Prog. Theor. Exp. Phys. 2012, 00000 (24 pages)
DOI: 10.1093/ptep/0000000000

Integrated Sachs Wolfe Effect and Rees Sciama
EffectAtsushi J. Nishizawa1
1

Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), The University ofTokyo, Chiba 277-8582, Japan∗
E-mail: atsushi.nishizawa@ipmu.jp...............................................................................

It has been around fifty years since R. K. Sachs and A. M. Wolfe predicted the existence ofanisotropy in the Cosmic Microwave Background (CMB) and ten years since the integrated
Sachs Wolfe effect (ISW) was first detected observationally. The ISW effect provides us witha unique probe of the accelerating expansion of the Universe. The cross-correlation between
the large-scale structure and CMB has been the most promising way to extract the ISW effectfrom the data. In this article, we review the physics of the ISW effect and summarize recent
observational results and interpretations.

.................................................................................................
Subject Index
Cosmological perturbation theory, Cosmic background radiations, Dark energy
and dark matter

1. Overview
After the discovery of the isotropic radiation of the Cosmic Microwave Background (CMB) of the
Universe by A. A. Penzias and R. W. Wilson in 1965 [100], R. K. Sachs and A. M Wolfe predict theexistence of anisotropy in the CMB associated with the gravitational redshift in 1967 [116]. They
fully integrate the geodesic equation in a perturbed Friedman-Robertson-Walker (FRW) metric in
the fully general relativistic framework. The Sachs-Wolfe (SW) is the first paper that predicts thepresence of the anisotropy in the CMB which now plays an important role for constraining cosmological models, the nature of dark energy, modified gravity, and non-Gaussianity of the primordial
fluctuation.
Let us begin by reviewing the history. In the first twenty years since the Sachs-Wolfe paper, mostof the works were focused on the extension of the Sachs-Wolfe calculation to non-linear collapsed
object [96, 113], or non-standard cosmological model such as topological defects [66]. Partridge
and Wilkinson 1967 first gave a glimpse of the existence of inhomogeneity in the CMB temperatureby using the Dicke Radiometer [143]. They found the temperature excess on the direction of the
known quasar cluster position and considered it as the Rees-Sciama effect [113]. In the age of theCOBE satellite, the Sachs-Wolfe paper attracted a huge attention. Most of the papers were focusedon the theoretical prediction that was related to the observation; prediction of the amplitude of thequadrupole power for the SW effect. [17, 46–48]. Crittenden & Turok 1996 pointed out that thegravitational potential may decay in the Λ dominated Universe at z < 1 to produce the ISW signal[24]. They also proposed a novel method to detect the ISW effect by cross correlating the largescale structure with the CMB. Kneissl et al. 1997 made an attempt to extract the ISW effect by crosscorrelating the CMB observed by the COBE with the ROSAT X-ray background [68], Boughn &Crittenden 2002 used the NVSS radio galaxies for the cross correlation [11], and Boughn et al. 1998
c The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.
°
This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.

used the HEAO1 A2 X-ray background [14] but none of them could find the significant detection.
The Sachs-Wolfe paper has attracted a renewed attention in the WMAP era. The first detection of
the ISW was finally achieved by cross-correlating the WMAP first-year data with the number countof radio galaxies from the NVSS data, as well as with the HEAO1 A1 X-ray data [12]. Subsequently
a lot of detections with various mass tracers have been reported. In the early 2000’s, much work was
focused on obtaining cosmological constraint on dark energy models from the ISW effect, while in
the late 2000’s to present, more and more works studied various systematic effects which may enterin different ways for different measurement methods.
In this paper, we review the ISW effect from theoretical derivation of the basic equations to thepresent cosmological interpretations. The paper is organized as follows. In section 2, we revisit
the derivation of the CMB anisotropy induced by the perturbation of the background geometry
decomposed into scalar, vector and tensor modes. In section 3, we discuss the statistical properties of
the ISW effect and the method to measure it in the cross correlation with the large-scale structure. We
also discuss the possible systematic effects that affect our interpretations. In section 4, we provide
cosmological applications of the ISW effect including constraints on dark energy and primordial
non-Gaussianity. In section 5, we give a summary.

2. Theory of the ISW effect In the Standard Cosmology
In this section, we derive the basic equations of the ISW effect based on the original paper [116].
We first write the line element in a spatially flat FRW metric,
ds2 = gµν dxµ dxν = a2 (τ )g̃µν dxµ dxν ,(1)

where a is the scale factor that depends solely on the conformal time τ and gµν and g̃µν are themetric and a conformally transformed metric, respectively. The metric consists of perturbed andunperturbed parts, i.e. g̃µν = ηµν + h̃µν , with the unperturbed metric ηµν = diag(−, +, +, +) andh̃µν ¿ 1. Here we ignore all the quantity of order O(h2 ) and higher. Introducing two affine parameters τ and λ to characterize the photon geodesic in the gµν and g̃µν metric respectively, we havedτ = a2 dλ since the action of the geodesic should be invariant under the rescaling of gµν → a2 gµνand dτ → a2 dλ. To simplify the calculation, we first work on the g̃µν system and then translate it to
the gµν system.
The metric perturbation can be decomposed into scalar, vector and tensor modes. h0i can bedivided into the contributions from scalar and vector while hij can be divided into the contributions
from scalar, vector, and tensor modes as,h̃00 = −2A(s)h̃0i =
h̃ij

(2)

(v)
−∂i B (s) − Bi

·
µ
¶
¸
1 2
(v)
(v)
(t)
(s)
(s)
= −2 D δij − ∂i ∂j − ∇ C+ (∂j Ci + ∂i Cj ) + Cij ,
3

(3)
(4)

where A, B, C and D are arbitrary functions and superscript with the parenthesis (s), (v) and (t) standfor the scalar, vector and tensor quantities respectively. The derivative ∂i denotes the 3 dimensional
covariant derivative. The scalar perturbation is not generated from the vector or tensor mode and thevector perturbation is not generated from the tensor mode thus we have constraints as(v)∂ i Bi

(5)

= 0,

(v)
∂ i Ci = 0,(t)iCi = 0,2/24(t)∂ i Cij = 0.

(6)

2.1. Scalar mode Linear Perturbation
We now need to fix a gauge degree of freedom. For the scalar perturbation, the conformal Newtonian
gauge (longitudinal gauge) is useful [e.g. 78, 87]. In the Newtonian gauge, all the gauge degrees
of freedom are used to eliminate the off-diagonal components of the perturbed metric. Then thevariables are fixed as B (s) = C (s) = 0, and the metric perturbation can be fully described by the two
scalar quantities of A = Φ, D = Ψ which are already gauge invariant. The metric turns out to be
ds2 = a2 [−(1 + 2Φ)dτ + (1 − 2Ψ)δij dxi dxj ],(7)

where Φ and Ψ are Newtonian potential and curvature perturbation respectively. Then we naturally
obtain the non vanishing connections,Γ̃000 = Φ0 , Γ̃00i = ∂i Φ, Γ̃0ij = −Ψ0 δij ,(8)where 0 ≡ ∂/∂τ . The photon geodesic x(λ) can be obtained by solving the geodesic equation,dk̃ α= Γ̃αµν k̃ µ k̃ ν ,
dλ

(9)

µ

where we introduce the 4-momentum k̃ µ = dx
dλ . It can be decomposed into the unperturbed andperturbed geodesic as k̃ µ = k̄ µ + δ k̃ µ . The photon energy is measured by the observer moving withthe fluid, thus the observed energy needs to be projected with the 4-velocity in the un-tilde gµν
system,E = gµν uµ k ν .

(10)

By definition gµν uµ uν = 1, the unperturbed component is uµ = a−1 (1, 0, 0, 0). The 4-velocity is
written as uµ = a−1 (1 − Φ, v i ). The spatial 3 dimensional velocity is already first order quantity,
hence we do not need to explicitly solve the spacial part of the geodesic equation, (9). The solution
for the unperturbed background is trivial, i.e. k̄ 0 = 1, k̄ i = ei , where e = (1, ei ) is the 4-tangent
vector of the geodesic. The time part of the first order solution of equation (9) is integrated as¯Z τ∗δ k̃ 0 ¯¯δ k̃ 0
=
+
2[Φ(τ
)
−
Φ(τ
)]
+
(Φ0 + Ψ0 )dτ,
(11)
¯
∗
0
a2
a2 ¯
τ0
τ∗

where τ∗ denotes the conformal time of the decoupling time, and τ0 today. Now the redshift isdefined by the photon energy ratio between emitter and receiver, 1 + z ≡ E(τ∗ )/E(τ0 ).Using thefact that under the rescaling of g̃µν → gµν , the 4-momentum scales as k̃ µ → a2 k µ ,1+z =k µ uµ |τ∗,k ν uν |τ0(12)

Since the temperature drops with redshift as T = T∗ /(1 + z), using equations (10), (11) and (12),observed temperature fluctuation over the sky is¯Z τ∗δT (s)
δT (s) ¯¯τ(Φ0 + Ψ0 )dτ,(13)=
¯ − Φ(τ0 ) + Φ(τ∗ ) + [v · e]τ∗0 +
T
T ¯
τ0
τ∗

apart from the isotropic temperature. The first term is the intrinsic photon fluctuation at the lastscattering surface other than those induced by the metric perturbation; δT /T |τ∗ = δγ (τ∗ )/4. The
second term is the gravitational redshift due to our gravitational potential which is the monopole
contribution and can not be observed. The third term represents the temperature anisotropy causedby the gravitational redshift due to the potential fluctuations at the decoupling epoch, which is called
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the naive or ordinary Sachs Wolfe effect in the literature. Since it also has the spacial dependence,
we shall write Φ(τ∗ ) = Φ(τ∗ , n̂) where n̂ denotes the angular position on the sky. The fourth termis the Doppler effect that is induced by the relative motion between the observer and the CMB lastscattering surface. The final integral term represents the temperature anisotropy caused by the time
variation of gravitational potential integrated along the line of sight, and this is the ISW effect. Since
the gravitational potential is static in the matter dominated Universe, i.e. Einstein de-Sitter Universe,
the ISW effect vanishes in the linear perturbation limit. Thus the ISW effect induces temperaturefluctuation at radiation dominated era or dark energy or curvature dominated Universe. The former
is called the early ISW and the latter the late ISW effect. We note that from the current observations
the matter radiation equality time, zeq ' 3300 is well before the decoupling, zdec = 1090 and thus
the temperature fluctuation of the early ISW is regarded as a part of the primary anisotropy. We
also note that the careful authors include the visibility function to the last scattering surface in the
integrand, e−τ , where τ is the optical depth; however, in the flat ΛCDM Universe, the redshift where
the ISW effect becomes important is at z < 1, and thus e−τ = 1 is a good approximation.

