiman et al. 2001). This is because redshift distribution of massive clusters is exponentially sensitive to the growth of structure history of the universe, which in turn, bears the signature of dark energy. The presence of EDE reduces structure formation, consequently the cluster abundances reduce. Thus, cluster counts can provide a smoking gun probe for detecting EDE. For example, Mantz et al. (2010) have used a flux limited X-ray cluster data to constrain the EDE component assuming the redshift of transition between 0 and 1. The current generation of SZ surveys will also measure the power spectrum of the cosmic microwave background (CMB) with an unprecedented accuracy down to scales of 1 arcmin. The SZ surveys have started measuring the SZ power spectrum which is the dominant signal at scales of few arcminutes (1; Fowler et al. 2010). The amplitude of the SZ power spectrum is proportional to the total number of objects that have formed and hence is extremely sensitive to the amount of dark energy present at early epoch.

The paper is organized as follows. In Section 2 we explain the EDE formalism and model used for this analysis. Section 3 expounds the nature of X-ray and microwave observations and mass-observable scaling relations. In Section 4 we show predictions from current and future observations, comparing our EDE models to a fiducial ΛCDM cosmology. Finally, Section 5 is devoted to conclusions and discussion.

2. EARLY DARK ENERGY

Dark energy perturbations for dynamic dark energy models have been studied in a number of works, usually under the formalism of a minimally coupled scalar field (see Ma et al. 1999; Hwang & Noh 2001; Hu 2002; Malquarti & Liddle 2002; Weller & Lewis 2003; Bean & Dore 2004; Dutta & Maor 2007; Mota et al. 2007; Novosyadlyj & Sergijenko 2008; Jassal 2009, and references therein). In this work, we follow the formalism of Weller & Lewis (2003). First-order perturbations in a homogeneous and isotropic large-scale universe described by the Friedman–Lemaitre–Robertson–Walker metric take the form

$\begin{equation} ds^2 = a^2(\eta) [(1+2\Psi ({\bf x}, \eta)) d\eta ^2 - (1+2\Phi ({\bf x}, \eta)) \delta _{\alpha \beta }dx^{\alpha } dx^{\beta }], \end{equation} \tag{ 1 }$

where η is the conformal time, x is the length element, a(η) is the scale factor, and Φ, Ψ are the Bardeen potentials. If proper isotropy of the medium is zero, then Φ = −Ψ.

Along with the matter and radiation components, we consider dark energy to be an additional fluid component, so that the dark energy perturbations are characterized by an equation of state and an adiabatic sound speed—-

$\begin{equation} w_{\rm DE} = \frac{p_{{\rm DE}}}{\rho _{{\rm DE}}}, \end{equation} \tag{ 2 }$

$\begin{equation} c_{\rm a,DE}^2 = \frac{\dot{p}_{{\rm DE}}}{\dot{\rho }_{{\rm DE}}}. \end{equation} \tag{ 3 }$

Defining the frame-invariant quantity c²_s,i (the fluid sound speed in the frame comoving with the fluid), the evolution equations for a fluid with equation-of-state w_i = p_i/ρ_i, and adiabatic speed of sound $c_{a,i}^2 = \dot{p}_i/\dot{\rho }_i$ , can be written as (prime denotes derivative with respect to η)

$\begin{eqnarray} \delta _i^{\prime } &=& -3 {\cal H}\big(c_{s,i}^2-w_i\big) \delta _i - 9 {\cal H}^2 \big(c_{s,i}^2-c_{a,i}^2\big) (1+ w_i) \frac{v_i}{k} \nonumber \\ && -(1+ w_i) k v_i - 3(1+ w_i) \Psi ^{\prime }, \end{eqnarray} \tag{ 4 }$

$\begin{eqnarray} v_i^{\prime } &=& - {\cal H}\big(1-3 c_{s,i}^2\big) v_i + \frac{k c_{s,i}^2 \delta _i }{(1 + w_i)} - kA, \end{eqnarray} \tag{ 5 }$

where ${\cal H}= a^{\prime }/a$ is the conformal Hubble parameter and A is the acceleration (A = 0 in the synchronous gauge and A = −Ψ in the Newtonian gauge). Adiabatic initial conditions are considered. For the matter component, w_m = c²_a = c²_s = 0. For the dark energy component, a fluid with varying w_DE ⩾ −1 has c²_a,DE = w_DE − [dw_DE/d(ln a)]/3(1 + w_DE). For scalar field like dark energy models, c²_s,DE = 1. For a more general class of models, such as k-essence, c²_s,DE could be variable as well.

