EVIDENCE OF QUASI-LINEAR SUPER-STRUCTURES IN THE COSMIC MICROWAVE BACKGROUND AND GALAXY DISTRIBUTION

For simplicity, in this section, we assume that super-structures are modeled by spherically symmetric homogeneous compensated voids/clusters with an infinitesimally thin shell. The background spacetime is assumed to be a flat FRW universe with matter and a cosmological constant Λ.

Let κ and ξ be the curvature and the physical radius of a void/cluster in unit of the Hubble radius H⁻¹, and δ_H as the Hubble parameter contrast, t denotes the cosmological time. We describe the angle between the normal vector of the shell and the three dimensional momentum of the CMB photon that leaves the shell by ψ. We assume that the comoving radius r_v of the void in the background coordinates satisfies r_v ∝ t^β, where β is a constant.

Up to order O((r_v /H⁻¹)³) and O(κ²), the temperature anisotropy of the CMB photons that pass through spherical homogeneous compensated voids in the flat FRW universe can be written as (Inoue & Silk 2007)

$$\begin{eqnarray} \frac{\it {\Delta T}}{T}&=&\frac{1}{3 }\big [\xi ^3 \cos {\psi } \big (-2 \delta _H^2-\delta _H^3+(3+4 \delta _m)\delta _H \Omega _m \nonumber \\ \nonumber &&+\delta _m \Omega _m(-6 \beta /s +1)+ (2\delta _H^2+\delta _H^3+\delta _m \Omega _m \\ &&+(3+ 2\delta _m) \delta _H \Omega _m)\cos {2 \psi }\big)\big ], \\ t&=&s H^{-1}, s=\frac{2}{3\sqrt{1-\Omega _m}} \ln \biggl [\frac{\sqrt{1-\Omega _m}+1}{\sqrt{\Omega _m}} \biggr ], \\ &&\nonumber \end{eqnarray} \tag{ 1 }$$

where δ_m and Ω_m denote the matter density contrast of the void and the matter density parameter, respectively. The variables ξ, ψ, δ_m , δ_H , and Ω_m are evaluated at the time the CMB photon leaves the shell. It should be noted that formula (1) is valid even if |δ_m | or |δ_H | is somewhat large as long as the normalized curvature κ is sufficiently small. Formula (1) can be also applied to spherical compensating clusters with a density contrast δ_m > 0 corresponding to a homogeneous spherical cluster with an infinitesimally thin "wall." This approximation holds only in weakly nonlinear regime since the amplitude of the density contrast corresponding to a negative mass cannot exceed 1. We examine this approximation in Sections 2–4 in detail.

Because we are mainly interested in linear |δ_m | ≪ 1 and quasi-linear |δ_m | = O(0.1) regime, we expand δ_H in terms of δ_m up to the second order as

$$\begin{equation} \delta _H=\Omega _m \delta _m (1+1/f(w))/2-\epsilon \delta _m^2, \end{equation} \tag{ 3 }$$

where w is an equation-of-state parameter, is a constant that describes the nonlinear effect and

$$\begin{equation} f(w)=-\frac{3}{5}(1+w)^{1/3}{}_2 F_1\biggl [ \frac{5}{6},\frac{1}{3},\frac{11}{6},-w \biggr ], \end{equation} \tag{ 4 }$$

where ₂ F₁ is Gauss' hypergeometric function. can be estimated from numerical integration of the Friedmann equation inside the shell as we shall show later. In a similar manner, for the shell expansion, we assume the following relation for the wall peculiar velocity normalized by the background Hubble expansion

$$\begin{equation} \tilde{v}=s^{-1}\beta, \beta =-\frac{f(\Omega _m)}{6} \delta _m+\nu \delta _m^2, \end{equation} \tag{ 5 }$$

where ν represents a constant that describes the nonlinear effect (Inoue & Silk 2007). f(Ω_m ) is written in terms of the scale factor a and the growth factor D as

$$\begin{equation} f(\Omega _m)=\frac{a}{D}\frac{\it dD}{da}\sim \Omega _m^{0.6}. \end{equation} \tag{ 6 }$$

In quasi-linear regime, simplification = η = 0 can be verified, which will be shown in Sections 2.3 and 2.4.

