Probing dark energy with steerable wavelets through correlation of WMAP
  and NVSS local morphological measures
Mon. Not. R. Astron. Soc. 000, 1–12 (2007) Printed 25 October 2018 (MN LaTEX style file v2.2)
Probing dark energy with steerable wavelets through correlation of
WMAP and NVSS local morphological measures
J. D. McEwen1⋆, Y. Wiaux2, M. P. Hobson1, P. Vandergheynst2, A. N. Lasenby1
1Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, UK
2Signal Processing Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Accepted 27 November 2007. Received 27 November 2007; in original form 12 April 2007
ABSTRACT
Using local morphological measures on the sphere defined through a steerable wavelet analy-
sis, we examine the three-year WMAP and the NVSS data for correlation induced by the inte-
grated Sachs-Wolfe (ISW) effect. The steerable wavelet constructed from the second deriva-
tive of a Gaussian allows one to define three local morphological measures, namely the signed-
intensity, orientation and elongation of local features. Detections of correlation between the
WMAP and NVSS data are made with each of these morphological measures. The most sig-
nificant detection is obtained in the correlation of the signed-intensity of local features at a
significance of 99.9%. By inspecting signed-intensity sky maps, it is possible for the first time
to see the correlation between the WMAP and NVSS data by eye. Foreground contamination
and instrumental systematics in the WMAP data are ruled out as the source of all significant
detections of correlation. Our results provide new insight on the ISW effect by probing the
morphological nature of the correlation induced between the cosmic microwave background
and large scale structure of the Universe. Given the current constraints on the flatness of the
Universe, our detection of the ISW effect again provides direct and independent evidence for
dark energy. Moreover, this new morphological analysis may be used in future to help us to
better understand the nature of dark energy.
Key words: cosmic microwave background – cosmology: observations – methods: data anal-
ysis – methods: numerical.
1 INTRODUCTION
A cosmological concordance model has emerged recently, explain-
ing many observations of our Universe to very good approxi-
mation. In this model our Universe is dominated by an exotic
dark energy component that may be described by a cosmologi-
cal fluid with negative pressure, interacting only gravitationally
to counteract the attractive gravitational nature of matter. Dark
energy may be strongly inferred from observations of the cos-
mic microwave background (CMB), such as the Wilkinson Mi-
crowave Anisotropy Probe (WMAP) data (Bennett et al. 2003b;
Hinshaw et al. 2007), together with either observations of Type
Ia Supernovae (Riess et al. 1998; Perlmutter et al. 1999) or of the
large scale structure (LSS) of the Universe (e.g. Allen et al. 2002).
Although dark energy dominates the energy density of our Uni-
verse, we know very little about its origin and nature. Indeed, a
consistent model to describe dark energy in the framework of par-
ticle physics is lacking.
The integrated Sachs-Wolfe (ISW) effect (Sachs & Wolfe
1967) provides an independent physical phenomenon that may be
⋆ E-mail: mcewen@mrao.cam.ac.uk
used to detect and probe dark energy. As CMB photons travel to-
wards us from the surface of last scattering they pass through gravi-
tational potential wells due to the LSS. If the gravitational potential
evolves during the photon propagation, then the photon undergoes
an energy shift. The ISW effect is the integrated sum of this en-
ergy shift along the photon path. In a matter-dominated Einstein-de
Sitter universe the gravitational potential remains constant with re-
spect to conformal time, thus there is no ISW effect. However, if the
universe deviates from matter domination due to curvature or dark
energy then an ISW effect is induced. Strong constraints have been
placed on the flatness of the Universe by WMAP (Spergel et al.
2007), hence a detection of the ISW effect may be inferred as di-
rect and independent confirmation of dark energy.
It is difficult to isolate the ISW contribution to CMB
anisotropies directly, therefore Crittenden & Turok (1996) sug-
gested that the ISW effect be detected by cross-correlating CMB
anisotropies with tracers of the local matter distribution, such
as the nearby galaxy density distribution. A positive large-scale
correlation will be induced by the ISW effect as a consequence
of decaying gravitational potentials due to dark energy. First
attempts to detect the ISW effect using Cosmic Background
Explorer-Differential Microwave Radiometer (COBE-DMR) data
c© 2007 RAS
http://arxiv.org/abs/0704.0626v2
2 McEwen et al.
failed, concluding that greater sensitivity and resolution than
that provided by COBE were required (Boughn & Crittenden
2002). Fortunately, the WMAP mission soon provided suitable
CMB data. Correlations indicative of the ISW effect have now
been detected between both the first- and three-year WMAP data
and a large number of tracers of the LSS. Detections have been
made using NRAO VLA Sky Survey (NVSS; Condon et al. 1998)
data (Boughn & Crittenden 2004; Nolta et al. 2004; Vielva et al.
2006; McEwen et al. 2007b; Pietrobon et al. 2006), hard X-ray
data provided by the High Energy Astronomy Observatory-
1 satellite (HEAO-1; Boldt 1987) (Boughn & Crittenden
2005, 2004), APM galaxy survey (Maddox et al. 1990) data
(Fosalba & Gaztañaga 2004), Sloan Digital Sky Survey (SDSS;
York D. G. et al. (SDSS collaboration) 2000) data (Scranton et al.
2003; Fosalba et al. 2003; Padmanabhan et al. 2005; Cabré et al.
2006; Giannantonio et al. 2006), the Two Micron All Sky Survey
Extended Source Catalogue (2MASS XSC; Jarrett et al. 2000)
(Afshordi et al. 2004; Rassat et al. 2006) and combinations of the
aforementioned data sets (Gaztañaga et al. 2006). Furthermore, a
number of other works have focused on theoretical detectability,
future experiments and/or error analyses related to ISW detections
(Afshordi 2004; Hu & Scranton 2004; Pogosian 2005, 2006;
Pogosian et al. 2005; Corasaniti et al. 2005; LoVerde et al. 2007;
Cabré et al. 2007).