2.2. Vector and Tensor mode Perturbations
The photon geodesic is perturbed also by the vector and tensor modes. For the vector mode, thegeodesic perturbation can be characterized by Bi and Ci . The temperature anisotropy induced by
the vector mode is given by¯Z£ i ¤τ∗ 1 τ∗δT (v)
δT (v) ¯¯=dτ (∂i Vj + ∂j Vi )ei ej ,¯ + V ei τ 0 +
¯
T
T
2 τ0

(14)

τ∗

where the first term is intrinsic vector type temperature fluctuation, Vi is a rotational componentof the velocity and Vi = Ci0 + Bi . In the standard scenario of the inflation, the vacuum fluctuation
generates no super-horizon vector mode perturbation [74]. Even if it is generated with some exotic
mechanisms, vector mode has a only decaying mode solution which can be negligible at later times;
thus we shall assume that the vector mode does not exist [73]. In practice, it makes negligible
contribution to the CMB temperature observation so we assume it is absent.
The tensor mode, in other words the gravitational wave, is given by the traceless transverse rank
two tensor, Cij in equation (4). The temperature fluctuation induced by the tensor mode metricperturbation is¯ZδT (t)
δT (t) ¯¯
1 τ∗
0(t)
=
dτ Cij ei ej .
(15)
¯ −
T
T ¯
2 τ0
τ∗

The tensor mode decays on the scales smaller than the horizon at the decoupling, say ∼ 1 degree,
and the significant contribution comes from the largest scales. Recently, the BICEP2 experimentreport that they detect a signature of large amplitude of the gravitational wave which is observed by
the B-mode polarization power spectrum of the CMB [8]. This is indeed the tensor ISW effect! The
best fit value of the tensor to scalar ratio is r = 0.16 after removing the foreground components.

2.3. Spectrum of the ISW effect
In this section, we consider the power spectrum for the scalar mode perturbation. Here we assume
that the energy contents of the Universe have no anisotropic stress, which relates Φ and Ψ as Φ = Ψ.
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Dℓ [µK2 ]

103

102

scalar+tensor
primary scalar
ISW (z < 0.5 only)
ISW (all)tensor (r=0.16)
Planck 1yr low-ℓ

101

101
multipole ℓ

Fig. 1 Temperature anisotropy at low multipoles. The thick solid line shows the total anisotropy
integrated from today to the last scattering surface including scalar and tensor contributions. The thindashed line shows the scalar contribution of the primary anisotropy at the last scattering surface,
the thick dashed and the dot-dashed lines show contributions from the ISW effect generated atz < 0.5 and z < 100, respectively and the horizontal thick purple line below 100[µK2 ] shows thetensor contribution with r = 0.16. Also shown by points with errorbars is the CMB power spectrum
obtained by the first year low-` Planck data [106].
Then the integrated Sachs Wolfe effect is written asZδTISW (n̂)∂Φ(n̂r, τ )
=2
dτ.
(16)
T
∂τ
Now the potential Φ can be related to the density fluctuation through the Poisson equation. Working
in the Fourier space makes things simple. In the Fourier space, the Poisson equation in the flat FRW
Universe is,µ ¶ ·¸
3 Ωm H0 2 3aH
Φ(k, τ ) =θ(k, τ ) + δ(k, τ )
(17)
2 a
k
k2
where δ and θ are the density contrast and divergence of velocity of the matter, respectively andΩm and H0 are the matter density and Hubble parameter today, respectively. In the Newtonian limit
where the scale of interest is well smaller than the horizon, i.e. k À aH , equation (17) is reduced to
the well known form, Φ = 3Ωm /2a(H0 /k)2 δ . We usually expand the Fourier basis into the sphericalharmonics and spherical Bessel function to obtain the full sky expression of the fluctuation in a
direction n̂,
µ
¶
Z
Z
X
dkdΩk̂ ∂ δ(k, τ )∗2ljl (kr)Ylm (n̂)Ylm
(k̂)
(18)
ΘISW (n̂) = 12πΩm H0(−i) dτ(2π)3 ∂τa
ΘISW (n̂) ≡lm

The spherical harmonic counterpart is obtained using the orthogonality of the spherical harmonics,
µ
¶
Z
Z
Z
dkdΩk̂ ∂ δ(k, τ )∗2l∗Θlm = dΩn̂ Θ(n̂)Ylm (n̂) = 12πΩm H0 (−i) dτjl (kr)Ylm(k̂).
(2π)3 ∂τ
a
(19)
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The angular power spectrum is then calculated as,
ClISW = hΘlm Θ∗lm i =18 2 4
Ω H
π m 0

·Z

Z
dkP (k)¸2
drD(f − 1)Hjl (kr) ,(20)

where f is the velocity factor, f ≡ d ln D/d ln a and H = a0 /a is the conformal Hubble parameter.
The numerical calculation of equation (20) would be formidable because of the oscillatory behaviorof the Bessel function. The Hankel or also known as Fourier-Bessel transform can be helpful insteadof the direct integration [51, 97]. The Limber’s approximation is used in the literature but it isaccurate only up to ∼ 10% on scales larger than l < 10 [e.g. 35].
Figure 1 shows the power spectra of the primary CMB temperature fluctuation and the ISW effectfor the best fit model to the Planck first year data combined with the WMAP polarization and high-`
CMB experiments [107]. We also show the anisotropy generated from the tensor mode with tensor
to scalar ratio r = 0.16 which is recently suggested by the discovery of the BICEP2 experiment[8]. We use the publicly available CAMB code 1 to compute the scalar and the tensor spectra. The
ISW component can be easily calculated by the slight modification to the ‘equations.f90‘. Most of
the signal at l > 20 comes from the primary anisotropy while on large scales, significant fraction ofanisotropy is generated at low redshifts, z < 0.5. We also show the first year low-` Planck data 2 .
Because the large r value constrained by the B-mode power spectrum, ClBB of the BICEP2
enhances the amplitude of the temperature fluctuation, ClTT on large scales, it is required to makesome modifications of the model to keep the current observational constraints from the WMAP or
the Planck unchanged: either the smaller (i.e. negatively larger) value of running of scalar spectral
index αs [8] or isocurvature component of the initial fluctuation is required [67] to suppress the large
scale power of the CMB temperature power spectrum. However, the amplitude of tensor mode withr = 0.16 model is subdominant at ` < 10 compared to the ISW component. Thus there still remains
some possibilities that the non-standard gravity models or the nonlinearity of the local large-scale
structure can alter the amplitude of the ISW effect, and thus the constraint on the negatively large
value of the running or the fraction of the isocurvature perturbation component might be reduced.

2.4. Non Linear ISW Effect
M. J. Rees and D. W. Sciama extended the Sachs-Wolfe calculation to non-linear collapsed objects
in 1968 [113]. In 1967, they were investigating an apparent large-scale clustering of quasars [112]reported by Strittmatter & Faulkner [129]. They consider the possibility that inhomogeneity in thematter distribution inferred from the large-scale clustering of quasars creates anisotropy in the CMB
as predicted by Sachs & Wolfe and estimates the amplitude of the temperature fluctuation induced
by a spherically symmetric collapse of the objects in an expanding Universe [113]. Note that here
we use the terminology of non-linear as the non-linear density fluctuations but the metric perturbations (hence the geodesic perturbations) are kept at the first order. The second order cosmologicalperturbation is treated in a consistent manner in [94, 134–136] but here we limit our discussion to
the first order geodesic equation and non-linearity is only included in density perturbations. Using
the Poisson equation, we see that there are two contributions to time evolution of a gravitational

1
2

CAMB code is available at http://camb.info/
Planck data can be retrieved from http://www.esa.int/Planck
6/24

potential3Φ0 = Ωm
2

µ

H0
k

¶2

1
(ik · p − Hδ) .
a(21)

Here we have used the continuity equation δ 0 = ik · p, where p is the momentum of the density
field p = v(1 + δ). From equation (21), we see that the ISW and RS effects consist of two components: one proportional to the Hubble flow and the other the momentum of the object. In the linear
perturbation limit, the momentum is equal to the velocity field so the linear ISW effect in principle
traces the statistical property of the large-scale velocity field [22]. In a weakly non-linear regime
or fully non-linear regime, a halo model approach is used to describe the non-linear time evolutionof the gravitational potential which is originally developed for describing the non-linear clusteringof the dark matter halo [22, 79, 120]. The dark matter power spectrum is described by the sum of
two contributions: a two-halo term where the pair is in different halos, and a one-halo term where
the pair is in the same halo [122]. Once we provide the mass function [e.g. 64, 125, 133, 142] andprofile of the dark matter halo [89] then we can immediately calculate the non-linear clustering of
the dark matter. Similarly, the velocity field can be decomposed into two components: the velocity
due to the virial motion about the center of mass of its parent halo, and that due to the motion of theparent halo itself [124] which provides the non-linear momentum power spectrum [79].
Another approach is the higher order perturbation theory. As we will see in section 3, the crosscorrelation between the ISW and density tracer is useful to isolate the ISW effect from the CMB[24]. Then the angular cross correlation power spectrum between the ISW and any tracers of thedensity field can be some function of the cross power spectrum of Φ0 and Φ,
9PΦ Φ (k) = Ω2m
4

µ

0

H0
k

¶4

1
(Pδδ0 − HPδδ ) ,
a2

(22)

where we employ the notation, hX(k)Y ∗ (k0 )i ≡ (2π)3 δ D (k − k0 )PXY (k). The continuity equation
is written as δ 0 = −θ(1 + δ) where θ is the divergence of the velocity. Then the perturbed variables
δ and θ can be expanded in a series,δ(k, τ ) =Xnθ(k, τ ) = Hf

Dn (τ )δn (k, τ ),X

(23)

Dn (τ )θn (k, τ ),

(24)

n

where the nth variable is written in terms of a product of linear fluctuations asZδn (k, τ ) =Zθn (k, τ ) = −

d3 qn
d3 q1
·
·
·
δ1 (q 1 ) · · · δ1 (q n )Fn (q 1 , · · · , q n )δ D
(2π)3
(2π)3
d3 qn
d3 q1
·
·
·
δ1 (q 1 ) · · · δ1 (q n )Gn (q 1 , · · · , q n )δ D
(2π)3
(2π)3

Ã
X
i

Ã
X

!