In order to describe the dynamics of local inhomogeneity, we adopt a homogeneous collapse model which consists of an inner FRW patch and a surrounding background flat FRW spacetime (Lahav et al. 1991). The size of the inner patch is assumed to be sufficiently smaller than the horizon H⁻¹ in the background spacetime.

We assume that both the regions have only dust and a cosmological constant Λ. The time evolution of either the inner patch or the background spacetime is described by the Friedmann equation

$$\begin{equation} \frac{H^2}{H_0^2}=\frac{\Omega _{m,0}}{a^3}+\Omega _{\Lambda,0}+\frac{1-\Omega _{\rm tot}}{a^2}, \end{equation} \tag{ 7 }$$

where a denotes the scale factor, Ω_m,0, Ω_Λ,0, Ω_tot are the present energy density parameters of non-relativistic matter, a cosmological constant Λ, and the total energy density, respectively. The scale factor at present for the background spacetime is set to a₀ = 1. In what follows, we describe variables in the inner patch by putting tilde "∼" on top of the variables and we consider only flat FRW universes with dusts and a cosmological constant Λ.

First, we calculate matter density contrast δ_m of the inner patch. Initially (z_i ≫ 1), we assume that the fluctuation of the matter perturbation δ_mi is so small that $\tilde{a}_i \approx a_i, \tilde{H}_i \approx H_i$. Then, the Friedmann equation (Equation (7)) yields

$$\begin{equation} \frac{\tilde{H}^2}{H_0^2}=\frac{\Omega _{m,0}(1+\delta _{mi})}{\tilde{a}^3}+\Omega _{\Lambda,0} -\frac{\delta _{mi} \Omega _{m,0}}{\tilde{a}^2 \tilde{a}_i}. \end{equation} \tag{ 8 }$$

In terms of physical radius of the patch $\tilde{R}=\tilde{a} r$, where r is the comoving radius measured in the background spacetime, Equation (8) can be written as

$$\begin{eqnarray} \biggl (\frac{\it d \tilde{R}}{d t} \biggr)^2\! &=&\! H_0^2\biggl [-\Omega _{m,0}(1+z_i)^3 \tilde{R}_i^2\delta _{mi}+\Omega _{m,0}(1+z_i)^3 \nonumber \\ &&\times (1+\delta _{mi})\frac{\tilde{R}_i^3}{\tilde{R}} + \Omega _{\Lambda,0}\tilde{R}^2 \biggr ], \end{eqnarray} \tag{ 9 }$$

where t is the cosmological time. The matter density contrast δ_m can be written as a function of a ratio of the present and the initial comoving radius η ≡ r/r_i as δ_m = η⁻³ − 1. From Equation (9), η as a function of redshift z is given by solving

$$\begin{eqnarray} \biggl (\frac{d \eta }{d z} \biggr)\! &=&\! -[-\Omega _{m,0}(1+z_i)\delta _{mi} +\Omega _{m,0}(\eta /(1+z))^{-1} \nonumber \\ &&+\Omega _{\Lambda,0}(\eta /(1+z))^2 ]^{1/2}[\Omega _{m,0}(1+z)^3+\Omega _{\Lambda,0}]^{-1/2} \nonumber \\ &&+\frac{\eta }{1+z}. \end{eqnarray} \tag{ 10 }$$

Note that right-hand side in Equation (10) does not depend on $\tilde{R}_i$. From numerical integration of Equation (10), the matter density contrast δ_m as a function of redshift z is obtained by setting initial density contrast δ_mi = δ_m (z_i ).