Many different analysis techniques have been employed in
these previous works to detect correlations between the data sets,
each of which have their own merits and limitations. In this
work we focus on wavelet-based techniques. Wavelets provide
an ideal analysis tool to search for the ISW effect. This is due
to the localised nature of the effect, in both scale and position
on the sky, and the simultaneous scale and position localisation
afforded by a wavelet analysis. Since the ISW effect is cosmic
variance limited, it is desirable to examine as great a sky cover-
age as possible. In this near full-sky setting the geometry of the
sphere should be taken into account, thus wavelet analyses on the
sphere are required. The first analysis using wavelets on the sphere
to search for correlations between the CMB and LSS was per-
formed by Vielva et al. (2006) using the axisymmetric spherical
Mexican hat wavelet. McEwen et al. (2007b) extended this anal-
ysis to directional wavelets, as correlated features induced by the
ISW effect may not necessarily be rotationally invariant. Indeed,
it is known that statistically isotropic Gaussian random fields are
characterised by local features that are not rotationally invariant
(Barreiro et al. 1997, 2001). Recently, Pietrobon et al. (2006) ap-
plied a new wavelet construction on the sphere called needlets to
search for correlations. All of these works examined the WMAP
and NVSS data (due to the large sky coverage of the NVSS data)
and made significant detections of the ISW effect. Moreover, in
each analysis the detection of the ISW effect was used to constrain
dark energy parameters.
In this paper we present a new search for correlation between
the three-year WMAP and the NVSS data using wavelets on the
sphere. However, the approach taken here differs fundamentally to
previous wavelet analyses. We use steerable wavelets on the sphere
to extract, from each of the two data sets, measures of the morphol-
ogy of local features, such as their signed-intensity, their orienta-
tion, or their elongation. We then correlate the WMAP and NVSS
data through these quantities. This represents the first direct anal-
ysis of local morphological measures on the sphere resulting from
the steerability of wavelets. A positive detection of correlation us-
ing any of these morphological probes would give a direct indi-
cation of the ISW effect, provided it were not due to unremoved
foregrounds or measurement systematics, which could again be in-
ferred as direct and independent evidence for dark energy. More-
over, such a detection would give an indication of the morphologi-
cal nature of the correlation. This further insight might help to bet-
ter understand the nature of dark energy. However, the derivation
of the theoretical correlation of the newly defined morphological
measures as a function of the dark energy content of the Universe
is not easily tractable. Consequently, corresponding constraints on
dark energy parameters are left to a future work.
The remainder of this paper is organised as follows. In sec-
tion 2 we present the physical and signal processing background
behind the analyses performed. The ISW effect is described and
the use of local morphological measures defined through a steer-
able wavelet analysis is presented as a probe to search for the ISW
effect. In section 3 the data and analysis procedures employed are
described in detail. Two distinct analysis techniques are proposed
to compute correlations from morphological measures: a local mor-
phological analysis and a matched intensity analysis. The results of
these analyses are presented in section 4 and section 5 respectively.
Concluding remarks are made in section 6.
2 ISW EFFECT AND STEERABLE WAVELETS
The existence of an ISW effect induces a cross-correlation between
the CMB and NVSS signals on the celestial sphere. From the spec-
tral point of view, the corresponding two-point cross-correlation
function may be expressed in terms of the cross-correlation angu-
lar power spectrum. In this paper, we go beyond this spectral anal-
ysis thanks to a decomposition of the CMB and NVSS signals with
steerable wavelets on the sphere. The wavelet analysis enables one
to probe local features at each analysis scale. In this section we
first review the ISW effect and the correlation between the CMB
and NVSS data that it induces. Secondly, we discuss wavelets on
the sphere and describe how the steerability of wavelets is used to
define local morphological measures on the sphere.
2.1 ISW effect
The secondary temperature anisotropy induced in the CMB by the
ISW effect is related to the evolution of the gravitational poten-
tial. Any recent acceleration of the scale factor of the Universe due
to dark energy will cause local gravitational potentials to decay.
CMB photons passing through over dense regions of decaying po-
tential suffer blue shifts, resulting in a positive correlation between
the induced anisotropy and the local matter distribution. It can be
shown that the relative temperature fluctuation induced in the CMB
is given by
T(ω) =
∆T (ω)
dη Φ̇(η, ω) (1)
(Sachs & Wolfe 1967; Nolta et al. 2004), where ∆T is the induced
temperature perturbation, T0 is the mean temperature of the CMB,
η is conformal time, Φ is the gravitational potential and the dot
represents a derivative with respect to conformal time. A point ω on
the sky is represented in spherical coordinates as ω = (θ, ϕ), with
co-latitude θ and longitude ϕ. The integral is computed over the
photon path from emission to observation, i.e. from today back to
the last scattering surface. In a matter-dominated Einstein-de Sitter
universe (with zero cosmological constant) the potential evolves as
Φ ∼ δ/R, where δ is the matter perturbation and R is the scale factor
of the universe. In this setting the matter perturbation evolves with
c© 2007 RAS, MNRAS 000, 1–12
Probing dark energy with steerable wavelets 3
the scale factor, δ ∝ R. Consequently, Φ̇ = 0 and there is no ISW
effect, as discussed previously.
We use the NVSS galaxy count distribution projected onto
the sky as a tracer of the local matter distribution. It is as-
sumed that the two corresponding relative fluctuations, respec-
tively δN(z, ω) and δ(z, ω), are related by the linear bias factor b(z):
δN(z, ω) = b(z) δ(z, ω), where z is redshift. Hereafter we take the
bias to be redshift independent since the redshift epoch over which
the ISW effect is produced is small. The galaxy source count fluc-
tuation observed on the sky is therefore given by
N(ω) = b
δ(z, ω) , (2)
where dN/ dz is the mean number of sources per steradian at red-
shift z and the integral is performed from today to the epoch of
recombination, i.e. last scattering.