If we consider dark energy without perturbations, the quantities δ_DE, δ'_DE = 0, and the matter density contrast is given by

$\begin{equation} \delta _m^{\prime \prime } - {\cal H}\delta _m^{\prime } -4 \pi G \rho _m \delta _m = 0, \end{equation} \tag{ 6 }$

thus the dark energy component appears only in the damping term, so that a non-negligible amount of dark energy would lead simply to a suppression of clustering of matter at large scales. Not taking into account the dark energy perturbations can lead to gauge dependent results, as shown in Park et al. (2009), in this paper we have used the commonly used comoving gauge. If dark energy perturbations are included self-consistently, the results are gauge independent.

To study EDE models under this formulation, we consider a w-parameterization which may represent a large class of varying dark energy models (Corasaniti et al. 2003)

$\begin{equation} w(a) = w_0+(w_m-w_0)\frac{1+e^{a_t/\Delta _t}}{1+e^{-(a-a_t)/\Delta _t}}\frac{1-e^{-(a-1)/\Delta _t}}{1-e^{1/\Delta _t}}, \end{equation} \tag{ 7 }$

where w₀ is the equation of state of dark energy today, w_m is the equation of state in the matter dominated era, a_t is the scale factor at which the transition between w₀ and w_m takes place, and Δ_t is the width of the transition. For studying EDE models with this parameterization, we choose w_m > − 0.1, to ensure the presence of adequate amount of dark energy at early times.

Current data put impressive constraints on the values of these parameters (see Alam 2010), however, many interesting EDE models still fall in the acceptable range. We choose two such EDE models in order to study the possibility of constraining these models further using future observations. The models chosen are

1.
Model 1: w₀ = −1.0, w_m = −0.05, a_t = 0.17, Δ_t = 0.17
2.
Model 2: w₀ = −0.9, w_m = −0.05, a_t = 0.1, Δ_t = 0.1.

Model 1 behaves like ΛCDM at present, while Model 2 has a higher value of the equation of state at present. We compare the results for these models with those for a ΛCDM model with identical values for the non-dark energy parameters. The non-dark energy parameters are chosen from the WMAP7 CMB+BAO+HST best fit, e.g., Ω_bh² = 0.0226, Ω_ch² = 0.1123, h = 0.704, andn_s = 0.963, as well as other parameters such as the scalar amplitude A_s and the reionization optical depth τ (Komatsu et al. 2010). We modified the publicly available COSMOMC code (Lewis & Bridle 2002) for generating the transfer functions, and therefore the power spectrum at different redshifts for these models.

Figure 1 shows the behavior of the equation of state of dark energy, and the matter power spectrum at redshift z = 0 for these two models. For comparison, we also plot the results for the matter power spectrum when dark energy perturbations are not considered. We see that when the perturbations are not considered, the EDE matter power spectrum, although there is a suppression the matter power spectrum, this effect is negligible, and for both models the matter power spectrum is very close to that for ΛCDM. However, when the perturbations are considered, the matter power spectrum is significantly different from ΛCDM, resulting in a σ₈ = 0.74 for EDE1, and a σ₈ = 0.76 for EDE2, whereas the ΛCDM has a σ₈ = 0.81. This behavior is typical of EDE models with rapid and large transition of the equation of state. This translates to a rapid change in the dark energy perturbations at high scales (low k), thus to enhancement in the transfer function at low k. When normalized to the CMB scale, this effect shows up as an apparent suppression of power at high k, and therefore a low σ₈. The effect is stronger in EDE1 (i.e., lower σ₈) because the difference between the equation of state at present and in the matter dominated era in this case is larger. Further details on how the change in equation of state affects the dark energy perturbations can be found in Paper I (Alam 2010).