The linearly perturbed matter density contrast δ^L _m in the FRW background spacetime is given by

$$\begin{equation} \delta _m^L(z)=\frac{3 \, \delta _{mi} H(z)}{5}\int _z^{\infty } du \frac{u+1}{H^3(1/u-1)}, \end{equation} \tag{ 11 }$$

where H²(z) = Ω_m,0(1 + z)³ + Ω_Λ,0 (Heath 1977). Constant 3/5 comes from our choice of initial condition that the peculiar velocity inside the initial patch is zero. The relation between δ_m and δ^L _m is shown in Figure 1.

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Because the relation does not change much even if one varies the cosmological parameters of the background spacetime, nonlinear isolated homogeneous spherical patches can be solely calculated from corresponding linear perturbations (Friedmann & Piran 2001). It should be noted, however, that the relation is valid only if δ_m ≳ δ_v,cut = −0.8 because of shell crossing (Furlanetto & Piran 2006).

Next, we calculate the Hubble parameter contrast $\delta _H=\tilde{H}/H-1$ in nonlinear regime. Plugging η = R(1 +  z)/(R_i (1 +  z_i )) into Equation (8), we have

$$\begin{eqnarray} &&\tilde{H}^2(\Omega _{m,0}(1+z)^3+\Omega _{\Lambda,0}) =H^2(\Omega _{m,0}(1+\delta _{mi})(1+z)^3 \eta ^{-3}\nonumber \\ &&\qquad+\Omega _{\Lambda,0} - \delta _{mi} \Omega _{m,0}(1+z_i)(1+z)^2\eta ^{-2}), \end{eqnarray} \tag{ 12 }$$

where η is given by solving Equation (10). From Equations (3) and (12), one can estimate the nonlinear parameter as

$$\begin{equation} \epsilon =\delta _{m}^{-1}\Omega _{m,0}(1+1/f(w))/2 -\delta _m^{-1}(\tilde{H}/H-1). \end{equation} \tag{ 13 }$$

In the previous section, we have seen that the nonlinear density contrast δ_m for a spherically symmetric homogeneous patch can be written in terms of corresponding linear density contrast δ^L _m . In order to calculate the ISW effect, we need to estimate the Hubble parameter contrast δ_H and the peculiar velocity $\tilde{v}$ of the wall. The nonlinear corrections to δ_H and $\tilde{v}$ can be characterized by two parameters and ν, respectively.

First, we consider the effect of nonlinear correction to the Hubble contrast δ_H . As one can see in Figures 2 and 3, is always negative and the amplitude is || < 0.16 for |δ_m | < 1.0. This represents a slight enhancement in the expansion speed within the inner patch due to nonlinearity. In low-density universes (Ω_m,0 < 1), || is smaller than that in high-density universes. In the Einstein–de Sitter (EdS) universe, depends only on δ_m (Figure 2). In contrast, in low-density universes, depends on the amplitude of the initial epoch as well (Figure 3). This is because the expansion speed inside the patch is suppressed when the energy component of the background universe is dominated by a cosmological constant Λ. We have found that the nonlinear contribution to δ_H is less than 10% for |δ_m | < 0.2 and Ω_m > 0.26.

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Second, we consider the effect of the nonlinear correction to the peculiar velocity of the wall. In the thin-shell limit, the motion of the spherically symmetric wall can be obtained by numerically solving a set of ordinary differential equations using Israel's method (Israel 1966; Maeda & Sato 1983). If the inner region and the outer region are described by the FLRW spacetime, the fitting formula for the peculiar velocity of an expanding wall normalized by the background Hubble expansion can be written as (K. Maeda et al. 2010, in preparation)

$$\begin{equation} \tilde{v}=\frac{\Omega _{m}^{0.56}}{6}\big(|\delta _m|+0.1 \delta _m^2+0.07|\delta _m|^3\big), \end{equation} \tag{ 14 }$$

for Ω_m + Ω_Λ = 1. We have confirmed that the accuracy of the fitting formula is within 1% for 0 < Ω_m ⩽ 1 and |δ_m | < 1 using numerically computed values. From Equation (14), we find that the contribution of the nonlinear effect is less than 5% for |δ_m | < 0.3. Thus an approximation $\delta _H \propto \delta _m \propto \tilde{v}$ can be validated in the quasi-linear regime.