(25)

qj − k!
qj − k ,(26)

iwhere δ D is the Dirac delta function. The functions Fn and Gn describe the mode coupling between
different wavevectors, and are explicitly given by [e.g. 63, 80]. Keeping all the terms which are less
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2
Log [ΔΦΦ'(k)]k [h/Mpc]

Fig. 2 The dimensionless cross power spectrum between Φ and Φ0 , ∆2ΦΦ0 (k) at different redshifts
computed using equations (22), (27), and (28). Dashed lines show a negative cross power, while thesolid lines positive. Figure is adapted from [91].
than fourth order of either δ1 or θ1 , we have [91]
Z
£
¤
d3 q
2
Pδδ (k) = D2 P (k) + 2D4
P
(q)
P
(|k
−
q|)F
(q,
k
−
q)
+
3P
(k)F
(q,
−q,
k)
, (27)
32(2π)3Pδδ0 (k) = Hf D2 P (k)
·
Z
d3 q
4
− Hf D P (k)P (q) (F3 (q, −q, k) + 3G3 (q, −q, k))
(2π)3
Z
d3 q
+2
P (q)P (|k − q|)F22 (q, k − q)G22 (q, k − q)
(2π)3
µ
Z
d3 q
+2
F2 (q, k − q)P (q)P (|k − q|) + G2 (q, q − k)P (k)P (|k − q|)
(2π)3
¶
¸
k·q
+ F2 (k, −q)P (k)P (q)
,
(28)
q2

where P (k) is the linear power spectrum of δ . More conveniently, Pδδ0 can be approximated by
1 ∂Pδ (k, τ ) [93, 139].
Pδδ0 (k, τ ) =2 ∂τFigure 2 shows the 3 dimensional cross correlation power spectrum of Φ and Φ0 at different redshifts in ΛCDM Universe. In the linear regime, gravitational potential is negative Φ < 0 in theoverdense region and it decays with time due to the accelerating expansion so that Φ0 > 0. ThusΦ and Φ0 show an anti-correlation. In the non linear regime, the gravitational potential grows, i.e.
the potential well gets deeper thus Φ0 < 0 which gives Φ and Φ0 a positive correlation. At redshift
z = 10, the Universe is close to the matter dominated so that the linear ISW effect is small andthe non-linear RS effect appears relatively prominently. At lower redshifts, the linear ISW effect issignificantly enhanced since the fraction of dark energy becomes more substantial but we can still
see the transition scale as the power turns from an anti-correlation to a correlation at k ∼ O(0.1)
Mpc/h. At z = 0, the linear ISW effect dominates at all scales.
The RS effect generates temperature anisotropy not only through the contraction and expansion of
the structure but also the bulk motion of the structure perpendicular to the line of sight [3, 4, 86, 115].
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Birkinshaw and Gull [9, 10] point out that the moving cluster which has transverse velocity to theline of sight may produce the local dipole structure on the CMB temperature. In order to explore thecomplete dynamics, N-body simulation with ray-tracing method is quite useful. Tuluie & Laguna[137, 138] carry out a ray tracing simulation for the maximum 360 Mpc/h box N-body simulation
to see the cumulative temperature fluctuation from z = 100 to today. They separate the sources ofanisotropy into the intrinsic change in gravitational potential of structure and the transverse bulk
velocity of structure; however, the box size is not large enough to see the dynamics of clusters orvoids whose sizes are larger than 100 Mpc/h. Cai et al. [20] use the Gigaparsec-size simulation to
construct a full sky map of the ISW. They argue that the non-linear RS and moving halo contributions
to the total are not significant, < 10% but the relative importance (relative to the linear ISW) is muchhigher if we go to higher redshift which is consistent with [19, 128] and with Figure 2. This is simply
because the linear ISW is negligible at high redshifts where the Universe is still matter dominated,while the non-linear RS effect exists regardless of dark energy.

3. Observing the ISW and RS effects
The ISW effect is first detected in the CMB measured by the WMAP cross correlated with thelarge-scale structure data traced by the X-ray background radiation and radio galaxies [12]. Several
attempts have been made to detect the ISW effect by cross correlating the CMB map measured
by the COBE with the ROSAT X-ray background [68], NVSS radio galaxies [11] or HEAO1 Xray background [14] but they could not detect the signal. However the non detection of the ISW
effect can put an upper limit on the amount of dark energy and surprisingly Boughn & Crittenden[11] present the limit ΩΛ < 0.74 which is fairly close to the current limit of dark energy parameter[108]. Subsequently a number of papers appeared. Table 1 presents a summary of the detectionof the ISW effect today. Some papers reach consistent conclusions while others show significant
inconsistencies. It is mainly due to either the wrong statistics used or the contamination of thesample due to an incomplete subtraction of systematics.
In the following sub sections, we review the observations of the ISW and RS effects. In sec. 3.1,
we define the cross correlation methods. In sec. 3.2, we discuss the possible systematic effects that
affect the significance of the detection of the ISW effect.

3.1. Cross Correlation with LSS
Since the ISW effect is generated when photons pass through a time varying gravitational potentialof the large-scale structure, it can be detected by cross correlating with some tracers of the large-scale
structure. We can write this idea in the equation ashΘ δLSS i = h(Θdec + Θfg + ΘISW + ΘSZ + Θlens + · · · )δLSS i,(29)

where Θ is a temperature fluctuation and δLSS is a density fluctuation of some tracers of the largescale structure, e.g. galaxy number count. The sources of the CMB temperature fluctuation include:Θdec is the primary anisotropy at or before the decoupling epoch, and Θfg is any astrophysicalforeground contamination from solar system or Galactic plane. The rest of the terms in equation(29) are attributed to the cosmological origin that generates secondary CMB anisotropy. ΘISW is
the ISW effect, ΘSZ is the Sunyaev-Zel’dovich (SZ) effect [130] and Θlens is the gravitational lenseffect [44, 59, 72]. Here we assume that the large-scale structure is not correlated with the primary
CMB at the decoupling epoch and we also assume that the ISW effect and the other secondary CMB
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AuthorKneissl et al. ’97 [68]
Boughn & Crittenden ’98 [14]Boughn & Crittenden ’02 [11]Boughn & Crittenden ’04,’05 [12, 13]
Fosalba & Gaztanaga ’04 [33]Fosalba et al. ’03 [34]

CMB
COBE4
COBE4
COBE4
WMAP1
WMAP1
WMAP1

Redshift
N/A
z'1
0<z<2
0<z<2
z ' 0.15
z ' 0.3, 0.5

Method
CCF
CCF
CCF
CCF
CCF
CCF

WMAP3
WMAP3

LSS data
ROSAT XRBHEAO1 XRB
NVSS
NVSS/HEAO1 XRB
APM
SDSS1(main, LRG)
combined
SDSS1(LRG)
NVSS
2MASS
SDSS4(LRG)
2MASS
APM
SDSS1(main,LRG)
NVSS+HEAO
SDSS4(main, LRG)
SDSS4(QSO)

Scranton et al. ’03 [119]Nolta et al. ’04 [95]Afshordi et al. ’04 [1]Padmanabhan et al. ’05 [97]Gaztanaga et al. ’06 [37]

WMAP1
WMAP1
WMAP1
WMAP1
WMAP1

Cabre et al. ’06 [18]Giannantonio et al. ’06 [39]

CCF
CCF
APS
APS
CCF

Detection
No detection
No detection
No detection
2-3σ
2.5σ
2.0, 3.0σ
3.6σ
> 2σ
2.6σ
2.5σ
2.5σ
4σ

DE Constraints––ΩΛ < 0.74–ΩΛ = 0.8+0.06−0.27 (2σ)–ΩΛ = 0.8+0.06−0.11 (2σ)–ΩΛ = 0.68−0.68 (2.2σ)–ΩΛ = 0.8+0.05−0.19 (2σ)ΩΛ = 0.70±0.05 (1σ)a)w = −1.02±0.17 (1σ)a)0.3 < zp < 0.80<z<20 < z < 0.2
0.2 < zp < 0.6
z ∼ 0.1z ∼ 0.15
z ∼ 0.3, 0.5z ∼ 0.9z ∼ 0.3, 0.5
0.1 < zp < 2.7

CCF
CCF

4.4σ
2 − 2.5σ

0<z<2
0<z<2

WLT
WLT

2.7σ
3.3σ

2MASS XSC
NVSS

0 < z < 0.2
0<z<2

APS
WLT

< 1σ
3.9σ b)ΩΛ ' 0.83
0.075 < Ωm < 0.475(1σ)−1.18 < w < −0.76(1σ)
0.41 < ΩΛ < 0.79(2σ)ΩΛ = 0.65+0.17−0.20 (1σ)
w = −0.70+0.35−0.50 (1σ)ΩΛ = 0.85+0.04 (2σ)ΩΛ = 0.63+0.18−0.17 (1σ)
w = −0.77+0.35−0.36 (1σ)

Pietrobon et al. ’06 [101]Vielva et al. ’06 [141]

WMAP3
WMAP1

NVSS
NVSS

Rassat et al. ’07 [111]McEwen et al. ’07 [84]

WMAP3
WMAP1

Giannantonio et al. ’08, ’12 [38, 42]

WMAP3,7

2MASS
SDSS6,8(main)
SDSS6,7(LRG)
SDSS6(QSO)
NVSS
HEAO
combined
NVSS
NVSS
2MASS
SDSS(LRG)SDSS(QSO)
NVSS
combined
SDSS6(LRG)
SDSS6(QSO)
NVSS
SDSS5 (LRG)
2SLAQ (LRG)
AAO (LRG)
NVSS
SDSS7
2MASS XSCWISE pre.
SDSS6 (LRG)
BOSS8d)SDSS8(main)
NVSS
SDSS6,7(Void)
BOSS8d)WISE full
WISE full (gal)
WISE full (AGN)z ∼ 0.2z ' 0.3
z ' 0.5
0<z<3
0<z<2
0<z<2

CCF

0<z<2
0<z<2
0 < z < 0.2
0.2 < zp < 0.6
0.5 < z < 2
0<z<3

CCF
WLT
CCF

0.4 < zp < 0.75
0 < zp < 30<z<30.2 < zs < 0.5
0.4 < zs < 0.7
0.5 < zs < 0.9
0<z<2

STK
CCF
C/A
CCF

0.5σ, 0.7σ
2.2σ, 2.2σ
2.2σ, 2.5σ
2.5σ, 2.3σ
3.3σ, 2.8σ
2.7σ, 2.4σ
4.5σ, 4.4σ
2.5σ
∼ 3σ
0.2 − 1.4σ
1.3 − 2.5σ
0.2 − 1.4σ
2.9σ
3.7σ
4.4σ
1.5 − 2.7σ
∼ 2 − 3σ
0.8σ
1.6σ
0.4σ
∼ 2σ
No detection
∼ 1σ
∼ 3σ
> 3σ
1.7σ
2.0σ
2.9σ
∼ 3σ
1.62 − 1.67σ∼ 1.0σ∼ 2.6σ∼ 1.2σRaccanelli et al. ’08 [110]McEwen et al. ’08 [85]Ho et al. ’08 [58]