3. OBSERVATIONS

Paper I (Alam 2010) explores the effects of EDE on the CMB power spectrum and other current observations; here we focus on forthcoming large-scale structure probes. We consider the influence of EDE on cluster counts and SZ power spectrum.

3.1. Cluster Counts

Redshift evolution of the cluster abundance provides a valuable insight into global dynamics of the universe. This abundance is sensitive to both the expansion and growth history of the universe, and as such can be used to constrain standard cosmological parameters like σ₈ or Ω_0m (see, e.g., Haiman et al. 2001), but also to explore possible mechanisms for observed acceleration, like modifying the law of gravity.

For a spatially flat cosmologies considered here (Ω_κ ≡ 0), the all-sky number of objects more massive than M_min in a redshift bin is

$\begin{equation} N(z) = \int _{M_{{\rm min}}}^{\infty } \int _{z_1}^{z_2} \frac{4 \pi r^2(z) dr(z)}{dz} \frac{dn(M,z)}{dM} dz \, dM \, \end{equation} \tag{ 8 }$

where the first term is the comoving volume and the second is the differential mass function.

From numerical simulations, we have a good quantification of the halo mass function. Although its precision is still not at the level needed to make it insignificant for future cosmological parameter studies (Cunha & Evrard 2010; Wu et al. 2010), that will not be relevant here as we want to explore the relative difference between our EDE models and the fiducial ΛCDM cosmology. The important point, on which we rely here, is that the mass function can be presented in a universal form, where the cosmology dependence comes from the linear power spectrum, and the linear growth factor. At a 20% level (often better than that), it is indeed proven to be the case, for many different cosmological models (Jenkins et al. 2001; Linder & Jenkins 2003; Jennings et al. 2010; Bhattacharya et al. 2010) and for a wide range of redshifts (Heitmann et al. 2006; Lukić et al. 2007; Reed et al. 2007). Since we are interested in modeling X-ray and SZ observations, we will use the mass function of halos defined as spheres enclosing a given overdensity, defined with respect to the critical density for the closure, ρ_c (for the analysis of different halo definitions; see, e.g., White 2002; Lukić et al. 2009).

One way of detecting groups and clusters of galaxies is via weak lensing maps (Marian & Bernstein 2006); this approach is appealing as masses are probed directly there, but on the downside the method can probe only the most massive systems, and only within limited redshifts (as galaxy shapes are very hard to measure beyond z ∼ 1). Promising alternatives to the weak lensing are mass measures through SZ effect (e.g., Carlstrom et al. 2002), X-ray emission (e.g., Kravtsov et al. 2006), and galaxy richness (High et al. 2010). For these indirect mass measurements, one first has to connect cluster mass to a relevant observable for a given survey, such as the X-ray flux (Mantz et al. 2010) or integrated Compton y-parameter (Vanderlinde et al. 2010; Sehgal et al 2010). In this paper, we will focus on X-ray and SZ surveys, and in the following we explain scaling relations we use.

In the self-similar theory (Kaiser 1986, 1991), the temperature of the intracluster medium (ICM) scales with gravitational potential (T ∝ M/R), and enclosed mass is M ∝ R³. This self-similar scaling was confirmed in hydrodynamical simulations (Mathiesen & Evrard 2001; Borgani et al. 2004; Kravtsov et al. 2006), as well as in observations (Vikhlinin et al. 2009). In the following, we will use M–T relation from Mathiesen & Evrard (2001):

$\begin{equation} \frac{M_{200}}{M_{\odot }h^{-1}} = 10^{15} \left(\frac{kT}{4.88\; {\rm keV}} \right)^{3/2} E(z)^{-1}\; \ \ \ E(z) \equiv H(z)/H_0. \end{equation} \tag{ 9 }$

We use the above relation, as it relates M₂₀₀ to the bolometric luminosity. While X-ray scaling relations are much tighter for M₅₀₀ (Kravtsov et al. 2006), here we want to use the same mass for both X-ray and SZ estimates, and whether it is the best observational approach is not of concern here.