We are now in a position to consider the correlation between
the galaxy count and CMB temperature fluctuations. We consider
the cross-power spectrum CNT
defined by the ensemble average of
the product of the spherical harmonic coefficients of the two signals
observed on the sky:
〈∆Nℓn ∆
ℓ′n′ 〉 = δℓℓ′δnn′ C
ℓ , (3)
where ∆ℓn = 〈Yℓn|∆〉 are the spherical harmonic coefficients of
∆(ω), 〈·|·〉 denotes the inner product on the sphere, Yℓn are the spher-
ical harmonic functions for multipole ℓ ∈ N, n ∈ Z, |n| 6 ℓ and δi j
is the Kronecker delta symbol. In writing the cross-correlation in
this manner we implicitly assume that the galaxy density and CMB
random fields on the sphere are homogeneous and isotropic, which
holds under the basic assumption of the cosmological principle.
Representing the gravitational potential and the matter density per-
turbation in Fourier space and substituting (1) and (2) into (3), it is
possible to show (e.g. Nolta et al. 2004) that
CNTℓ = 12π Ωm H0
δ(k) F
ℓ (k) F
ℓ (k) , (4)
where Ωm is the matter density, H0 is the Hubble parameter,
(k) = k3Pδ(k)/2π
2 is the logarithmic matter power spectrum,
Pδ(k) = 〈|δ(k)|2〉 is the matter power spectrum and the filter func-
tions for the galaxy density and CMB are given by
FNℓ (k) = b
D(z) jℓ[kη(z)] (5)
FTℓ (k) =
jℓ[kη(z)] (6)
respectively. The integration required to compute FT
(k) is per-
formed over z from zero to the epoch of recombination, whereas, in
practice, the integration range for FN
(k) is defined by the source
redshift distribution dN/ dz. D(z) is the linear growth factor for
the matter distribution: δ(z, k) = D(z)δ(k), with δ(k) = δ(0, k). The
function g(z) ≡ (1 + z)D(z) is the linear growth suppression factor
and jℓ(·) is the spherical Bessel function. We have represented in
harmonic space the expected correlation between the galaxy source
count fluctuations and the CMB temperature fluctuations induced
by the ISW effect. We next turn our attention to steerable wavelets
on the sphere as a potential tool for detecting this correlation.
2.2 Wavelets on the sphere
The WMAP and NVSS data may be understood as the sampling
of continuous signals on the sphere, which can be analysed in the
framework of a continuous wavelet formalism. The analysis of a
signal on the sphere with a wavelet, which is a local analysis func-
tion, yields a set of wavelet coefficients. These coefficients result
from the scalar products between the signal and the wavelet di-
lated at any scale, rotated around itself by any angle, and trans-
lated at any point on the sphere. The so-called steerable wavelets
allow, from these wavelet coefficients, the definition of local mor-
phological measures of the signal. In particular, the second Gaus-
sian derivative wavelet used in the present analysis gives access to
the measures of orientation, signed-intensity, and elongation of the
signal’s local features. The remainder of this section is devoted to
the explicit definition of these local morphological measures. The
reader not directly interested in these formal and technical details
may proceed directly to section 3.
The continuous wavelet formalism on the sphere originally
proposed by Antoine & Vandergheynst (1998) was recently further
developed in a practical approach by Wiaux et al. (2005). It de-
fines the wavelet decomposition of a signal on the sphere in the
following way. We consider an orthonormal Cartesian coordinate
system (o, ox̂, oŷ, oẑ) centred on the (unit) sphere, with the direc-
tion oẑ defining the North Pole. To relate this coordinate system to
the spherical coordinates ω = (θ, ϕ) defined previously, we let the
polar angle, or co-latitude, θ ∈ [0, π] represent the angle between
the vector identifying ω and the axis oẑ, and the azimuthal angle, or
longitude, ϕ ∈ [0, 2π) represent the angle between the orthogonal
projection of this vector in the plane (o, ox̂, oŷ) and the axis ox̂.
The signal to be analysed is represented by a square-integrable
function F(ω) on the sphere, i.e. F(ω) ∈ L2(S 2, dΩ), for the invari-
ant measure dΩ = d(cos θ) dϕ. A square-integrable analysis func-
tion Ψ(ω) is defined, the so-called mother wavelet, initially centred
at the North Pole. This wavelet may be dilated at any scale a > 0.
The dilation of a wavelet on the sphere in this formalism may be
uniquely related to the usual dilation in the tangent plane at the
north pole (as discussed in more detail below). The corresponding
Ψa can cover arbitrarily small or large regions of the sphere, re-
spectively corresponding to high or low frequencies. Through the
appropriate three-dimensional rotation, the wavelet may also be ro-
tated on itself by any angle χ ∈ [0, 2π) and translated to any point
ω0 = (θ0, ϕ0) on the sphere. The final dilated, rotated and translated
wavelet is denoted Ψω0 ,χ,a. At each scale, the wavelet coefficients
of the signal are defined by the directional correlation with Ψa, i.e.
by the simple scalar product
(ω0, χ, a) = 〈Ψω0 ,χ,a|F〉 ≡
dΩΨ∗ω0 ,χ,a (ω) F (ω) , (7)
where the superscript ∗ denotes complex conjugation. These
wavelet coefficients hence characterise the signal locally around
each point ω0, at scale a and orientation χ. The choice of possi-
ble mother wavelets is submitted to a condition ensuring that the
original signal can be reconstructed exactly from its wavelet co-
efficients. In practice, this wavelet admissibility condition on the
sphere is not easy to check for a given candidate function. It is
therefore difficult to build wavelets directly on the sphere.
A correspondence principle has been established (Wiaux et al.
2005), which states that the inverse stereographic projection of a
wavelet on the plane provides a wavelet on the sphere. The stereo-
graphic projection of a point ω = (θ, ϕ) on the sphere gives by defi-
nition a point ~x = (r, ϕ) on its tangent plane at the North pole, which
is co-linear with ω and the South Pole. It relates the radial variables
on the two manifolds through the bijection r(θ) = 2 tan(θ/2), and
simply identifies the angular variables ϕ (see Fig. 1). If the corre-
sponding unitary operator on functions is denoted Π, the correspon-
c© 2007 RAS, MNRAS 000, 1–12
4 McEwen et al.
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ϕ(r,  )
θ ϕ(  ,  )
Figure 1. Wavelets on the sphere can be defined through simple inverse
stereographic projection Π−1 of wavelets on the tangent plane at the North
pole. The corresponding functions are illustrated here by the shadow on the
sphere and the localised region on the plane. (Illustration reproduced from
Wiaux et al. 2005.)
dence principle states that, if ψ(~x) is a wavelet on the plane, then
Ψ(ω) = [Π−1ψ](ω) is a wavelet on the sphere.