WMAP3
WMAP3
WMAP3

Granett et al. ’08 [49]Xia et al. ’09 [145]
Hernandez-Monteagudo ’10 [53]Sawangwit et al. ’10 [117]

WMAP5
WMAP5
WMAP5
WMAP5

Lopez-Corredoira et al. ’10 [76]Francis & Peacock ’10 [35]Goto et al. ’12 [45]Flender et al. ’13 [32]Planck et al. XIX ’13 [105]

WMAP5
WMAP3
WMAP7
WMAP5
Planck1

Ilic et al. ’13 [62]Hernandez-Monteagudo et al. ’14 [54]Kovacs et al. ’14 [71]Ferraro et al. ’14 [31]

WMAP7
WMAP9
WMAP7
WMAP9

0 < zp < 0.3
0 < z < 0.3
0.4 < zp < 0.75
0 < zs < 0.7
0.1 < zp < 0.90<z<20 < zs < 0.7
0.15 < zp < 0.7
z ∼ 0.15
z ∼ 0.3z ∼ 1.1

CCF
APS
APS
STK
C/A/W

STK
C/A/W
APS
APS

Ωm = 0.20+0.19−0.11 (2σ)––ΩΛ = 0.746±0.09 (1σ)–Ωm = 0.273±0.019 (1σ)c)
–
–
–
–
–
–
–
–
–
–
–
–
–

–
–
–

Table 1 The score sheet for the ISW detection. If the redshift of the large-scale structure tracer
is measured spectroscopically, we denote the range with zs . We use zp for the range of photometric redshift, whereas we use z when the redshift is inferred by other methods like integrating theluminosity function, fitting the amplitude of cross correlation, or partial cross matching with the
known redshift sources. The numbers after survey name stand for the data release, e.g. WMAP1 is
the WMAP first-year data release, and SDSS4 is the SDSS fourth data release and so on. a) joint
constraints with the Type Ia supernovae (SNIa) data. b) include a posteriori selection c) WMAP5+BAO+SNIa+ISW d)
CMASS+LOWZ sample

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Fig. 3 Left panel shows the first detection of the ISW in 2004 by Boughn & Crittenden [12]. They
use the WMAP first-year data cross correlated with the NVSS radio galaxies and X-ray data and findthe ISW effect with 3.0 σ detection. Right panel shows the latest detection in 2013 by HernandezMonteagudo et al. [54, but see also [31]]. They use the WMAP ninth-year data cross correlated with
the LRG and CMASS galaxies observed by the BOSS and find the ISW effect with 1.6 σ detection.
sources can be distinguished since the ISW effect is only important at largest scales while the othereffects are dominant at much smaller scales.
3.1.1. Angular Power Spectrum. The observed quantities used in the literature can be classified into four: angular power spectrum, angular correlation function, wavelet and stacking. As weobserve the CMB temperature fluctuation projected on the sky, it can be expanded into the sphericalharmonic series
Z
aISW
lm =

∗
dΩn̂ ΘISW (n̂)Ylm
(n̂),

(30)

where n̂ is a unit vector pointing toward a two dimensional position on the sky, Ylm is the sphericalharmonic function, dΩn̂ is a volume element of the unit sphere and the integral is over the wholesky.
The expanded temperature fluctuation can be cross correlated with the density tracer X,
ZX∗
alm = dΩn̂ δX (n̂)Ylm
(n̂),
(31)
where X is the matter tracer field projected on to the sky byZ r∗
δX (n̂) =drδm (n̂r, τ )WX (r),

(32)

0

with the projection kernel WX . The upper bound of the integral is defined by the maximum distanceof the source that has a non-zero contribution to the projection. For the galaxy distribution, theprojection kernel is,
·Z
¸−1
2
2
Wgal (r) = b(r)r φ(r)drr φ(r)(33)
where φ is the radial selection function and b(r) is the galaxy bias which may depend on redshift
but not on the scale (by assumption). We have assumed that in the scale where the ISW effect isimportant, the galaxy number counts are linearly related to the underlying dark matter density but
strictly speaking the linear relation is valid only on very large scales. It breaks down on smaller
11/24

scales, introducing a possible source of systematics. However in most of the ISW studies it is typically enough to take into account the redshift evolution of the galaxy bias only and ignore a possible
scale dependence of the bias. The weak lensing convergence is another powerful tool to trace the
distribution of the dark matter, which has the kernel [see e.g. 6],
Z r∗
3
r0 − r
2r
Wκ (r) = Ωm H0
dr0 p(r0 ) 0 ,
(34)
2
a r
r
where p(r) is the distribution of galaxies whose shapes are measured and it is normalized asR
drp(r) = 1. From the statistical isotropy we can define the angular power spectrum (hereafter
denoted as APS) as
ISW−X
X∗δll0 δmm0 .
haISW
lm al0 m0 i = Cl

(35)

Here the ensemble average is over possible realizations of the Universe ; however, we have only one
Universe in reality. Thus Cl is estimated by averaging over 2l + 1 modes for each multipole l-mode,
m=lĈlISW−X =

X
1
X∗
aCMB
lm alm .
2l + 1(36)m=−lIn the practical observation, the whole sky is not observed. For example, we mask the region where
the signal is not reliable or significantly contaminated by the foreground such as Galactic dust or
synchrotron emission or zodiacal light of the solar system. The extra galactic radio sources should
also be masked. Then the cut sky harmonic coefficient, ãlm , is given nowZ∗ãlm = dΩn̂ W (n̂)ΘISW (n̂)Ylm
(n̂),
(37)
where W (n̂) is a window function ( or a mask function ) that takes 1 at the observed pixel and 0 at
the masked pixel. The power spectrum convolved with the mask function, C̃l ≡ hãlm ã∗lm i is called
the Pseudo power spectrum. It has a smaller amplitude typically by C̃l ' fsky Cl where fsky is thefraction of sky observed. The effect of the mask can be corrected in an unbiased manner with the
Pseudo Cl estimator (PCL) [29, 57], or quadratic maximum likelihood estimator (QML) [28, 131].
3.1.2. Angular Correlation Function. The cross correlation function (hereafter CCF) is definedas the Legendre transform of the power spectrum,C ISW−X (θ) =1 X
2(2l + 1)ClISW−X Pl (cos θ)bCMB
bXll pl ,
4π(38)lwhere pl and bl are the pixel and beam transfer functions, respectively. The inverse transform is
Z
¡
¢
ISW−X
2 −1
(39)
Cl
= 2π d cos θP (cos θ)C ISW−X (θ) bCMB
bX
l
l pl
The estimator of the correlation function is nothing but
P
ij ΘCMB (n̂i )δX (n̂j )wCMB (n̂i )wX (n̂j )
ISW−X
P
Ĉ
(θ) ≡
,
ij wCMB (n̂i )wX (n̂j )(40)

where cos θ = n̂i · n̂j . The function w is used to minimize the variance when the depth of the galaxy
survey or effective sensitivity to the CMB at each pixel is not uniform, at the expense of increasing
sample variance [33].
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3.1.3. Wavelet. Wavelet analysis is useful particularly when signals are localized in both configuration and frequency space. For the cross correlation study the covariance of the wavelet
coefficients (hereafter WLT) can be used [83, 101, 140, 141]. The WLT covariance at a given scalein configuration space R is defined as
1 X
ISW−X
CΨ
(R) =
ωCMB (R, n̂)ωX (R, n̂),
(41)
NR
n̂

where ω(R, n̂) is the wavelet coefficient at the position n̂. It can be obtained by convolving a map
with a wavelet function. Here we assume that the wavelet function is given by a Spherical Mexican
Hat (SMH) wavelet given by·³ y ´2 ¸2 ·³ y ´2 ¸122
Ψ(y, R) = √1+2−e−y /2R ,(42)
2R
2πN (R)p
where N (R) is a normalization constant, N (R) = R 1 + R2 /2 + R4 /4. The distance on thetangent plane is given by y = 2 tan(θ/2). Then the WLT coefficient is,ZωX (R, n̂) = dΩn̂0 X(n̂ − n̂0 )Ψ(θ0 , R),(43)
where the volume element dΩn̂ subtends an infinitesimal solid angle pointing toward n̂ = (θ, φ).
The observed WLT covariance can be compared with the theoretical calculation with this formula,
X 2l + 1
ISW−X,TH
ISW−X,THp2 Ψ2 (R)bCMB
bX
,
(44)
CΨ
(R) =
l
l Cl
4π l ll

where Ψl is the spherical harmonic coefficient of the SMH and ClISW−X,TH is the theoreticalprediction of the cross correlation APS between the ISW and mass tracer.
The choice of the wavelet function, equation (42), is not unique. Veilva et al. 2004 and Mukherjee& Wnag 2004 [88, 140] applied SMH wavelet and Cruz et al. 2006 [25] used the elliptical Mexican
Hat wavelet to find a prominent cold spot in the CMB map with more than 2 σ . However, Zhang &Huterer 2010 [147] claimed that the cold spot is found to be statistically significant only if the SMH
wavelet is used, but not with the tophat or Gaussian windows. As we mentioned the WLT is usefulfor finding localized features in the data while we should be careful for interpreting the results, given
the degree of freedom in choosing the wavelet functions.
3.1.4. Stacking. A stacking method is particularly useful for measuring the average temperature
profile of the CMB around clusters and voids [21, 49, 50, 62, 105] and for finding the correspondence
between the temperature fluctuation and the specific structure of the density field [49, 55, 98, 99].
The stacked CMB temperature can be expressed as
ZnX
S(R) =A−1dΩi ∆T (φi , θi )M (φi , θi )Ξ(θi , R)
(45)
i
i

R
where the normalization is Ai = dΩM (φi , θi ), M is the composite mask of the CMB and largescale structure, and Ξ is the filter function. For the compensated filter that eliminates the constantoffset of the temperature, the filter function is given as [see e.g. 21, 92],
 1, θi < R
Ξ(θi , R) =(46)
−1, R ≤ θi < cos−1 (2 cos R − 1) .
 0, otherwise

Here we take a local coordinate system in which the z-axis is pointing to the i-th cluster or void
center with θi , φi and dΩi denoting the polar and azimuthal angles and the volume element in this
13/24

local coordinate systems, respectively. The radius R is the physical scale instead of the angle which
satisfies the relation R = ϕi χ(zi ) where ϕi is the angle that subtends the size of a cluster or a void
lying at a comoving distance χ(zi ). In other words the CMB temperature profile is rescaled before
stacking, which increases the statistical significance of the correlation signal [62, 92]. Stacking the
CMB at the locations of clusters and voids gives a typical profile of the CMB temperature. Since thesign of the ISW temperature fluctuation is opposite at the voids and clusters, we take the difference
between the stacked temperature at voids and clusters, i.e.
S(R) = Scluster (R) − Svoid (R).