Self-similar theory fails to correctly predict the T–L relation (Markevitch 1998; Allen & Fabian 1998), and thus the mass–luminosity relation is inaccurate as well. The departure from self-similarity is due to the excess entropy in cluster cores which prevents gas from being compressed to very high densities (Ponman et al. 1999; Finoguenov et al. 2002). Therefore, in the following we will use the M–L relation from Bartelmann & White (2003):

$\begin{equation} L = 1.097 \times 10^{45} \left[ \frac{M_{200}}{10^{15}\,M_{\odot }h^{-1}} E(z) \right]^{1.554} {\rm erg\; s^{-1}}. \end{equation} \tag{ 10 }$

X-ray observations are sensitive in a given energy band, whose flux is

$\begin{equation} S_b = \frac{L_X f_b}{4\pi d_L(z)^2}, \end{equation} \tag{ 11 }$

where f_b is the band correction and d_L(z) is luminosity distance to a cluster:

$\begin{equation} d_L(z) = \frac{c(1+z)}{H_0} \int _0^z \frac{dz^{\prime }}{E(z^{\prime })}. \end{equation} \tag{ 12 }$

Emission from the ICM is predominantly thermal bremsstrah- lung, but for T < 2keV line emission from metals (clusters have metallicity ∼0.3 Z_☉) becomes non-negligible (Sarazin 1986). Thus, to calculate f_b we use the XSPEC package (Arnaud 1996) and we assume the Raymond–Smith (1977) plasma model for cluster emission. As the band correction depends on the plasma temperature, we solve iteratively for minimum mass and corresponding temperature using XSPEC and Equation (9). Galactic absorption is modeled with a constant column density of n_H = 10²¹cm^-2, roughly corresponding to Galactic mid-latitudes. In observational analysis one would use appropriate n_H for the line of sight to each cluster, and column density will generally be smaller than our number for high latitudes, and larger for low latitudes.

For the SZ observations, one has to relate the Compton y-parameter (integrated over the solid angle) to the mass of an object. Furthermore, it is necessary to renormalize such a relation to the directly observable SZ flux. In Fedeli et al. (2009), this was done using the Sehgal et al. (2007) Y₂₀₀–M₂₀₀ relation, and the assumption that SZ signal outside cluster virial radius can safely be neglected, resulting in the relation:

$\begin{equation} S_{\nu } = \frac{6.763 \times 10^7\; {\rm mJy}}{(d_A(z)/1\;{\rm Mpc})^2} \left(\frac{M_{200}}{10^{15}M_{\odot }} \right)^{1.876} | f(\nu) | E^{2/3}(z). \end{equation} \tag{ 13 }$

Here, d_A is the angular diameter distance to the object, and f(ν) is the spectral signature of the thermal SZ effect:

$\begin{equation} f(\nu) = \frac{x^4 e^x}{(e^x-1)^2} \left[ x \frac{e^x+1}{e^x-1} - 4 \right]; \ \ \ x \equiv \frac{h\nu }{kT_{\gamma }}\, \end{equation} \tag{ 14 }$

and T_γ is the CMB temperature.

3.2. Sunyaev–Zeldovich Power Spectrum

The current generation of CMB experiments will measure the power spectrum of the CMB with an unprecedented accuracy down to scales of 1 arcmin. While the primary CMB fluctuations dominate the power spectrum at a degree scale, at scales of a few arcminutes the secondary fluctuations arising from the SZ effect and lensing of the CMB become the dominant signal.