In literature, the thin-shell approximation has been often used to describe almost empty voids with δ ∼ −1 (Maeda & Sato 1983). In quasi-linear regime, however, we also need to consider the effect of thickness of the wall and inhomogeneity of the matter distribution because quasi-linear voids are not in the asymptotic regime. Nonlinearity of the wall may significantly affect the CMB photons that pass through it. Moreover, it seems not realistic to apply the thin-shell approximation to spherically symmetric clusters since the mass of the wall cannot be negative.

In order to estimate the validity of the thin-shell approximation, we have compared the ISW signal with those obtained by using the second-order perturbation theory (Tomita & Inoue 2008) and by using the Lemaitre–Tolman–Bondi (LTB) solution (Sakai & Inoue 2008), which yields exact results without recourse to the cosmological Newtonian approximation. We have assumed top-hat-type matter distribution (for linear matter perturbation) for void/cluster for calculation using second order perturbation theory and a smooth distribution specified by a certain polynomial function for calculation using the LTB solution. The voids/clusters are assumed to be compensated so that the gravitational potential outside the cutoff radius (r₁ for the perturbative analysis and r_out for the LTB-based analysis) is constant. For detail, see Appendices A and B.

As an example, we have computed temperature fluctuations ΔT generated from a compensated void/cluster using the three types of methods. The density contrast, the comoving radius and the redshift of the center of a void/cluster are set to |δ_m | = 0.1, r_v = 200 h⁻¹ Mpc, r₁ = r_out = 210 h⁻¹ Mpc, and z = 0.2, respectively. The width of the wall is assumed to be 1/10 of the cutoff radius. As one can see in Figure 4, the three methods agree well for low density universes in which the linear effect is dominant. In contrast, the discrepancy becomes apparent for high density universes in which the nonlinear effect is dominant. This discrepancy is partially due to a slight difference in the assumed density profile (top-hat-type for the perturbative analysis, polynomial type for the LTB). In order to demonstrate the role of nonlinearity, we have plotted the first-order (linear ISW effect) and the second-order (RS effect minus linear ISW effect) contributions to the ISW signal (Figure 5). The first-order effect makes the CMB temperature negative (positive) for a void (cluster) but the second-order effect makes the CMB temperature negative near the center and positive near the boundary regardless of the sign of the density contrast. As a result, the amplitude of temperature fluctuation for a void (cluster) is enhanced (suppressed) in the direction near the center but it is suppressed (enhanced) in the direction near the boundary. These nonlinear effects become much more apparent for models with higher background density because the linear ISW effect becomes less effective. However, if we take into account the thickness of the wall, these nonlinear effects can be less conspicuous since the amplitude of the gravitational potential becomes smaller for a fixed outer radius (Figure 6). In the Λ-dominated universe, a quasi-linear compensated void can be recognized as a cold spot surrounded by a very weak hot ring, whereas a quasi-linear compensated cluster can be recognized as a hot spot possibly with a dip at the center of it. In the EdS universe, either a compensated quasi-linear void or a cluster can be identified as a cold spot surrounded by a hot ring.

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In order to fully utilize information of the three dimensional distribution, we consider a temperature anisotropy ΔT/T obtained from stacked images on the CMB sky that corresponds to most prominent voids/clusters in a galaxy catalog. First, we fix an angular radius θ_out of a circular region on the CMB sky that will be used in the stacking analysis. Then, the corresponding smoothing scale r_s in comoving coordinate for the corresponding fluctuation at z is r_s = (1 + z)D_A (z)θ_out, where D_A (z) is the angular diameter distance to the galaxy. The corresponding initial smoothing scale is rⁱ _s = η⁻¹ r_s .