The dilation of a wavelet on the sphere actually corresponds
to the conjugate, through the stereographic projection, of the usual
dilation in the tangent plane at the north pole. This conjugation
relation also trivially holds for the rotation of wavelets by χ, which
only affects the longitude, while the stereographic projection leaves
it invariant. In other words, the dilation and rotation of the wavelet
on itself may be performed through the natural dilation and rotation
on the plane before projection:
Ψχ,a(ω) = [Π
−1ψχ,a](ω) . (8)
Only the translation by ω0 must be performed explicitly on the
sphere through the appropriate rotation in three dimensions. As
wavelets on the plane are well-known and may be easily built as
simple zero-mean filters, the correspondence principle enables one
to define a large variety of wavelets on the sphere by simple projec-
tion. This principle also allows one to transfer wavelet properties
from the plane onto the sphere, such as the notion of steerability
discussed in the next section.
The wavelet analysis on the sphere described above is compu-
tationally demanding. The directional correlation described by (7)
requires the evaluation of a two-dimensional integral over a three-
dimensional domain. Direct computation of this integral by sim-
ple quadrature is not computationally feasible for practical CMB
data, such as the currently available 3 megapixel WMAP maps and
the forthcoming 50 megapixel Planck (Planck collaboration 2005)
maps. Fast algorithms to compute the wavelet decomposition of
signals on the sphere have been developed and implemented by
Wiaux et al. (2006) and McEwen et al. (2007a) (for a review see
Wiaux et al. invited contribution, 2007). In this work we utilise the
implementation described by McEwen et al. (2007a).
2.3 Steerability and morphological measures
Firstly, we review the notion of steerable wavelets on the sphere.
Just as on the plane, a function on the sphere is said to be ax-
isymmetric if it is invariant under rotations around itself. Any non-
axisymmetric function is called directional. The directionality of a
wavelet is essential in allowing one to probe the direction of local
features in a signal. A wavelet is said to be steerable if its rota-
tion around itself by any angle χ may be written as a simple linear
combination of non-rotated basis filters. On the sphere, for M basis
filters Ψm, a steerable wavelet may be written, by definition, as
Ψχ(ω) =
km(χ)Ψm(ω) (9)
(Wiaux et al. 2005). The weights km(χ), with 1 6 m 6 M, are called
interpolation functions. In particular cases, the basis filters may be
specific rotations by angles χm of the original wavelet: Ψm = Ψχm .
Steerable wavelets have a non-zero angular width in the azimuthal
angle ϕ. This property makes them sensitive to a range of directions
and enables them to satisfy their relation of definition. In spherical
harmonic space, this non-zero angular width corresponds to an az-
imuthal angular band limit N in the integer index n associated with
the azimuthal variable ϕ:Ψℓn = 0 for |n| > N. Typically, the number
M of interpolating functions is of the same order as the azimuthal
band limit N. Notice that the steerability is a property related to the
azimuthal variable ϕ only. Again as the stereographic projection
only affects the radial variables, steerable wavelets on the sphere
can easily be built as simple inverse stereographic projection of
steerable wavelets on the plane.
The derivatives of order Nd in direction x̂ of radial functions
φ(r) on the plane are steerable wavelets. The corresponding steer-
able wavelets on the sphere produced by inverse stereographic pro-
jection may be rotated in terms of M = Nd + 1 basis filters, and are
band-limited in ϕ at N = Nd + 1. The second derivative Ψ
x̂ of any
radial function has a band limit N = 3, and contains the frequen-
cies n = {0,±2} only. Ψ∂
x̂ may be rotated in terms of three basis
filters: the second derivatives in directions x̂ and ŷ, Ψ1 = Ψ
x̂ and
Ψ2 = Ψ
∂2ŷ , and the cross derivativeΨ3 = Ψ
∂x̂∂ŷ , with the correspond-
ing weights k1(χ) = cos
2 χ, k2(χ) = sin
2 χ, and k3(χ) = sin 2χ.
Fig. 2 illustrates the explicit example of the second derivative of a
Gaussian, which is the wavelet employed in the subsequent analy-
sis. In this case, the original radial function on the plane is given
as φ(Gauss)(r) = −
4/3πe−r
2/2, with proper normalisation ensuring
that the mother wavelet Ψ∂
(Gauss) on the sphere is normalised to
unity. The explicit basis filters obtained by differentiation are ana-
lytically defined by Wiaux et al. (2006). Recall that the natural dila-
tion on the tangent plane at the North Pole sends the radial variable
r(θ) onto r(θ)/a. The value of the scale a therefore identifies with
the dispersion of the Gaussian. Consequently, we define the angu-
lar size of our second Gaussian derivative wavelet on the sphere as
twice the half-width of the wavelet, where the half-width is defined
by θhw = 2 arctan(a/2), which is closely approximated by a at small
scales.
We use wavelet steerability to define local morphological
measures of real signals F on the sphere. Local geometrical quanti-
ties have previously been defined in real space, and have been used
to analyse CMB data and test cosmological models (Barreiro et al.