Predictions for the SZ power spectrum amplitude C_l,SZ (henceforth C_l) can be made using the halo model and estimates of the radial pressure profile of intracluster gas. Assuming that the cluster gas resides in hydrostatic equilibrium in the potential well of the host dark matter halo, Komatsu & Seljak (2002) demonstrated that the ensemble-averaged power spectrum amplitude C_l has an extremely sensitive dependence on σ₈, where C_l ∝ σ⁷₈(Ω_bh²)². The kinetic SZ power spectrum contributes roughly 10% to the total SZ angular power spectrum and hence we consider only the thermal component of the SZ power spectrum (tSZ hereafter) here. The cosmological information of the tSZ power spectrum comes from the total number of halos of mass M ⩾ 10¹³ M_☉h⁻¹ in the survey area. Thus, the SZ power spectrum contains a significant contribution from low-mass clusters and group-mass objects. Note that measurements of the mass function usually probe only massive halos (M ⩾ 10¹⁴ M_☉) and not group size halos.

The SZ power spectrum can be calculated by simply summing up the squared Fourier-space SZ profiles, $\tilde{y}(M,z,\ell)$ of all clusters:

$\begin{equation} C_{\ell } = g_\nu ^2 \int dz {dV \over dz } \int d \ln M {dn(M,z) \over d \ln M} \tilde{y}(M,z,\ell)^2, \end{equation} \tag{ 15 }$

where V(z) is the comoving volume per steradian, n(M, z) is the number density of objects of mass M at redshift z, and g_ν is the frequency factor of the SZ effect. In this study, we show the power spectrum at 150 GHz, where g_ν = −1. Note that whilst this calculation assumes that clusters are Poisson distributed, Shaw et al. (2009) have shown that including halo clustering modifies the power spectrum (compared to the Poisson case) by less than 1%.

The number density of halos n(M, z) can be calculated for a ΛCDM cosmology using the fitting functions provided by Jenkins et al. (2001) and more recently (Tinker et al. 2008). For the SZ profiles (also mass and redshift dependent), previous studies have frequently used either a simple β-model or the hydrostatic model of Komatsu & Seljak (2002). Note that both models simply assume that the gas resides in hydrostatic equilibrium in the potential well of a Navarro–Frenk–White-like halo (and is isothermal in the case of the β-model). Neither model accounts for the fraction of hot gas that will have cooled and been converted into stars or any non-thermal energy input into the ICM. As shown in White et al. (2002), Bhattacharya et al. (2008), and Battaglia et al. (2010), these effects change the SZ power spectrum by 10%–30%. One possible way to include the uncertainty in gas physics is the semi-analytic approach of modeling gas physics (Bode et al. 2007; 2009; Shaw et al. 2010) and calibration of the parameters using X-ray observations of individual clusters. The other source of uncertainty is the non-Gaussian nature of the SZ power spectrum (Shaw et al. 2009). All these effects need to be taken into account for precision modeling of the power spectrum. This will be essential to harness the full merit of the upcoming data sets. In the current study, we choose to use the model by Komatsu & Seljak (2002) to assess the impact of dark energy on the SZ power spectrum.

4. RESULTS

4.1. Cluster Counts

We consider two observational campaigns: the upcoming X-ray survey eROSITA and the ongoing South Pole SZ survey. The first one, due in 2012, will cover almost half a sky (f_sky ≈ 0.49), and have a flux limit of F_min = 3.3 × 10⁻¹⁴ erg s^-1cm⁻² in the energy band [0.5, 5.0] keV. SPT will scan 4000 deg² (f_sky ≈ 0.1) with a limit of S_min ≈ 5 mJy at frequency ν₀ = 150 Hz. Using the mass-observable scaling relations described in Section 3.1 we can find the minimum detectable mass for each survey, as well as the redshift volume element being observed (Figures 2 and 3). For the theoretical prediction of cluster abundance, we use the Tinker et al. (2008) mass function for the overdensity of 200ρ_c (Δ ≈ 778 in their notation), and we use their fitting formulas for the parameters of f(σ) as the function of log Δ. The total number density of objects with masses above the given threshold as predicted by Tinker et al. (2008) is shown in the lower left panels of Figures 2 and 3, for eROSITA and SPT survey, respectively.