We assume that the probability distribution function (PDF) of linear density contrast δ^L _m at redshift z is given by a Gaussian distribution function

$$\begin{equation} P^L\big(\delta _m^L;\sigma \big(r^i_s,z\big)\big)=\frac{1}{\sqrt{2 \pi }\sigma \big(r^i_s,z\big)}\exp \biggl [-\frac{\big(\delta _m^L\big)^2}{2 \sigma ^2\big(r^i_s,z\big)} \biggr ], \end{equation} \tag{ 15 }$$

where σ²(rⁱ _s , z) is the variance of the linearly extrapolated density contrast at redshift z smoothed by a spherically symmetric top-hat-type window function with an initial comoving radius rⁱ _s . Note that σ(rⁱ _s , z) depends on cosmological parameters such as Ω_m,0, σ₈ and the spectrum index n. Then, the PDF of nonlinear density contrast of the inner patch δ_m is given by

$$\begin{equation} P^{NL}\big(\delta _m;\sigma \big(r^i_s,z\big)\big)=\alpha P^L\big(\delta _m^L(\delta _m);\sigma \big(r^i_s,z\big)\big)\frac{d \delta _m^L}{d \delta _m}, \end{equation} \tag{ 16 }$$

where α is a constant that normalizes the PDF.

As shown in Figure 7, the PDF of δ_m is positively skewed in comparison with the PDF of δ^L _m because of nonlinearity. For a sample region at redshift z with a total comoving volume V, the total number of voids or clusters with a radius r_s is approximately N_t ≈ 3V/(4πr³ _s ). In what follows, we assume that the number of prominent voids/clusters (N_v /N_c ) determines the corresponding threshold of density contrast δ_m,th (z), which is given by

$$\begin{equation} N_v/N_t=\int _{\delta _{v,cut}}^{\delta _{m,th}} P^{NL}\big(\delta _m;\sigma \big(r^i_s,z\big)\big)d \delta _m, \end{equation} \tag{ 17 }$$

and

$$\begin{equation} N_c/N_t=\int _{\delta _{m,th}}^{\delta _{c,cut}} P^{NL}\big(\delta _m;\sigma \big(r^i_s,z\big)\big)d \delta _m, \end{equation} \tag{ 18 }$$

respectively. From Equations (1), (17), and (18) the mean temperature fluctuation within an angular radius θ_out for a stacked N_v or N_c images corresponding to prominent quasi-linear voids/clusters at redshift ∼z can be approximately written as

$$\begin{equation} \langle \!\it {\Delta T}\rangle \!=\frac{\int \!\!\! \int W(\theta ;\theta _{\rm in})\it {\Delta T}(\delta _m, \psi (\theta)) P^{NL}(\delta _m;\sigma)d^{2}{\boldsymbol{\theta }} d\delta _m}{ \pi \theta _{th}^2 \int P^{NL}(\delta _m;\sigma)d\delta _m }, \end{equation} \tag{ 19 }$$

where 0 ⩽ θ ⩽ θ_out, W(θ; θ_in) is a compensating window function that satisfies

$$\begin{equation} \int _0^{\theta _{\rm in}} 2 \pi \theta W(\theta ;\theta _{\rm in}) d\theta =-\int _{\theta _{\rm in}}^{\theta _{\rm out}} 2 \pi \theta W(\theta ;\theta _{\rm in}) d\theta \end{equation} \tag{ 20 }$$

and

$$\begin{equation} \psi (\theta)=\theta + \arcsin [h^{-1} \cos \theta \sin \theta - \sqrt{\sin ^2\!\theta - h^{-2} \sin ^4 \!\theta } ], \end{equation} \tag{ 21 }$$

where h = r_s (1 + z)/D_A (z).