(47)

The interpretation of the significance of the detection of the ISW with the stacking analysis needs
careful treatment. As pointed out in Ref [55], if we look at the particular scale, the significance isprominent and have tension to the standard ΛCDM prediction but the significance goes down if we
combine all scales; it is so called a posteriori selection effect [7, 55, 62, 105].

3.2. systematic errors
As seen in table 1, the reported detection significances vary among papers and are often not consistent with each other, even if they use the same CMB and large-scale structure data sets. It is partly
because of the difference in the statistics they used and partly because of the imperfect control ofthe systematics. In this section, we discuss possible contamination which may affect the analysisof the ISW effect. The main contaminant of the CMB is foreground Galactic dust and synchrotron
emissions. On the other hand, the galaxy distribution as a tracer of the large-scale structure has also
uncertainty including the incomplete star-galaxy separation, magnification bias, and redshift distribution uncertainties. Furthermore, the SZ effect and point source contamination can be a source ofsystematics which is discussed in the later section.
3.2.1. Magnification bias. Suppose that we correlate the CMB with the number count of galaxies.
The observed number count of galaxies reflects the true, underlying clustering of galaxies, while it
can also generate an apparent (artificial) clustering in the sky due to various effects. The gravitationallens effect alters the number count of galaxies through two effects. It locally changes the area ofthe sky observed hence the number of galaxies observed. It also magnifies the light of the distantfaint galaxies, and thus enhances the number of galaxies observed in the vicinity of massive low-z
galaxies [e.g. 15]. Given that we observe the galaxy number density at a certain redshift bin, zi , theobserved number density can be written as a sum of two components, [77],
µ
obs
s
δgal
(n̂, zi ) = δgal(n̂, zi ) + δgal(n̂, zi ),(48)

s is the intrinsic galaxy or, in other words, unlensed galaxy number density and δ µ is thewhere δgal
gal
magnification bias correction. Now we make a modification to the equation (33). The underlying
three dimensional dark matter fluctuations projected into a redshift bin zi with the kernel is,Zsδgal (n̂, zi ) = dzb(z)W s (z, zi )δm [n̂χ(z)],(49)

where b(z) is a galaxy bias which depends on redshift and W s (z, zi ) is a kernel that projects galaxies
at z onto a redshift bin of zi . The explicit form of the kernel depends on the model assumed. If weassume that each galaxy has a redshift measured with a Gaussian error of σ(z), the kernel can be
14/24

written as

·
µ
¶
µ
¶¸
1
(i − 1)∆z − zi∆z − z√W (z, zi ) = N (z) erfc
− erfc √
,
2
2σ(z)
2σ(z)
s

(50)

where ∆z is the width of the redshift bin, erfc is the complementary error function and N (z)2isZ redshift distribution of galaxies which satisfies, N (z)dz = p(r)dr = r φ(r)dr normalized as
dzN (z) = 1.
µNow we derive the expression of δgal
. The gravitational lens deflects light and thus changes the
position of galaxies asn̂s = n̂ + δ n̂,(51)

where the superscript s denotes the quantity for unlensed galaxies, and δ n̂ is deflection angle. The
magnification can be given by the Jacobian of the mapping of n̂s → n̂: i.e.
¯ s¯¯ ∂ n̂ ¯−1¯.
A ≡ ¯¯(52)∂ n̂ ¯
The galaxy flux is magnified by
f = Af s ,(53)

where f is the observed flux and f s is the intrinsic galaxy flux. In the weak lensing limit, themagnification can be approximated as A ≈ 1 + 2κ, where κ is the so-called “convergence field”,
which satisfies |κ| ¿ 1. In this limit, the observed number density of galaxies can be expressed interms of the number density of unlensed galaxies as,
obs
s
s
δgal
(n̂, z) = δgal(n̂ + δ n̂, z) + (5α(z) − 2)κ[1 + δgal(n̂ + δ n̂, z)],(54)

where α is the logarithmic slope of the cumulative number counts of galaxies at the faint end¯
d log N (> m) ¯¯
.
(55)
α=
¯
dm
mlim
This definition of α is only true for galaxy populations that have a linear slope in the relation between
the logarithmic cumulative number and the magnitude. Though in most cases it is true, one cangeneralize the definition of α by introducing an efficiency function [61]. In the weak lensing limit,s (n̂ + δ n̂) and keeping thethe deflection angle δ n̂ is also a first order small quantity. Expanding δgal
first order term gives
obs
s
δgal
(z, n̂) = δgal(z, n̂) + (5α − 2)κ,(56)µ
where the second term of the RHS denotes the δgal(z, n̂). Using the expression of the convergenceκ for the sources distributed around the zi bin, we obtainZ
dz
3Ωm H02µ(5α(zi ) − 2)g(z, zi )(1 + z)δm [n̂χ(z)],δgal (n̂, zi ) =2
H(z)where χ is the comoving distance and the lensing efficiency function g is given byZ ∞χ(z 0 ) − χ(z)g(z, zi ) = χ(z)dz 0W (z 0 , zi ).
0)
χ(z
z

(57)

(58)

It has been shown that the magnification bias in the ISW analysis does not affect the constraints ondark energy parameters inferred from low-z samples; however, it can have a significant impact when
we cross-correlate the CMB with galaxy samples at z > 2 [77]. Thus, we need to be careful wheninterpreting the null detection of the ISW at high redshift.
15/24

3.2.2. Extinctions. The Galactic dust extinction may be a major source of systematics in themeasurement of the ISW effect [38, 42]. In the region where the Galactic dust extinction is high, thegalaxy colors tend to be red and alter the large-scale distribution of the galaxy sample dependingon the wavelength observed. In the high extinction region, we also expect the observed temperature
map to contain microwave emission from the Galactic dust which then produce a spurious correlations between the temperature map and the galaxy count data. The advantages to use the near- to
mid-infrared are to minimize the amount of the dust extinction, which enables us to mitigate the systematics. Correcting the effect of dust is essential for the cross-correlation study because it allows us
to use more sky area. As the ISW effect appears only on large angular scales where both the CMB
and galaxy data are almost sample variance limited, increasing the usable sky area is the only way
to increase statistical significance of the cross-correlation signal. [43, 102–104, 118, 146]. However
the dust correction contains uncertainty and it may affect the precise measurement. A more conservative approach is to simply mask out the contaminated region of the sky. Giannantonio et al. [41]exclude the high extinction region with Ar > 0.18 and 0.08 which leaves the usable sky fractionof fsky = 0.22 and 0.11, respectively. The latter has less contamination but larger statistical error
and is statistically consistent with the former to within 1 σ . Therefor the former cut appears to be
sufficient for this data set.
3.2.3. SZ and point sources. A correlation between the CMB and the large-scale structure is also
generated by the SZ effect and point sources. As high energy electron scatters CMB photon throughthe inverse Compton scattering in a high temperature cluster of galaxies, the SZ effect distorts theshape of the blackbody CMB spectrum. For that reason, the temperature change induced by theSZ effect depends on the frequency we observe; temperature decreases at low frequency while it
increases at high frequency and no change occurs at 217 GHz. Thus the SZ signal in the cross correlation can in principle be isolated with multi-frequency observations. Furthermore, the characteristic
angular scale that the SZ effect correlates with the large-scale structure is smaller than that of the
ISW effect. In the harmonics space, the ISW peaks at l = 2 and becomes negligible at l = 30 while
the SZ effect becomes important at l > 100 [1, 56, 119]. The point sources found in the CMB map
are extra-galactic sources such as AGNs and dusty galaxies. They are bright in low (AGNs) and high(dusty galaxies) frequency bands. They are also visible in optical and infrared bands, where many
galaxy survey data are available. They also produce a correlation between the CMB and large-scale
structure; however, it can be distinguished from the ISW signal because point sources produce a
correlation at much smaller scales than that of the ISW effect [1, 30, 60]. This discrimination by
scale is possible only for the power spectrum in harmonics space but impossible for the correlation
function in the configuration space because, in the harmonic space, multipole scales where ISW
and point sources signals become prominent are different while in the configuration space, different
angular scales and the different physical origins do not have one to one correspondance [53, 54].
Although the point sources can be removed by masking if they are resolved by CMB experiments,
unresolved galaxies may still bring an extra-correlation signal.

4. Application to the Cosmology
4.1. Constraints On Dark Energy in Λ- and w-CDM modelAs first pointed out by Crittenden & Turok 1996 [24], the ISW effect can be used as a powerfulprobe of dark energy. In this section we describe the background Universe by the flat FRW metricwith matter and generalized time varying dark energy. Then we can write the Friedmann equation
16/24

(D/a)’

Fig. 4 Dependence of the ISW effect on the cosmological models, (D/a)0 , as a function of redshift
is shown. The less amount of dark energy makes Universe more close to the EdS, and thus produces
weaker ISW effect (black solid and red dashed). Less negative equation of state, w (green dotdashed) makes dark energy dominate at earlier epoch and enhances the ISW effect. That model canbe tested by the cross correlation with high-z objects like QSOs or radio galaxies.
as
H 2 (z) = H02 [Ωm (1 + z)3 + ΩDE ζ(z, w)],(59)

where the function ζ is given by
· Z z
¸
0
0 1 + w(z )ζ(z, w) = exp 3dz,
1 + z00(60)

where w is in general a function of redshift but it is parametrized as w = const. or w(z) = w0 +z
where w0 and wa are constants [e.g. 75]. The explicit cosmological dependence of the ISW
wa 1+zeffect originates from ∂τ (D/a) = HD(f − 1) and the matter power spectrum. When we combine
other cosmological data sets such as galaxy clustering or weak lensing, the ISW effect provides
information on dark energy via HD(f − 1) term. This quantity is shown in Figure 4 as a functionof redshift. Less amount of dark energy makes Universe more close to the EdS Universe, and thus
produces weaker ISW effect (the black solid and the red dashed lines). Larger value of equation ofstate, i.e. w > −1 (the green dot-dashed line), makes dark energy prominent at earlier time which
enhances the ISW effect. Such an early dark energy models can be tested by the cross correlation
with the high-z density tracers like quasars, which lie at z < 3 [114].
Table 1 shows the summary of reported detections of the ISW effect and the constraints on darkenergy models. Boughn and Crittenden 2002 [11] put the first constraint on dark energy parameterby cross-correlating the NVSS galaxies number counts with the CMB temperature observed by the
COBE. Although they do not find a significant detection of the ISW effect, it gives an upper limitof the amount of dark energy, ΩΛ < 0.74. The redshift distribution of the NVSS galaxies is inferredfrom an integral of a luminosity function of the radio galaxies [27]. With the inferred redshift distribution, the galaxy auto-correlation power spectrum can nicely account for the observed clusteringof the radio galaxies but it is not sufficient to validate the true redshift distribution. Fosalba & Gaztanaga 2004 [33] present more stringent constraints on the ΛCDM model with the APM galaxy
survey and the WMAP data. They fixed the Hubble parameter, σ8 and the constant bias parameter
17/24