**Figure 2.** eROSITA survey, from upper left panel in clockwise direction: minimal detectable mass (dashed line shows instrument capability and solid line shows minimal mass for galaxy clusters only with mass threshold of 10¹⁴ M_☉), luminosity distance, comoving volume element covered by the survey, and comoving cluster abundance as a function of redshift.
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**Figure 3.** South Pole SZ survey, from upper left panel in clockwise direction: minimal detectable mass (dashed line shows instrument capability—same as in Figure 2), angular diameter distance, comoving volume element covered by the survey, and comoving cluster abundance as a function of redshift.
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Figures 4 and 5 show the expected redshift distribution of galaxy clusters for the cosmologies considered here, where we have taken redshift bins of Δz = 0.1. As expected, the SZ survey can detect a meaningful number of clusters up to higher redshift, but has fewer objects at lower redshifts, due to the smaller sky coverage. The shaded area shows a Poisson error bar plus a systematic 10% uncertainty which describes our confidence in the universal formula of the mass function. The dashed lines show expectations for EDE models if perturbations would be neglected. We see that the presence of perturbations significantly affects the growth of structure in the universe, effectively reducing σ₈. Since the amount of this reduction is redshift dependent, it is not possible to mimic it with ΛCDM cosmology with different σ₈ (i.e., it is possible to mimic it at a particular redshift, but not at a redshift range). Most importantly, we see that perturbations cannot be neglected when considering predictions from dynamical EDE models.

**Figure 4.** Expected redshift distribution of galaxy clusters for eROSITA Telescope. In black is our fiducial ΛCDM model, red and blue are EDE1 and EDE2, respectively, and in dashed lines we show results of EDE models when perturbations are neglected. Shaded regions are statistical Poisson errors plus 10% systematic error coming from non-universality of the mass function.
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**Figure 5.** Same as Figure 4, but for the South Pole Telescope.
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Finally, Figures 4 and 5 showcase that future large-scale structure results will be able to constrain EDE models significantly better than is currently the case. Both the EDE1 and EDE2 models, which are (1) indistinguishable from ΛCDM at present (EDE1), or (2) at the upper limit of w₀ (EDE2), are distinguishable from ΛCDM cosmology with high significance (2σ or more). Constraining power is larger on the EDE1 model which behaves the same as ΛCDM today but has a later transition from EDE-like behavior (at redshift z ≃ 5), as compared to the EDE2 model which is different from ΛCDM today but has a much earlier transition from EDE-like behavior (at redshift z = 9). Thus, the cluster counts may constrain not only the current equation of state of dark energy, but also the redshift at which the transition from EDE behavior occurs.

4.2. SZ Power Spectrum

In this section, we show the impact of the dark energy perturbation on the SZ power spectrum. The presence of perturbations in the dark energy changes the transfer function, volume factor, and the growth function of the universe, consequently the abundance of the halos changes. Since the SZ power spectrum is proportional to the total abundance of the halos that have formed in the universe and volume of the universe, it depends strongly on the perturbations of dark energy. The impact of different EDE models on the SZ power spectrum is shown in Figure 6. To assess the merit of the upcoming data sets to constrain EDE, we assume two fiducial surveys.

1.
Planck-like full-sky survey. We assume 75% of the full-sky coverage to exclude contaminations due to galactic foregrounds, 7' resolution and 7 μK per pixel as the sensitivity.
2.
ACT/SPT-like higher resolution surveys. We assume a coverage of 4000 deg², 1.2' angular resolution, and a sensitivity of 20 μK per pixel.