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A cross correlation analysis using a stacked image built by averaging the CMB surrounding distinct voids/clusters has been done by Granett et al. (2008). They have used 1.1 million luminous red galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS) catalog covering 7500 deg². The range of redshifts of the LRGs is 0.4 < z < 0.75, with a median of ∼0.5. The total volume is ∼5 h⁻³ Gpc³. They used the so-called ZOBOV (ZOnes Bordering On Voidness; Neyrinck 2008) algorithm to find supervoids and superclusters in the LRG catalog and made a stacked image from an inversely variance weighted WMAP 5 year (Q, V, and W) map. In order to reduce contribution from CMB fluctuations on scales larger than the objects, they used a top-hat-type compensating filter

$$\begin{equation} W_{th}(\theta ;\theta _{\rm in})= \left\lbrace \begin{array}{@{}ll} 1 & (\theta <\theta _{\rm in}) \\ \! -1 & (\theta _{\rm in}\le \theta \le \theta _{\rm out}), \\ \end{array} \right. \end{equation} \tag{ 22 }$$

where θ_out = cos⁻¹(2cos θ_in − 1).

First, using the developed tools based on thin-shell approximation and homogeneous collapse model in Section 2, we estimate the expected amplitude of the ISW signal for prominent structures in a concordant ΛCDM model with Gaussian primordial fluctuations and compare with the observed values obtained from the SDSS–LRG catalog. The number of most distinct voids or clusters N and the cutoff radius θ_in are chosen as free parameters. At redshift z = 0.5, the mean density contrast filtered by a top-hat-type function with radius r = 130 h⁻¹ Mpc corresponding to θ_out ∼ 56 is just 〈δ^L _m 〉 = 0.046 and the background density parameter is Ω_m (z = 0.5) = 0.54. Because the influence of non-linear ISW effect is weaker than that of the linear ISW effect in this setting, we expect that details of nonlinear calculations will not affect the result much. In what follows, we use an approximation $\delta _H \propto \delta _m \propto \tilde{v}$, where δ_m is determined from the homogeneous collapse model in Section 2.

As shown in Table 1, it turned out that the expected values of the ISW (Rees–Sciama) signal are typically of the order of O(10⁻⁷) K. As expected, the amplitude gets larger as the number of stacked image decreases, and the amplitude for voids systematically becomes larger than those for clusters by 5%–10% (Tomita & Inoue 2008; Sakai & Inoue 2008). On the other hand, the order of the observed amplitudes are extremely large as O(10⁻⁶) K. It turns out that the discrepancy remains at 3σ–4σ level for N = 30 and N = 50.

Second, we reconstruct the mean density profile from the observed ISW signal for the SDSS–LRG catalog using our LTB model. From Figure 7, one can notice a hot ring around a cold spot for the stacked image of voids and a dip at the center of a hot spot for the stacked image of clusters. Although the amplitude of the hot-ring cannot be reproduced well, the observed dip at the center of the hot spot can be qualitatively reproduced in our LTB models. We have found that the dip at the center of a compensated cluster can be generated only if the linear ISW effect balances the nonlinear ISW effect in a limited parameter region. Thus the observed features in stacked images strongly imply that the corresponding super-structures are not linear but quasi-linear or nonlinear objects. The density fluctuations which are necessary to produce the observed ISW signals are found to be tremendously large. In Figure 8, we plot the ISW signal from a compensated cluster with δ_m ∼ 7σ and that from a compensated void with |δ_m | ∼ 10σ at z = 10 (see the radial density profiles in Figure A1 in Appendix A). Even for these very rare objects, the amplitudes of ISW signals in our LTB models are much smaller than the observed ones. In fact, the mean temperatures for a compensating filter θ_in = 4° are 3.6 μK(1.4σ) and −3.1 μK(2.6σ) for the cluster and the void, respectively. On the other hand, the probability of generating these fluctuations is as extremely small as 10⁻¹² in standard inflationary models that predict primordial Gaussianity.

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Thus, the observed large ISW signals for the stacked image strongly suggest a presence of super-structures on scales O(100 h⁻¹ Mpc) with anomalously large density contrast O(0.1) which cannot be produced in the concordant LCDM model.