with assuming a flat geometry and obtained 0.53 < ΩΛ < 0.86 (2σ). Fosalba et al. 2003 [34] findsimilar result from the combined analysis of the APM galaxies, the SDSS main galaxies and the
LRG sample, 0.69 < ΩΛ < 0.87 (2σ). They utilize the cross-correlation function for the measurement of the ISW effect. The covariance matrix is estimated in two ways: a Jack Knife resampling
and a Monte Carlo simulation. The Jack Knife (JK) error is consistent with the Monte Carlo (MC) on
large scales while the JK error is significantly under-estimated on scales smaller than θ < 2 degrees.
Nolta et al. 2004 [95] revisit the NVSS cross correlation with the WMAP. Thanks to the greater
sensitivity of the WMAP, they find the 2.6σ detection of the ISW effect with CCF, and obtain theconstraint of ΩΛ > 0 at 2σ . They also reject the closed Universe at 3σ significance in which the
ISW signal shows a negative cross correlation with the large-scale structure. The covariance matrix
is estimated by the MC simulations for randomly generated CMB spectra, while keeping large-scale
structure unchanged. This may underestimate the error since the large-scale structure and primary
CMB as well as the cross correlation between them all contribute to the covariance matrix as shownin equation (62).
For a joint analysis of the ISW and the galaxy clustering APS, the logarithm of the likelihood
function, χ2 ≡ −2 ln L, is given byχ2 = (xiobs − hxi i)[C−1 ]ij (xjobs − hxj i),(61)

where xobs is the observed APS, ClgT , and the index i represents both the angular l-bin and sample
used, when we use multiple samples for the large-scale structure tracers. The covariance matrix C
can be estimated in different ways. The Fisher matrix formalism gives the theoretical estimate of
the covariance matrix, while the Jack Knife (JK) or the bootstrap (BS) resampling give the estimateof the covariance based on the observational data itself. The latter tends to underestimate the trueerror due to the lack of the modes larger than the survey volume. The Monte Carlo simulation isparticularly useful when the data are not Gaussian, or the processes such as complicated radial or
angular selection functions are taken into account. For the Gaussian case, the covariance matrix canbe simply calculated by the Fisher formula,C=

1[C̃ g C̃ T + (ClgT )2 ],fsky (2l + 1) l l(62)

where C̃ g is the power spectrum of the galaxies with the shot noise, C̃lg = Clg + n̄−1
g where ng
is the number density of galaxies and C̃lT = ClT + NlT where NlT is the noise power spectrum of
the CMB. Padmanabhan et al. 2005 [97] use the fourth data release of the SDSS LRG photometricsample with accurate photometric redshifts ∆zp = 0.03. The cross-correlation APS is used as an
estimator and they introduce a quadratic estimator [121, 131] which enables us a nearly maximum
likelihood estimation. Gaztanaga et al. 2006 [37] explore wCDM model with multiple large-scale
structure tracers: 2MASS, APM, SDSS, NVSS and HEAO. They point out that the parameter degeneracy between ΩΛ and Ωm obtained from the ISW effect is perpendicular to those from SNIa
observation and thus the combination of the ISW effect with other cosmological probes can bea powerful tool to constrain those parameters. Combining the ISW data with the SNIa data, theyobtain the cosmological constraints of ΩΛ = 0.7 ± 0.05 and w = −1.02 ± 0.17. Using the samedata set, Corasaniti et al. 2005 [23] extend the analysis to constraining the model where dark energy
has clustering with a finite sound speed but do not find a meaningful constraint on the sound speedof dark energy. Some works [84, 101, 141] use wavelet for constraining dark energy models. As we
mentioned before, the wavelet analysis has the advantage of detecting localized signals. Therefore, it
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Fig. 5 The cross correlation power spectrum of the ISW and NVSS galaxy. The lines show thebest fit fNL model (black solid), fNL = 100 (blue dotted), fNL = −100 (red dashed), and fNL =800 (green dot-dashed) while keeping the other cosmological parameters unchanged. The figure isadopted from Slosar et al. 2008 [126].
has the higher significance than other method in terms of the signal to noise ratio of finding a signal
at a given scale [141]. They find the signal of the ISW effect with higher significance of ∼ 3σ thanother methods. The error covariance for the wavelet is derived in a straightforward manner using
equations (44) and (62),·hi2 X 2l + 1³´2 ¸¡ CMB X ¢2
ISW−X,TH
ISW−X
4 4
CMB X
∆ CΨ
(R) =
p Ψ (R) bl
bl
ClCl + Cl.
(63)
16π 2 l llThey present consistent results with the current ΩΛ constraint [e.g. 107] but slightly lower values arefavored. The equation of state parameter favors larger (less negative) value but it is still consistent
with a cosmological constant within 1σ level.

4.2. Non Gaussianity
Although the recent observation of the CMB by the Planck shows that the non Gaussianity (NG)of the primordial fluctuation is consistent with zero, it is still worth discussing it for the future
observations which would give much tighter constraints on (or potentially detect) NG. In this section,
we describe the possible impacts of the NG of the primordial fluctuation on the ISW effect. We
assume that the NG has the form of [36, 70],
Φ = φ + fNL (φ2 − hφ2 i)(64)

with the parameter fNL which describes the amplitude of NG of the primordial fluctuation, and φ isa random Gaussian variable.
The ISW effect basically plays two roles in the context of NG. First, the ISW effect can be a proxyof mass of the large-scale structure especially on large scales. It is difficult to discriminate the primordial NG from those which originate from the non linear gravitational evolution of the structure.
However, the non-zero fNL alters the number of rare objects formed responding to the initial gravitational potentials. The number of rare objects is enhanced if the fNL is positive but depressed forthe negative fNL . Thus the effect of non-zero fNL on the halo or galaxy power spectrum becomes
19/24

prominent on large scales [26]. As a result, the NG bias acquires a correction of b → b + ∆b, where∆b(k) = 2(1 − b)fNL δc

3Ωm H02,
2D(a)k 2(65)

where δc is the critical density, b is the usual Eulerian constant bias. Thus the NG can be constrained through the ISW-galaxy cross correlation. The galaxy-galaxy auto-correlation is mainly
used for constraining the NG and the ISW-galaxy cross correlation has relatively lower statistical power. However, the cross correlation is less sensitive to the systematic effects and thus givesa robust measurement; they are complementary to each other. Figure 5 compares the cross correlation power spectrum of the ISW and the NVSS galaxies with the standard ΛCDM prediction
and NG predictions keeping all the parameters other than fNL unchanged [126]. Slosar et al. 2008[126] found no significant detection of the NG. They obtained a constraint of fNL = 105+647−337 fromusing only the ISW-NVSS cross correlation, while Afshordi & Tolley 2008 [2] found a 2 σ hint
for the NG, fNL = 236 ± 127. Xia et al. 2011 [144] also found a weak evidence for the positivefNL , fNL = 74 ± 40(1σ) by combining the auto correlation of the NVSS and the NVSS-ISW crosscorrelation. They used the redshift distribution of the NVSS galaxies which was directly measured
by the limited number of spectroscopic galaxies [16] whereas Slosar et al. 2008 [126] used one
measured from the NVSS clustering itself, which made significant difference in the fNL . Lately
Giannantonio et al. 2014 [41] point out that the NVSS catalog contains a serious systematic error:
the number density of galaxies depends on declination and right ascension. This systematics largely
affects auto correlation of the NVSS galaxies while the impact of the systematics on the cross correlation is small. Giannantonio and Percival 2014 [40] revisit the NG constraints via cross correlationof the ISW effect with a suite of galaxy samples and find no evidence of the NG, fNL = 46 ± 68 byusing the ISW-galaxy cross correlation only. They also take into account the cross correlation of the
CMB-lensing with galaxy samples and obtain a joint constraint of fNL = 12 ± 21 in the end. Thisis consistent with the CMB bispectrum analysis of the Planck [109].
Second, the ISW effect may bring a systematic effect to the measurement of the CMB bispectrum
through the ISW-lensing correlation [44, 109]. The observed CMB temperature can be decomposed into three parts; primordial CMB at last scattering surface, lensed-CMB, and the secondary
anisotropy that is generated at low redshifts such as the ISW or the SZ. For the Gaussian field, all
the information is encoded into the power spectrum or two point correlation function so that thebispectrum or three point function is the lowest statistics that describes the NG clustering. However,the ISW effect and the CMB-lensing are correlated with each other because the large-scale structurethat makes CMB photon deflect via gravitational lens also induces the ISW effect at low redshift.
Therefore it mimics the primordial NG to produce a non-zero signal in the CMB bispectrum. The
ISW effect is important on large scales and the CMB-lensing appears on much smaller scales, so
the ISW-lensing bispectrum peaks at a squeezed configuration of the triangle. Thus the correlation
between the CMB-lens and the ISW effect brings a bias on the measurement of the primordial NG
especially for the local type by ∆fNL ' 10 [20, 69] which may seriously affect the current measurement; i.e. σfNL = 5.8 [109]. On the other hand, the ISW-lensing bias to the other configurationsof the bispectrum is negligible [52, 65, 81, 82, 123, 127].