1997, 2001; Monteserı́n et al. 2005; Gurzadyan et al. 2005). Our
approach in wavelet space is completely novel. By linearity of the
wavelet decomposition, the steerability relation also holds on the
wavelet coefficients of a signal F,
(ω0, χ, a) =
km (χ) W
(ω0, a) , (10)
where the coefficients WF
(ω0, a) result from the correlation of F
with the non-rotated (χ = 0) filter Ψm at scale a. Technically these
M coefficients thus gather all the local information retained by the
steerable wavelet analysis at the point ω0 and at the scale a. One
c© 2007 RAS, MNRAS 000, 1–12
Probing dark energy with steerable wavelets 5
(a) Ψ∂
x̂(Gauss) (b) Ψ
(Gauss)
(c) Ψ∂x̂∂ŷ(Gauss)
(d) Ψ∂
(Gauss) rotated by χ = π/4
Figure 2. Mollweide projection of second Gaussian derivative wavelet on
the sphere for wavelet half-width θhw ≃ 0.4 rad. The rotated wavelet illus-
trated in panel (d) can be constructed from a sum of weighted versions of
the basis wavelets illustrated in panels (a) through (c).
may be willing to reorganise this information in terms of M quan-
tities with an explicit local morphological meaning. For M = 1,
the steerable wavelet is actually axisymmetric, and only a measure
of signed-intensity IF(ω0, a) of local features in the signal is ac-
cessible (given by the wavelet coefficient WF
(ω0, a) directly). For
M > 2, the steerable wavelet is directional and the orientation,
or direction, DF (ω0, a) of local features may be accessed through
the value which maximises the wavelet coefficient. The signed-
intensity IF (ω0, a) of the local feature is given by that coefficient
of maximum absolute value. Additional local morphological mea-
sures may be defined for M > 3, notably a measure of the elonga-
tion of local features, probing their non-axisymmetry, in addition to
their signed-intensity and orientation. As the wavelet coefficient in
each orientation naturally probes the extension of the local feature
in the corresponding direction, this elongation EF (ω0, a) can sim-
ply be measured from the absolute value of the ratio of the wavelet
coefficient of maximum absolute value, to the wavelet coefficient
in the perpendicular direction. The higher the number M of basis
filters, the wider the accessible range of local morphological mea-
sures.
Any second derivative of a radial function, such as the sec-
ond Gaussian derivative, is defined from M = 3 basis filters. The
signed-intensity, orientation, and elongation are therefore accessi-
ble in this case. Because the mother wavelet Ψ∂
x̂ oscillates in the
tangent direction x̂, it actually detects features oriented along the
tangent direction ŷ. The orientation of local features is therefore
given as
6 DF (ω0, a) ≡ χ0 +
, (11)
where the orientation χ0 ≡ χ0(ω0, a) that maximises the wavelet co-
efficient can be analytically defined from (10). Notice that the sec-
ond derivative of a radial function is invariant under a rotation by π
around itself. Consequently, the local orientations define headless
vectors in the tangent plane atω0. By convention the corresponding
vectors are chosen to point towards the Northern hemisphere only,
and the local orientations are therefore in the range [π/2, 3π/2).
For a second derivative of a radial function, as for any real steer-
able wavelet, the signed-intensity of local features is given by the
wavelet coefficient in the direction χ0:
IF (ω0, a) ≡ WF
(ω0, χ0, a) . (12)
The elongation of local features analysed by the second derivative
of a radial function is explicitly defined by
0 6 EF (ω0, a) ≡ 1 −
ω0, χ0 +
(ω0, χ0, a)
6 1 . (13)
Numerical tests performed on elliptical Gaussian-profile features
show that this elongation measure for the second Gaussian deriva-
tive increases monotonously in the range [0, 1] with the intrinsic
eccentricity e ∈ [0, 1] of the features. While it is possible to de-
fine alternative elongation measures, these numerical tests indicate
that the chosen definition is not an arbitrary measure of the non-
axisymmetry of local features, but represents a rough estimate of
the eccentricity of a Gaussian-profile local feature.
3 ANALYSIS PROCEDURES
The data and analysis procedures that we use in an attempt to detect
the ISW effect are described in detail in this section. We propose
two distinct analysis techniques based on the correlation of the local
morphological measures on the sphere defined in section 2.3. After
describing the data, we discuss the generic procedure to search for
any correlation between the NVSS and WMAP data. We then de-
scribe in detail the two approaches proposed, motivating the analy-
ses and highlighting the differences between them.
3.1 Data and simulations
In this work we examine the three-year release of the WMAP data
and the NVSS data for correlations induced by the ISW effect. Here
we briefly describe the data, preprocessing of the data and the sim-
ulations performed to quantify any correlations detected in the data.
We examine the co-added three-year WMAP sky map. This
map is constructed from a noise weighted sum of the maps ob-
served by the Q-, V- and W- channels of WMAP, in order to
enhance the signal-to-noise ratio of the resultant map. The co-
added map was first introduced by the WMAP team in their non-
Gaussianity analysis (Komatsu et al. 2003) and has since been
used in numerous analyses. We use the template based foreground
cleaned WMAP band maps (Bennett et al. 2003a) to construct the
co-added map. The conservative Kp0 mask provided by the WMAP
team is used to remove remaining Galactic emission and bright
point sources. Since the ISW effect is expected to induce correla-
tions on scales > 2◦ (Afshordi 2004), we downsample the co-added
map to a pixel size of ∼55′ (a HEALPix1 (Górski et al. 2005) reso-
lution of Nside = 64). This reduces the computation requirements of
the subsequent analysis considerably, while ensuring the analysis
remains sensitive to ISW induced correlations.
The large sky coverage and source distribution of the NVSS
data make it an ideal probe of the local matter distribution to use
when searching for the ISW effect. Sources in the catalogue are
thought to be distributed in the range 0 < z < 2, with a peak dis-
tribution at z ∼ 0.8 (Boughn & Crittenden 2002). This corresponds
closely to redshift regions where the ISW signal that we hope to
detect is produced (Afshordi 2004). The NVSS source distribution
is projected onto the sky in a HEALPix representation at the same
resolution as the WMAP co-added map considered (i.e. Nside = 64)
and an important systematic in the data is corrected. The correc-
tion of this systematic and the resulting preprocessed NVSS data
examined here are both identical to that analysed in McEwen et al.
(2007b), hence we refer the interested reader to this work for more
1 http://healpix.jpl.nasa.gov/
c© 2007 RAS, MNRAS 000, 1–12
http://healpix.jpl.nasa.gov/
6 McEwen et al.
(a) WMAP
(b) NVSS
Figure 3. WMAP co-added three-year and NVSS maps (Mollweide projec-
tion) after application of the joint mask. The maps are downsampled to a
pixel size of ∼55′. The WMAP temperature data are reported in mK, while
the NVSS data are reported in number-of-counts per pixel.
details of the data preprocessing. Not all of the sky is sufficiently
observed in the NVSS catalogue. We construct a joint mask to ex-
clude from our subsequent analysis those regions of the sky not
observed in the catalogue, in addition to the regions excluded by
the WMAP Kp0 mask. The WMAP and NVSS data analysed sub-
sequently, with the joint mask applied, are illustrated in Fig. 3.