**Figure 6.** Impact of dark energy perturbations on the SZ power spectrum. Different early dark energy models are explained in Section 2. The error bars represent a Planck-like full-sky survey and ACT/SPT-like higher resolution survey. The error bars include cosmic variance including the non-Gaussian contribution from the SZ trispectrum and the resolution of the surveys. We assume perfect removal of astrophysical contaminations from infrared and radio galaxies and primary CMB.
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The error on the SZ power spectrum for closely separated bins of size Δl in l space can be written as

$\begin{eqnarray} \sigma ^2(C_l) &=& f_{{\rm sky}}^{-1} \left[ \frac{2 C_l^2}{(2l+1)\Delta l} + \frac{2}{(2l+1)\Delta l}\right. \nonumber \\ &&\times \left. \left(f_{{\rm sky}}w^{-1}\exp \left[\frac{l^2\theta _{{\rm fwhm}}^2}{8\ln 2}\right]\right)^2 + \frac{T_{ll}}{4\pi } \right], \end{eqnarray} \tag{ 16 }$

where f_sky is the fraction of the sky covered by the assumed surveys, w⁻¹ = [σ_pixθ_fwhm/T_CMB]², σ_pix is the noise per pixel, θ_fwhm is the resolution at full width at half-maximum assuming a Gaussian beam, and T_CMB is the CMB temperature (Jungman et al. 1996).

The trispectrum T_ll (Poisson term) is given by

$\begin{equation} C_{\ell } = g_\nu ^4\int dz {dV \over dz } \int d \ln M {dn(M,z) \over d \ln M}\tilde{y}(M,z,\ell)^4. \end{equation} \tag{ 17 }$

As shown in Figure 6, the upcoming measurements of the SZ power spectrum will be able to distinguish between ΛCDM and different EDE models (that are not ruled out by current observations) with high significance. As in the case of the galaxy cluster counts, the SZ power spectrum is also sensitive to the redshift of transition from EDE-like to present-day ΛCDM-like behavior. Current constraints on the redshift and width of transition are z_t ≳ 4, Δ_t ≲ 0.2 (Alam 2010), while future observations of the SZ power spectrum may even rule out EDE models with z_t ≳ 9, Δ_t ≲ 0.1 (EDE2). These results are of course for a fixed value of other non-dark energy parameters, and we expect degeneracies with these in a full analysis. In the next section we attempt to constrain the EDE parameters using future simulated SZ power spectrum measurements and cluster counts.

4.3. Likelihood Analysis

In this section we study the effect of the degeneracies between the different cosmological parameters for EDE models. We simulate the data according to WMAP7+BAO+H₀ ΛCDM model (Komatsu et al. 2010). We generate the primary CMB TT power spectrum simulated to match the Planck survey resolution and sky coverage. We also simulate the SZ power spectrum data as obtainable from ACT/SPT surveys with the survey parameters discussed in detail in Section 4.2. For studying the cluster data, we simulate an SPT survey, assuming a Tinker et al. (2008) number density of clusters; the details are given in Section 4.1. The results for eROSITA are expected to be similar.

Typically, the cluster data alone are not enough to break the degeneracies between different parameters and we need the constraints from the primary CMB power spectrum. We add CMB data in the form of scalar Cl's simulated as per Planck specifications. We also apply a Gaussian prior on H₀ as H₀ = 70.4 ± 3.6 km s^–1 Mpc^–1, where the error is consistent with the recent results of the Hubble constant from the SHOES (Supernovae and H₀ for the Equation of State) program (Riess et al. 2009). We do not add any other possible future data sets such as the supernovae or Baryon Acoustic Oscillations.

In analyzing the data, we keep w_m ⩾ −0.1 since we are interested in constraining models which have sufficient amount of dark energy at early times. The true ΛCDM model cannot be exactly reproduced by the EDE class of models since w_m ≠ −1, but in the limit w₀ = −1, a_t → 0(z_t → ∞), Δ_t → 0 these models replicate ΛCDM-like behavior.

As shown in Figure 7, CMB + cluster count analysis can constrain the EDE parameters to w₀ ≲ −0.9, a_t ≳ 0.2(z_t ≳ 4), Δ_t ≲ 0.3, which implies that cluster counts + CMB can constrain any deviation from ΛCDM behavior up to at least z = 4. For the SZ power spectrum data, we find that the EDE parameters are constrained to w₀ ≲ −0.85, a_t ≲ 0.15(z_t ≳ 5.5), Δ_t ≲ 0.15 (see Figure 8). The SZ power spectrum provides complementary information about dark energy parameters compared to cluster counts. For example, the SZ power spectrum puts tighter constraints on the dark energy transition parameters compared to cluster counts. The matter density is reasonably reconstructed for both the cluster counts and the SZ power spectrum.