5. Summary
In this article we review the ISW effect induced by linear and non-linear structures of the Universe.
It is well known that the Sachs-Wolfe effect and the ISW effect are simultaneously derived from
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the cosmological perturbation theory. The ISW effect can be generated by scalar, vector and tensor
mode fluctuations of the metric but the significant contribution comes from the scalar mode, whichis related to the time variation of the matter density fluctuation. The recent detection of the tensormode CMB polarization claimed by the BICEP2 collaboration [8] can indeed be the polarization
generated by the tensor-mode ISW effect from primordial gravitational waves. The time variation of
the gravitational potential in the standard ΛCDM Universe comes from: 1) the coherent decay of thegravitational potential due to the accelerating expansion of the Universe, 2) the isotropic clustering
inflow of mass lump toward the center of mass, and 3) the transverse motions of the clusters. The
latter two effects are too tiny to detect with the current CMB experiments but it might be possible todetect with future experiments having finer angular resolutions such as ACTPol [90], SPTPol [5] or
COrE [132].
We also review the observational studies of the ISW effect. The ISW effect was first detectedby the cross correlation of the CMB observed by the WMAP with the number counts of the radiogalaxies measured by the NVSS. The significance of detection was 2 − 3σ [12]. Subsequently anumber of detections have been reported with various tracers of dark matter using a variety ofstatistical methods. As the ISW effect reflects the large-scale fluctuations in the Universe, they canbe used for constraining the cosmological models especially at low redshifts (z < 1). The statisticalerror of the ISW effect is mostly dominated by sample variance but the detection significances vary
among papers and are often not consistent with each other. It is partly due to the consequence ofthe different statistics they used, different treatments of the foreground, or different estimates orassumptions on the redshift distribution of galaxies. As we acquire more knowledge on those issues,
we may be able to understand inconsistencies seen in table 1. All sky polarization data from thePlanck will be useful for further studying the dust model of our Galaxy as well as the confirmationof the large amplitude of tensor mode fluctuation discovered by the BICEP2. In addition to this,complete BOSS spectroscopic galaxies and QSOs samples can be used for calibrating the redshift
distribution of galaxies and QSOs. Therefore both the Planck and the BOSS data which are going to
be delivered soon would allow us to extensively study the ISW effect with better understandings ofthe systematics.

acknowledgement
The author would like to thank Eiichiro Komatsu for the careful reading of this article and giving
many useful comments. This work is supported in part by the FIRST program ”Subaru Measurements of Images and Redshifts (SuMRe)” initiated by the Council for Science and Technology
Policy (CSTP).
References
[1]
[2]
[3]
[4]
[5]

N. Afshordi, Y.-S. Loh, and M. A. Strauss, Phys. Rev. D, 69(8), 083524 (April 2004).
N. Afshordi and A. J. Tolley, Phys. Rev. D, 78(12), 123507 (December 2008).
N. Aghanim, S. Prunet, O. Forni, and F. R. Bouchet, Astronomy & Astrophysics, 334, 409–419 (June 1998).
O. Aso, M. Hattori, and T. Futamase, ApJL, 576, L5–L7 (September 2002).
J. E. Austermann, K. A. Aird, J. A. Beall, D. Becker, A. Bender, B. A. Benson, L. E. Bleem, J. Britton, et al., SPTpol: an instrument for CMB polarization measurements with the South Pole Telescope, In Society of Photo-Optical
Instrumentation Engineers (SPIE) Conference Series, volume 8452 of Society of Photo-Optical Instrumentation
Engineers (SPIE) Conference Series (September 2012).
[6] M. Bartelmann and P. Schneider, Phys.Rep., 340, 291–472 (January 2001).
[7] C. L. Bennett, R. S. Hill, G. Hinshaw, D. Larson, K. M. Smith, J. Dunkley, B. Gold, M. Halpern, et al., ApJS,
192, 17 (February 2011).
[8] BICEP2 Collaboration, P. A. R Ade, R. W. Aikin, D. Barkats, S. J. Benton, C. A. Bischoff, J. J. Bock, J. A. Brevik,

21/24and et al., ArXiv 1403.3985 (March 2014).
[9] M. Birkinshaw, Moving Gravitational Lenses, In J. M. Moran, J. N. Hewitt, and K.-Y. Lo, editors, Gravitational
Lenses, volume 330 of Lecture Notes in Physics, Berlin Springer Verlag, page 59 (1989).
[10] M. Birkinshaw and S. F. Gull, Nature, 302, 315–317 (March 1983).
[11] S. P. Boughn and R. G. Crittenden, Physical Review Letters, 88(2), 021302 (January 2002).
[12] S. P. Boughn and R. G. Crittenden, Nature, 427, 45–47 (January 2004).
[13] S. P. Boughn and R. G. Crittenden, New Astron. Rev., 49, 75–78 (May 2005).
[14] S. P. Boughn, R. G. Crittenden, and N. G. Turok, New Astron., 3, 275–291 (July 1998).
[15] T. J. Broadhurst, A. N. Taylor, and J. A. Peacock, ApJ, 438, 49–61 (January 1995).
[16] M. H. Brookes, P. N. Best, J. A. Peacock, H. J. A. Röttgering, and J. S. Dunlop, MNRAS, 385, 1297–1326 (April2008).
[17] E. F. Bunn and N. Sugiyama, ApJ, 446, 49 (June 1995).
[18] A. Cabré, E. Gaztañaga, M. Manera, P. Fosalba, and F. Castander, MNRAS, 372, L23–L27 (October 2006).
[19] Y.-C. Cai, S. Cole, A. Jenkins, and C. Frenk, MNRAS, 396, 772–778 (June 2009).
[20] Y.-C. Cai, S. Cole, A. Jenkins, and C. S. Frenk, MNRAS, 407, 201–224 (September 2010).
[21] Y.-C. Cai, M. C. Neyrinck, I. Szapudi, S. Cole, and C. S. Frenk, ArXiv 1301.6136 (January 2013).
[22] A. Cooray and R. Sheth, Phys.Rep., 372, 1–129 (December 2002).
[23] P.-S. Corasaniti, T. Giannantonio, and A. Melchiorri, Phys. Rev. D, 71(12), 123521 (June 2005).
[24] R. G. Crittenden and N. Turok, Physical Review Letters, 76, 575–578 (January 1996).
[25] M. Cruz, M. Tucci, E. Martı́nez-González, and P. Vielva, MNRAS, 369, 57–67 (June 2006).
[26] N. Dalal, O. Doré, D. Huterer, and A. Shirokov, Phys. Rev. D, 77(12), 123514 (June 2008).
[27] J. S. Dunlop and J. A. Peacock, MNRAS, 247, 19 (November 1990).
[28] G. Efstathiou, MNRAS, 348, 885–896 (March 2004).
[29] G. Efstathiou, MNRAS, 349, 603–626 (April 2004).
[30] H. K. Eriksen, G. Huey, R. Saha, F. K. Hansen, J. Dick, A. J. Banday, K. M. Górski, P. Jain, and et al., ApJ, 656,
641–652 (February 2007).
[31] S. Ferraro, B. D. Sherwin, and D. N. Spergel, ArXiv 1401.1193 (January 2014).
[32] S. Flender, S. Hotchkiss, and S. Nadathur, JCAP, 2, 13 (February 2013).
[33] P. Fosalba and E. Gaztañaga, MNRAS, 350, L37–L41 (May 2004).
[34] P. Fosalba, E. Gaztañaga, and F. J. Castander, ApJL, 597, L89–L92 (November 2003).
[35] C. L. Francis and J. A. Peacock, MNRAS, 406, 2–13 (July 2010).
[36] A. Gangui, F. Lucchin, S. Matarrese, and S. Mollerach, ApJ, 430, 447–457 (August 1994).
[37] E. Gaztañaga, M. Manera, and T. Multamäki, MNRAS, 365, 171–177 (January 2006).
[38] T. Giannantonio, R. Crittenden, R. Nichol, and A. J. Ross, MNRAS, 426, 2581–2599 (November 2012).
[39] T. Giannantonio, R. G. Crittenden, R. C. Nichol, R. Scranton, G. T. Richards, A. D. Myers, R. J. Brunner, A. G.
Gray, and et al., Phys. Rev. D, 74(6), 063520 (September 2006).
[40] T. Giannantonio and W. J. Percival, MNRAS, in press (ArXiv 1312.5154) (March 2014).
[41] T. Giannantonio, A. J. Ross, W. J. Percival, R. Crittenden, D. Bacher, M. Kilbinger, R. Nichol, and J. Weller,
Phys. Rev. D, 89(2), 023511 (January 2014).
[42] T. Giannantonio, R. Scranton, R. G. Crittenden, R. C. Nichol, S. P. Boughn, A. D. Myers, and G. T. Richards,
Phys. Rev. D, 77(12), 123520 (June 2008).
[43] B. Gold, N. Odegard, J. L. Weiland, R. S. Hill, A. Kogut, C. L. Bennett, G. Hinshaw, X. Chen, and et al., ApJS,
192, 15 (February 2011).
[44] D. M. Goldberg and D. N. Spergel, Phys. Rev. D, 59(10), 103002 (May 1999).
[45] T. Goto, I. Szapudi, and B. R. Granett, MNRAS, 422, L77 (May 2012).
[46] N. Gouda and N. Sugiyama, ApJL, 395, L59–L63 (August 1992).
[47] N. Gouda, N. Sugiyama, and M. Sasaki, ApJL, 372, L49–L52 (May 1991).
[48] N. Gouda, N. Sugiyama, and M. Sasaki, Progress of Theoretical Physics, 85, 1023–1039 (May 1991).
[49] B. R. Granett, M. C. Neyrinck, and I. Szapudi, ApJL, 683, L99–L102 (August 2008).
[50] B. R. Granett, M. C. Neyrinck, and I. Szapudi, ApJ, 701, 414–422 (August 2009).
[51] A. J. S. Hamilton, MNRAS, 312, 257–284 (February 2000).
[52] D. Hanson, K. M. Smith, A. Challinor, and M. Liguori, Phys. Rev. D, 80(8), 083004 (October 2009).
[53] C. Hernández-Monteagudo, Astronomy & Astrophysics, 520, A101 (September 2010).
[54] C. Hernández-Monteagudo, A. J. Ross, A. Cuesta, R. Génova-Santos, J.-Q. Xia, F. Prada, G. Rossi, M. Neyrinck,
and et al., MNRAS, 438, 1724–1740 (February 2014).
[55] C. Hernández-Monteagudo and R. E. Smith, MNRAS, 435, 1094–1107 (October 2013).
[56] J. C. Hill and D. N. Spergel, ArXiv 1312.4525 (December 2013).
[57] E. Hivon, K. M. Górski, C. B. Netterfield, B. P. Crill, S. Prunet, and F. Hansen, ApJ, 567, 2–17 (March 2002).
[58] S. Ho, C. Hirata, N. Padmanabhan, U. Seljak, and N. Bahcall, Phys. Rev. D, 78(4), 043519 (August 2008).
[59] W. Hu, Phys. Rev. D, 62(4), 043007 (August 2000).
[60] K. M. Huffenberger, H. K. Eriksen, and F. K. Hansen, ApJL, 651, L81–L84 (November 2006).