In order to constrain the statistical significance of any detec-
tion of correlation between the WMAP and NVSS data we perform
Monte Carlo simulations. 1000 simulations of the WMAP co-added
map are constructed, modelling carefully the beam and anisotropic
noise properties of each of the WMAP channels and mimicking
the co-added map construction procedure (including the downsam-
pling stage). The analyses subsequently performed on the data are
repeated on the simulations and compared.
3.2 Generic procedure
Two different analysis procedures are applied to test the data for
correlation. The first procedure correlates the local morphological
measures of signed-intensity, orientation and elongation, extracted
independently from the two data sets. This procedure does not
rely on any assumption about the correlation in the data. The sec-
ond procedure correlates local features in the WMAP data that are
matched in orientation to local features extracted from the NVSS
data. This approach explicitly assumes that local features of the
LSS are somehow included in the CMB. In the absence of any ISW
effect one would not expect to detect any significant correlation in
either of these analyses. The analysis procedures are described in
more detail in the following subsections. Firstly, we describe the
generic part of the procedure common to both techniques.
We consider only those scales where the ISW effect
is expected to be significant (Afshordi 2004), i.e. scales
approximately corresponding to wavelet half-widths θhw of
{100′, 150′, 200′, 250′ , 300′, 400′ , 500′ , 600′}. The local morpho-
logical measures defined in section 2.3 are computed for the
WMAP and NVSS data and are used to compute various correlation
statistics (see the following subsections for more detail). Identical
statistics are computed for the Monte Carlo simulations in order
to measure the statistical significance of any correlation detected
in the data. A statistically significant correlation in the data appar-
ent from any of these statistics is an indication of the ISW effect,
provided it is not due to foreground contributions or systematics.
The procedure described previously would be sufficient for
data with full-sky coverage. Unfortunately this is not the case and
the application of the joint mask must be taken into account. The
application of the mask distorts those morphological quantities
constructed from wavelets with support that overlaps the mask ex-
clusion regions. The associated local morphological measures must
be excluded from the analysis. An extended mask is computed
for each scale to remove all contaminated values. The extended
masks are constructed by extending the central masked region by
the wavelet half-width, whilst maintaining the size of point source
regions in the mask. We use identical masks to those applied by
McEwen et al. (2007b) and refer the interested reader to this work
for more details.
Before proceeding with our morphological analyses it is im-
portant to check that the WMAP and NVSS data do not contain pre-
dominantly axisymmetric features, corresponding to local elonga-
tion trivially equal to zero and for which no local orientation might
be defined. To do this we examine the distribution of our morpho-
logical measure of elongation defined in section 2.3. In Fig. 4 we
plot the histograms of the elongation measures computed from the
data (outside of the extended exclusion masks). These histograms
are built from all scales but the distributions obtained for each indi-
vidual scale also exhibit similar structure. Many elongation values
computed from both data sets lie far from zero, thereby justifying
the morphological analyses that we propose.
In order to give some intuition on the magnitude of various
statistics computed in the remainder of this paper, we also quote the
orders of magnitude for the means and standard deviations of the
different local morphological measures, as estimated from the two
data sets. Firstly, the mean and standard deviation of the elonga-
tion in the WMAP data are respectively µ̂TE = 0.61 and σ̂
E = 0.27,
when data from all scales are gathered. The corresponding values
for the NVSS data are µ̂NE = 0.62 and σ̂
E = 0.26. Secondly, re-
call that the galaxy density and CMB random fields on the sphere
are assumed to be homogeneous and isotropic. The local orienta-
tions computed from the data sets should therefore reflect a uni-
form distribution in the range [π/2, 3π/2). Consistently, the mean
and standard deviation of the orientation in the WMAP data are re-
spectively µ̂TD = 3.10 and σ̂
D = 0.93, again gathering data from all
scales. The corresponding values for the NVSS data are µ̂ND = 3.10
and σ̂ND = 0.90. Finally, notice that the local signed-intensity of a
signal is expressed, a priori on the whole real line, in the same units
as the signal itself. The signed-intensity of the WMAP temperature
fluctuation data is given in mK, while the signed-intensity of the
NVSS galaxy source count fluctuation data is given in number-
of-counts per pixel (see Fig. 3). The mean and standard devia-
tion of the signed-intensity in the WMAP data are respectively
µ̂TI = −2×10
−4 and σ̂TI = 6×10
−3, still accounting for data from all
scales. The corresponding values for the NVSS data are µ̂NI = 0.001
and σ̂NI = 0.01.
c© 2007 RAS, MNRAS 000, 1–12
Probing dark energy with steerable wavelets 7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PSfrag replacements
(a) WMAP
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PSfrag replacements
(b) NVSS
Figure 4. Histograms of the elongation values computed from the WMAP
and NVSS data for all scales. Many elongation values lie far from zero,
thereby justifying the morphological analyses that we propose.
3.3 Local morphological analysis
In the local morphological analysis that follows we correlate the
WMAP and NVSS data through each of the morphological mea-
sures provided by the second derivative of a Gaussian wavelet sep-
arately, i.e. the signed-intensity, orientation and elongation of lo-
cal features. Local features are extracted independently from the
WMAP and NVSS data. We then compute the correlation by
XNTS i (a) =
S Ni (ω0, a) S
i (ω0, a)
S Ni (ω0, a)
S Ti (ω0, a)
, (14)
where S i = {I, D, E} is the morphological measure examined, with
superscript N or T representing respectively the NVSS and WMAP
data (note that the morphological measures are computed at Np dis-
crete points on the sky only). This analysis provides the most gen-
eral approach to computing correlations and does not make any
assumption about possible correlation in the data. In the absence
of an ISW effect the WMAP and NVSS data should be indepen-
dent and none of the correlation estimators defined by (14) should
exhibit a significant deviation from zero.