**Figure 8.** Confidence levels for a combination of primary CMB power spectrum from Planck survey +SZ power spectrum obtainable from ACT/SPT survey.
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5. CONCLUSIONS

In this work, we have studied EDE models in light of future galaxy cluster data. We studied two EDE models, both of which are allowed within the current observational constraints. The first model (EDE1) transits from EDE behavior at redshift z ≳ 5 to ΛCDM-like behavior at present, while the second model (EDE2) transits at a higher redshift (z ≃ 9) to an equation-of-state w₀ = −0.9, which is close to but not identical to ΛCDM. For both the models, as well as for the fiducial ΛCDM cosmology, we show the predictions for cluster abundance and the SZ power spectrum. For that purpose, we consider two instruments—eROSITA for X-ray surveys, and ACT/SPT for SZ effect. We also do a likelihood analysis of data simulated replicating the SPT and eROSITA survey specifications, along with the simulated Planck CMB data, to obtain the constraints on the EDE parameters from future surveys, and find that future galaxy cluster counts and SZ power spectrum can put competitive constraints on these parameters.

It is worth noting some apparent differences between our work here and some other recent works on EDE signatures. For example, Fedeli et al. (2009) also considered several EDE models, but their cluster counts are higher than for the fiducial ΛCDM model, although σ₈ for EDE models are commonly lower in their work (as is the case here as well). That is because for their choice of EDE models Hubble parameters, E(z) differs significantly from the corresponding ΛCDM model. This affects the minimum observable mass as a function of redshift; their EDE models have much lower mass threshold than ΛCDM, thus resulting in higher observable cluster counts. In contrast, we have chosen models which have E(z) similar to that of the corresponding ΛCDM model, to highlight the difference coming from the perturbations rather than the background expansion. A similar difference from this work, due to the radical departure of E(z) from ΛCDM, can be seen in Waizmann & Bartelmann (2009), for the effects of EDE on the SZ power spectrum. In some other cases differences can arise depending whether one uses low-k, CMB normalization of the power spectrum as done here, or σ₈ normalization as in Sadeh et al. (2007), although it is clear that a successful model has to fulfil both constraints.

We show that the inclusion of dark energy perturbations has a major effect on the matter power spectrum, therefore increasing the possibility of discriminating EDE models from ΛCDM using large-scale structure probes. Neglecting the dark energy perturbations leads to severe underestimation of the imprint which the sharp transition in dark energy equation of state leaves on the dark energy perturbations and therefore on the matter power spectrum. We show that the models considered here which are allowed by the current observations can be ruled out using the future galaxy cluster probes. It is also interesting that both the cluster counts and the SZ power spectrum are sensitive to the redshift at which the transition between early and present-day dark energy occurs. We expect to put strong constraints on the equation of state of dark energy at present using low redshift geometric observables (such as the luminosity distance of type Ia SNe and Baryon Acoustic Oscillations peaks). These probes of geometry of the universe, however, are insensitive to the high redshift behavior of dark energy. The galaxy cluster observables such as their redshift abundance and the SZ power spectrum, although at low redshift, are sensitive to the perturbations in the EDE models, and hence would be able to put constraints on the redshift of transition from EDE behavior to present-day, ΛCDM-like behavior. With the ongoing and future cluster surveys in microwave, optical, and X-ray wavebands, galaxy clusters will be able to provide strong constraints on the dynamical dark energy sector.

We thank Konstantin Borozdin, Salman Habib, and Katrin Heitmann for useful discussions. We also thank the referee for his useful suggestions. The authors acknowledge support from Los Alamos National Laboratory and the Department of Energy via the LDRD program at LANL.

GALAXY CLUSTERS AS A PROBE OF EARLY DARK ENERGY

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ABSTRACT

1. INTRODUCTION

2. EARLY DARK ENERGY