22/24

[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
[78]
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]

[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
[100]
[101]
[102]
[103]
[104]
[105]
[106]

L. Hui, E. Gaztañaga, and M. Loverde, Phys. Rev. D, 76(10), 103502 (November 2007).
S. Ilić, M. Langer, and M. Douspis, Astronomy & Astrophysics, 556, A51 (August 2013).
B. Jain and E. Bertschinger, ApJ, 431, 495–505 (August 1994).
A. Jenkins, C. S. Frenk, S. D. M. White, J. M. Colberg, S. Cole, A. E. Evrard, H. M. P. Couchman, and N. Yoshida,
MNRAS, 321, 372–384 (February 2001).
V. Junk and E. Komatsu, Phys. Rev. D, 85(12), 123524 (June 2012).
N. Kaiser and A. Stebbins, Nature, 310, 391–393 (August 1984).
M. Kawasaki, T. Sekiguchi, T. Takahashi, and S. Yokoyama, ArXiv 1404.2175 (April 2014).
R. Kneissl, R. Egger, G. Hasinger, A. M. Soltan, and J. Truemper, Astronomy & Astrophysics, 320, 685–695(April 1997).
E. Komatsu, Classical and Quantum Gravity, 27(12), 124010 (June 2010).
E. Komatsu and D. N. Spergel, Phys. Rev. D, 63(6), 063002 (March 2001).
A. Kovács and I. Szapudi, ArXiv 1401.0156 (December 2014).
A. Lewis and A. Challinor, Phys.Rep., 429, 1–65 (June 2006).
A. R. Liddle and D. H. Lyth, Phys.Rep., 231, 1–105 (August 1993).
A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure, June 2000).
E. V. Linder, Physical Review Letters, 90(9), 091301 (March 2003).
M. López-Corredoira, F. Sylos Labini, and J. Betancort-Rijo, Astronomy & Astrophysics, 513, A3 (April 2010).
M. Loverde, L. Hui, and E. Gaztañaga, Phys. Rev. D, 75(4), 043519 (February 2007).
C.-P. Ma and E. Bertschinger, ApJ, 455, 7 (December 1995).
C.-P. Ma and J. N. Fry, Physical Review Letters, 88(21), 211301 (May 2002).
N. Makino, M. Sasaki, and Y. Suto, Phys. Rev. D, 46, 585–602 (July 1992).
A. Mangilli and L. Verde, Phys. Rev. D, 80(12), 123007 (December 2009).
A. Mangilli, B. Wandelt, F. Elsner, and M. Liguori, Astronomy & Astrophysics, 555, A82 (July 2013).
E. Martı́nez-González, J. E. Gallegos, F. Argüeso, L. Cayón, and J. L. Sanz, MNRAS, 336, 22–32 (October 2002).
J. D. McEwen, P. Vielva, M. P. Hobson, E. Martı́nez-González, and A. N. Lasenby, MNRAS, 376, 1211–1226(April 2007).
J. D. McEwen, Y. Wiaux, M. P. Hobson, P. Vandergheynst, and A. N. Lasenby, MNRAS, 384, 1289–1300 (March
2008).
S. M. Molnar and M. Birkinshaw, ApJ, 537, 542–554 (July 2000).
V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Phys.Rep., 215, 203–333 (June 1992).
P. Mukherjee and Y. Wang, ApJ, 613, 51–60 (September 2004).
J. F. Navarro, C. S. Frenk, and S. D. M. White, ApJ, 490, 493 (December 1997).
M. D. Niemack, P. A. R. Ade, J. Aguirre, F. Barrientos, J. A. Beall, J. R. Bond, J. Britton, H. M. Cho, et al.,
ACTPol: a polarization-sensitive receiver for the Atacama Cosmology Telescope, In Society of Photo-Optical
Instrumentation Engineers (SPIE) Conference Series, volume 7741 of Society of Photo-Optical Instrumentation
Engineers (SPIE) Conference Series (July 2010).
A. J. Nishizawa, Ph.D. Thesis (March 2008).
A. J. Nishizawa and K. T. Inoue, in preparation (2014).
A. J. Nishizawa, E. Komatsu, N. Yoshida, R. Takahashi, and N. Sugiyama, ApJL, 676, L93–L96 (April 2008).
H. Noh and J.-C. Hwang, Phys. Rev. D, 69(10), 104011 (May 2004).
M. R. Nolta, E. L. Wright, L. Page, C. L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, and et al., ApJ,
608, 10–15 (June 2004).
J. P. Ostriker and E. T. Vishniac, ApJL, 306, L51–L54 (July 1986).
N. Padmanabhan, C. M. Hirata, U. Seljak, D. J. Schlegel, J. Brinkmann, and D. P. Schneider, Phys. Rev. D, 72(4),043525 (August 2005).
P. Pápai and I. Szapudi, ApJ, 725, 2078–2086 (December 2010).
P. Pápai, I. Szapudi, and B. R. Granett, ApJ, 732, 27 (May 2011).
A. A. Penzias and R. W. Wilson, ApJ, 142, 419–421 (July 1965).
D. Pietrobon, A. Balbi, and D. Marinucci, Phys. Rev. D, 74(4), 043524 (August 2006).
Planck Collaboration, P. A. R. Ade, N. Aghanim, M. I. R. Alves, C. Armitage-Caplan, M. Arnaud, M. Ashdown,
F. Atrio-Barandela, J. Aumont, C. Baccigaluppi, and et al., ArXiv 1303.5073 (March 2013).
Planck Collaboration, P. A. R. Ade, N. Aghanim, M. I. R. Alves, M. Arnaud, M. Ashdown, F. Atrio-Barandela,J. Aumont, C. Baccigalupi, A. J. Banday, and et al., ArXiv 1307.6815 (July 2013).
Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. AtrioBarandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al., ArXiv 1303.5072 (March 2013).
Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. AtrioBarandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al., ArXiv 1303.5079 (March 2013).
Planck collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. AtrioBarandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al., ArXiv 1303.5075 (March 2013).

23/24

[107] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. AtrioBarandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al., ArXiv 1303.5076 (March 2013).
[108] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. AtrioBarandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al., ArXiv 1303.5077 (March 2013).
[109] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. AtrioBarandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al., ArXiv 1303.5084 (March 2013).
[110] A. Raccanelli, A. Bonaldi, M. Negrello, S. Matarrese, G. Tormen, and G. de Zotti, MNRAS, 386, 2161–2166(June 2008).
[111] A. Rassat, K. Land, O. Lahav, and F. B. Abdalla, MNRAS, 377, 1085–1094 (May 2007).
[112] M. J. Rees and D. W. Sciama, Nature, 213, 374–375 (January 1967).
[113] M. J. Rees and D. W. Sciama, Nature, 217, 511–516 (February 1968).
[114] N. P. Ross, A. D. Myers, E. S. Sheldon, C. Yèche, M. A. Strauss, J. Bovy, J. A. Kirkpatrick, G. T. Richards, andet al., ApJS, 199, 3 (March 2012).
[115] J. A. Rubiño-Martı́n, C. Hernández-Monteagudo, and T. A. Enßlin, Astronomy & Astrophysics, 419, 439–447(May 2004).
[116] R. K. Sachs and A. M. Wolfe, ApJ, 147, 73 (January 1967).
[117] U. Sawangwit, T. Shanks, R. D. Cannon, S. M. Croom, N. P. Ross, and D. A. Wake, MNRAS, 402, 2228–2244(March 2010).
[118] D. J. Schlegel, D. P. Finkbeiner, and M. Davis, ApJ, 500, 525 (June 1998).
[119] R. Scranton, A. J. Connolly, R. C. Nichol, A. Stebbins, I. Szapudi, D. J. Eisenstein, N. Afshordi, T. Budavari, andet al., ArXiv astro-ph/0307335 (July 2003).
[120] U. Seljak, ApJ, 460, 549 (April 1996).
[121] U. Seljak, ApJ, 506, 64–79 (October 1998).
[122] U. Seljak, MNRAS, 318, 203–213 (October 2000).
[123] P. Serra and A. Cooray, Phys. Rev. D, 77(10), 107305 (May 2008).
[124] R. K. Sheth and A. Diaferio, MNRAS, 322, 901–917 (April 2001).
[125] R. K. Sheth and G. Tormen, MNRAS, 308, 119–126 (September 1999).
[126] A. Slosar, C. Hirata, U. Seljak, S. Ho, and N. Padmanabhan, JCAP, 8, 31 (August 2008).
[127] K. M. Smith and M. Zaldarriaga, MNRAS, 417, 2–19 (October 2011).
[128] R. E. Smith, C. Hernández-Monteagudo, and U. Seljak, Phys. Rev. D, 80(6), 063528 (September 2009).
[129] P. Strittmatter and J. Faulkner, Nature, 212, 1441–1442 (December 1966).
[130] R. A. Sunyaev and Y. B. Zeldovich, Astrophys. & Space Science, 7, 3–19 (April 1970).
[131] M. Tegmark, Phys. Rev. D, 55, 5895–5907 (May 1997).
[132] The COrE Collaboration, C. Armitage-Caplan, M. Avillez, D. Barbosa, A. Banday, N. Bartolo, R. Battye,
J. Bernard, et al., ArXiv 1102.2181 (February 2011).
[133] J. L. Tinker, B. E. Robertson, A. V. Kravtsov, A. Klypin, M. S. Warren, G. Yepes, and S. Gottlöber, ApJ, 724,
878–886 (December 2010).
[134] K. Tomita, Phys. Rev. D, 72(10), 103506 (November 2005).
[135] K. Tomita, Phys. Rev. D, 72(4), 043526 (August 2005).
[136] K. Tomita and K. T. Inoue, Phys. Rev. D, 77(10), 103522 (May 2008).
[137] R. Tuluie and P. Laguna, ApJL, 445, L73–L76 (June 1995).
[138] R. Tuluie, P. Laguna, and P. Anninos, ApJ, 463, 15 (May 1996).
[139] L. Verde and D. N. Spergel, Phys. Rev. D, 65(4), 043007 (February 2002).
[140] P. Vielva, E. Martı́nez-González, R. B. Barreiro, J. L. Sanz, and L. Cayón, ApJ, 609, 22–34 (July 2004).
[141] P. Vielva, E. Martı́nez-González, and M. Tucci, MNRAS, 365, 891–901 (January 2006).
[142] M. S. Warren, K. Abazajian, D. E. Holz, and L. Teodoro, ApJ, 646, 881–885 (August 2006).
[143] D. T. Wilkinson and R. B. Partridge, Nature, 215, 719 (August 1967).
[144] J.-Q. Xia, C. Baccigalupi, S. Matarrese, L. Verde, and M. Viel, JCAP, 8, 33 (August 2011).
[145] J.-Q. Xia, M. Viel, C. Baccigalupi, and S. Matarrese, JCAP, 9, 3 (September 2009).
[146] K. Yahata, A. Yonehara, Y. Suto, E. L. Turner, T. Broadhurst, and D. P. Finkbeiner, PASJ, 59, 205–219 (February2007).
[147] R. Zhang and D. Huterer, Astroparticle Physics, 33, 69–74 (March 2010).

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