3.4 Matched intensity analysis
In the matched intensity analysis that follows we first compute
the orientation and signed-intensity of local features for the NVSS
data. Using the local orientations extracted from the NVSS data,
we compute the signed-intensity of local features in the WMAP
data that are matched in orientation to the local features in NVSS
data. For this matched intensity analysis it no longer makes sense
to examine the correlation of orientations any further since they are
identical in both data sets, nor to examine the elongation measures
any further since the elongation does not measure the eccentricity
of local features in the WMAP data when the orientation is ex-
tracted from the NVSS data. We therefore only correlate, through
(14) again, the signed-intensity of features extracted from the data.
This analysis is based on the assumption that local features in the
NVSS data are somehow included in the WMAP data. This is a
conceptually different analysis to the local morphological analysis
described in the preceding subsection.
Before proceeding with this analysis it is important to check
that it differs in practice to the local morphological analysis. To do
this we examine the difference in the orientation of local features,
extracted independently in each data set, at each position on the
sky. From relation (11) it can be seen that this difference techni-
cally lies in the range [−π, π). It is however defined modulo π, and
the range is explicitly restricted to ∆D ∈ [−π/2, π/2). If orientation
differences are predominantly zero, then we know a priori that the
two analysis procedures are essentially the same and would give
the same result for the correlation of signed-intensities. In Fig. 5
we plot the histogram of the differences in orientation between the
two data sets (outside of the extended exclusion masks), built from
all scales (the distributions obtained for each individual scale ex-
hibit similar structure). The orientation differences are broadly dis-
tributed about zero, thus the two analysis techniques proposed are
indeed different practically, as well as conceptually. Moreover, the
relative flatness of the distribution suggests no obvious correlation
of orientations between the WMAP and NVSS data. This is exam-
ined in detail through the local morphological analysis when ap-
plied to orientations.
Notice that the local morphological measure of the orienta-
tion defined by (11) depends on the coordinate system consid-
ered through the definition of the origin χ = 0 at each point on
the sphere. The estimator for the correlation of orientations be-
tween the two data sets given by (14) also depends on the coordi-
nate system. This dependence does not affect the statistical signifi-
cance of any correlation detected, which simply measures a possi-
ble anomaly between the data and the simulations. One could sub-
stitute the estimator (14) by a measure of flatness for the distribu-
tion of the difference in the orientation ∆D between the two data
sets. This difference is indeed independent of the coordinate sys-
tem and has a direct physical meaning. However, this would simply
provide an alternative measure of the statistical significance of any
correlation detected in the orientation of local features. Either es-
timator is therefore suitable; for consistence we choose the former
case.
4 LOCAL MORPHOLOGICAL CORRELATIONS
In this section we present the results obtained from the local mor-
phological analysis described in section 3.3 to examine the WMAP
and NVSS data for possible correlation. Correlations are detected
and are examined to determine whether they are due to foreground
contamination or instrumental systematics, or whether they are in-
deed induced by the ISW effect.
c© 2007 RAS, MNRAS 000, 1–12
8 McEwen et al.
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
PSfrag replacements
∆D (rad)
Figure 5. Histogram of the difference in orientation of local features ex-
tracted independently in the WMAP and NVSS data for all scales. This
difference ∆D is defined modulo π, in the range [−π/2, π/2). The distribu-
tion is not concentrated about zero, indicating that the local morphological
and matched intensity analyses are indeed different practically, as well as
conceptually.
4.1 Detections
The correlation statistics computed from the WMAP and NVSS
data for each morphological measure are displayed in Fig. 6. Sig-
nificance levels computed from the 1000 Monte Carlo simulations
described in section 3.1 are also shown on each plot. A non-zero
correlation signal is clearly present in the signed-intensity of local
features. A strong detection at 99.3% significance is made in the
correlation of signed-intensities for wavelet half-width θhw ≃ 400′ .
This correlation deviates from the mean of the Monte Carlo sim-
ulations by 2.8 standard deviations. Moderate detections are also
made in the correlation of the orientations and elongations of local
features for θhw ≃ 400′ at 93.3% significance and for θhw ≃ 600′ at
97.2% significance respectively.
The statistical significance of these findings is based on an
a posteriori selection of the scale corresponding to the most sig-
nificant correlation. Nevertheless, such statistics have been widely
used to quantify detections of the ISW effect, and are satisfactory
provided that the a posteriori nature of the analysis is acknowl-
edged. However, in this work we also consider a χ2 analysis to
examine the significance of correlation between the data when the
correlation statistics for all scales are considered in aggregate. This
a priori statistic is more conservative in nature; we consider it as a
simple alternative detection measure which takes into account the
correlation between scales. For the correlation computed for each
local morphological measure we compute the χ2 defined by
χ2S i = X
C−1S i X
, (15)
where XNTS i is a concatenated vector of X
(a) for all scales, the
matrix CS i is a similarly ordered covariance matrix of the mor-
phological correlation statistics computed from the Monte Carlo
simulations and † represents the matrix transpose operation. Com-
paring the χ2 values computed from the data to those computed
from the Monte Carlo simulations, it is possible to quantify the sta-
tistical significance of correlation in the data. Moderate detections
are made in the correlation of signed-intensities and orientations of
local features at 94.5% and 93.2% significance respectively. The
detection in the correlation of elongations drops considerably to
84.5% significance.
The steerable wavelet analysis performed allows one to lo-
calise on the sky those regions that contribute most significantly
to the correlation detected in the data. This type of localisation
was performed by McEwen et al. (2007b). However, it was found
in this work that, although the localised regions were a significant
source of correlation between the data, they by no means provided
the sole contribution of correlation. Similar findings were also ob-
tained by Boughn & Crittenden (2005) using a different analysis
technique. These results are consistent with intuitive predictions
based on the ISW effect, namely that one would expect to observe
correlations weakly distributed over the entire sky, rather than only
a few highly correlated localised regions. For these reasons we do
not repeat the localisation performed by McEwen et al. (2007b)
for the current analysis (especially since we shown in section 4.2
that the detection is likely due to the ISW effect and not contam-
ination or systematics). Instead, we display in Fig. 7 the signed-
intensity values on the sky for the scale (θhw ≃ 400′) correspond-
ing to the most significance detection of correlation. Due to the
strength of the correlation, it is possible to see it by eye between
the WMAP (Fig. 7 (a)) and NVSS data (Fig. 7 (c)). We consider
only the signed-intensity of local features here since this morpho-
logical measure corresponds to the most significant detection of
correlation made in these local morphological analyses.