Francis & Peacock (2010) estimated the local density field in redshift shells using photometric redshifts for the Two Micron All Sky Survey (2MASS) galaxy catalog. They reconstructed the CMB anisotropies due to the ISW effect from the local density field δ_m . They approximated the bias in each redshift shell by a linear bias relation δ_g = b δ_m and assumed that the bias is independent of scale and redshift in each shell. In order to obtain the bias parameter b, a maximum likelihood analysis of the galaxy catalog was performed.

There are two prominent spots in the reconstructed CMB anisotropy. One is a hot spot due to a supercluster around the Shapley concentration at redshifts 0.1 ⩽ z ⩽ 0.2. Another one is a cold spot due to a supervoid at redshifts 0.2 ⩽ z ⩽ 0.3 in the direction to (l, b) ∼ (0, − 30°). The angular radii of both structures are θ = 20°–30°. The temperatures near the center of both structures are ∼20 μK. The position of the supervoid is very close to the one predicted in Inoue & Silk (2007), (l, b) ∼ (330°, − 30°).

Based on the methods developed in Section 2, we have estimated the expected density contrast δ_m and the corresponding temperature profile (Figure 9) due to a most prominent object in the shell. We have assumed the same cosmological parameters and primordial Gaussianity as those discussed in Section 3. In order to compute the temperature profile, we have used a homogeneous thin-shell model. As shown in Table 2, the observed density contrasts are larger by 4–7 times the expected values. If the comoving radius of the structure is ∼200 h⁻¹ Mpc, the absolute value of the density contrast should be |δ_m | = 0.2–0.3, which implies the presence of anomalous quasi-linear super-structures. Our result is consistent with the power spectrum analysis in Francis & Peacock (2010) where a noticeable excess of the observed power at low multipoles 2 ⩽ l ⩽ 4 was reported.

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The most striking CMB anomaly is the presence of an apparent cold spot in the Wilkinson Microwave Anisotropy Probe (WMAP) data in the Galactic southern hemisphere (Vielva et al. 2004; Cruz et al. 2005; see Figure 10). The cold spot has a less than 2% probability of being generated as random Gaussian fluctuations (Cruz et al. 2007a), if one uses spherical mexican-hat-type wavelets as filter functions (see also Zhang & Huterer 2010). Assuming that it is not a statistical artifact, a variety of theoretical explanations have been proposed, such as galactic foreground(Cruz et al. 2006), texture (Cruz et al. 2007b), and Sunyaev–Zeldovich (SZ) effect. However, these models failed to explain other large-angle anomalies by the same mechanism.

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Inoue & Silk (2006, 2007) proposed that the cold spot may be produced by a supervoid at z < 1 in the line of sight due to the ISW effect and have shown that another pair of supervoids that are tangential to the Shapley concentration can explain the alignment between the quadrupole and the octopole in the CMB. Subsequently, Rudnick et al. (2007) found a depression in source counts in the NRAO VLA Sky Survey(NVSS) in the direction to the cold spot, although the statistical significance has been questioned (Smith & Huterer 2010). Recent optical observations (Granett et al. 2010; Bremer et al. 2010), however, revealed that any noticeable supervoids at 0.35 < z < 1 in the line-of-sight are ruled out. These observations suggest that the angular size of the supervoid may be larger or smaller than expected and that it resides at low redshifts z < 0.35 or at high redshifts z > 1.

In order to test such a possibility, we have calculated the average temperature around the cold spot (see Figure 11) using a spherical top-hat compensating filter W_th (θ; θ_in).

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Interestingly, we have discovered two peaks in the plot of the filtered mean temperature around the cold spot as a function of inner radius of the filter (Figure 12). The inner and the outer peaks are observed at θ_in = 4°–5°(θ_out = 6°–7°) and θ_in = 12°–13° (θ_out = 16°–17°). The outer peak corresponds to a hot ring, which is visible by eyes (see Figure 10).

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In order to estimate the statistical significance of the peaks, we have used a WMAP 7 year internal linear combination (ILC) map smoothed at 1° scale with a Galactic sky cut |b| < 20° and a combination of the Q, V, and W band frequency maps smoothed at 1° scale averaged with weights inversely proportional to the noise variances with a "standard" Galactic sky cut made by the WMAP team. In order to reduce possible residual contamination from the Galactic foreground, we further cut a region |b| < 35° for the Q+V+W map. In order to estimate the errors, firstly, we generated 1000 random positions on the ILC(|b|>20°) and the Q+V+W(|b|>35°) maps, and then computed variances σ² for the filtered mean temperature. Second, we have calculated expected σ² for the filtered temperature using the angular power spectrum C_l for the WMAP 7 year data obtained by the WMAP team (see Appendix A). Note that we have computed pseudo-C_l 's from the C_l ' for each sky map. As shown in Figure 11, the observed standard deviations are 1σ = 19 ∼ 20 μK for θ_in = 4° and 1σ = 14 ∼ 16 μK for θ_in = 14°. Our result for θ_in = 4° is roughly consistent with the values for stacked images in Granett et al. (2008) assuming no correlation between voids/clusters in a particular configuration. In the Q+V+W map, a slight suppression in σ is observed at θ_in > 8°. Theoretically calculated values are found to be systematically lager than the observed values by 4%–16% for 4° < θ_in < 14°. These discrepancies represent an uncertainty due to the Galactic foreground emission.

As one can see in Figure 12, a deviation corresponding to the inner peak is roughly 4σ and that corresponding to the outer peak is 4 ∼ 4.5σ. Assuming that the filtered mean temperature obeys Gaussian statistics, the statistical significances are P(>4σ) = 6 × 10⁻³ and P(>4.5σ) = 7% × 10⁻⁴%. The total solid angle of the ILC map (|b|>20°) is 8.27 sr and that of the Q+V+W map (|b|>35°) is 5.36 sr. The total number of the independent patches is roughly given by a ratio between the solid angle of the map and the area of the spherical patch with angular radius θ_out. Therefore, for θ_out = 7°, we have ∼180 samples for the ILC and ∼110 samples for Q+V+W map, yielding (110–180) × P(>4σ) = 0.7%–1%. In a similar manner, for θ_out = 17°, one can easily show that the statistical significance is ∼0.01%–0.2% corresponding to ∼3σ if the likelihood function is a Gaussian one.

Thus, the cold spot surrounded by a hot ring at scale ∼17° is more peculiar than the cold spot at scale ∼7°. Therefore, the real size of the supervoid is expected to be larger than the apparent size of the cold region. Because 1σ deviation (corresponding to 15–20 μK) due to a supervoid is enough to make the signal non-Gaussian, it is reasonable to assume that the contribution from a supervoid is less important than that from other effects due to acoustic oscillation or Doppler shift at the last scattering surface. For instance, a supervoid with a density contrast δ_m = −0.3 with a comoving radius r = 200 h⁻¹ Mpc at a redshift z = 0.2 corresponding to an angular radius θ = 20° would yield a temperature decrease ΔT ∼ 20 μK in the direction to the center. Moreover, if the supervoid is not compensating, a wall surrounding the supervoid could generate the observed hot ring. It may consist of just an ordinary underdense region surrounded by massive superclusters. Further observational study is necessary for checking the validity of the "local supervoid with a massive wall" scenario.

We thank B. R. Granett for providing us the data of the radial profile of stacked images for the SDSS-LRG catalog. We also thank I. Szapudi, H. Kodama, M. Sasaki, A. Taruya, T. Matsubara, and P. Cabella for useful conversations. Some of the results presented here have been derived using the Healpix package (Goŕski et al. 2005). We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA)⁴ . Support for LAMBDA is provided by the NASA Office of Space Science. Numerical calculations were partly carried out on the computer center at YITP in Kyoto University. This work is in part supported by a Grant-in-Aid for Young Scientists (B) (20740146) and Grant-in-Aid for Scientific Research on Innovative Areas (22111502) of the MEXT in Japan.