4.2 Foregrounds and systematics
To test whether the correlation detected in the preceding subsection
is perhaps due to foreground contamination or instrumental system-
atics in the WMAP data, we examine the correlation signals com-
puted using individual WMAP bands and difference maps in place
of the WMAP co-added map. Again, we consider only the signed-
intensity of local features here since this morphological measure
corresponds to the most significant detection of correlation made in
these local morphological analyses. In Fig. 8 (a) we plot the mor-
phological signed-intensity correlation of the NVSS data with each
of the individual WMAP band maps (constructed from the mean
signal observed by all receivers in that band), while in Fig. 8 (b)
we plot the correlation for maps constructed from band differences
(constructed from differences of signals observed by various re-
ceivers).
Any correlation induced by unremoved foreground contami-
nation in the WMAP data is expected to exhibit a frequency de-
pendence, reflecting the emission law of the foreground compo-
nent. However, the same correlation signal is observed in each of
the WMAP bands and no frequency dependence is apparent (see
Fig. 8 (a)). Furthermore, the difference map W−V−Q is clearly
contaminated by foreground contributions, while having a minimal
CMB contribution. No correlation is detected in this difference map
(see Fig. 8 (b)). Consequently, we may conclude that foreground
contamination is not the source of the correlation detected.
The same correlation signal is observed in each WMAP band
(see Fig. 8 (a)). Furthermore, this signal is not present in any of the
difference maps constructed for each given band (see Fig. 8 (b)),
which contain no CMB or foreground contributions. These findings
suggest that the correlation detected is not due to systematics in the
WMAP data.
The preliminary tests performed in this subsection indicate
that it is unlikely that either foregrounds or systematics are respon-
sible for the correlation detected in the signed-intensity of local
features. This strongly suggests that the ISW effect is responsible
for the statistically significant correlation detected in this section
between the WMAP and NVSS data.
c© 2007 RAS, MNRAS 000, 1–12
Probing dark energy with steerable wavelets 9
100 150 200 250 300 350 400 450 500 550 600
PSfrag replacements
Wavelet half-width θhw (arcmin)
(a) Signed-intensity
100 150 200 250 300 350 400 450 500 550 600
−0.15
−0.05
PSfrag replacements
Wavelet half-width θhw (arcmin)
(b) Orientation
100 150 200 250 300 350 400 450 500 550 600
−0.015
−0.01
−0.005
0.005
0.015
PSfrag replacements
Wavelet half-width θhw (arcmin)
(c) Elongation
Figure 6. Correlation statistics computed for each morphological measure
in the local morphological analysis, from the WMAP co-added map and
the NVSS map. Significance levels obtained from the 1000 Monte Carlo
simulations are shown by the shaded regions for 68% (yellow/light-grey),
95% (magenta/grey) and 99% (red/dark-grey) levels. The different ranges
of the correlations for the signed-intensity, orientation, and elongation are
consistent with our rough estimations of the standard deviations for each
local morphological measure in each of the two data sets (see section 3.2).
5 MATCHED INTENSITY CORRELATION
In this section we present the results obtained from the matched in-
tensity analysis described in section 3.4 to examine the WMAP and
NVSS data for possible correlation. In this setting the orientation
of local features extracted from the WMAP data at each position
on the sky are matched to orientations extracted from the NVSS
data. It therefore no longer makes sense to correlate the orientation
or elongation of features between the two data sets. Correlations
are detected between the signed-intensity of matched features and
are examined to determine whether they are due to foreground con-
tamination or instrumental systematics, or whether they are indeed
induced by the ISW effect.
(a) WMAP for independent features
(b) WMAP for features matched to NVSS orientations
(c) NVSS
Figure 7. Morphological signed-intensity maps (Mollweide projection) cor-
responding to the scale (θhw ≃ 400′) on which the maximum detections of
correlation are made. In panel (a) signed-intensities are shown for local fea-
tures extracted independently from the WMAP co-added data, whereas in
panel (b) signed-intensities are shown for local features in the WMAP co-
added data that are matched in orientation to local features in the NVSS
data. Due to the strength of the correlation in the data, it is possible to ob-
serve the correlation both between maps (a) and (c) and between maps (b)
and (c) by eye.
5.1 Detections
The correlation statistics computed from the WMAP and NVSS
data for the signed-intensity of matched features are displayed in
Fig. 9. Significance levels computed from the Monte Carlo simula-
tions are again shown on each plot. A non-zero correlation signal
is clearly present. A strong detection at 99.2% significance is made
in the correlation of signed-intensities of local features for wavelet
half-width θhw ≃ 400′. This correlation deviates from the mean of
the Monte Carlo simulation by 2.6 standard deviations. Note that
these significance measures are again based on an a posteriori scale
selection.
We also compute the a priori significance of the detection of
correlation when considering all scales in aggregate, taking into
account the correlation between scales. This is based on the χ2 test
described in section 4.1. For this case we detect a highly significant
correlation at a level of 99.9%.
Although it is possible to localise regions on the sky that con-
c© 2007 RAS, MNRAS 000, 1–12
10 McEwen et al.
100 150 200 250 300 350 400 450 500 550 600
PSfrag replacements
Wavelet half-width θhw (arcmin)
(a) Individual band maps
100 150 200 250 300 350 400 450 500 550 600
W−V−Q
Q1−Q2
V1−V2
W1−W2+W3−W4
PSfrag replacements
Wavelet half-width θhw (arcmin)
(b) Difference maps
Figure 8. Correlation statistics computed for the signed-intensity of features
in the local morphological analysis, from the WMAP individual band and
difference maps and the NVSS map. Significance levels obtained from the
1000 Monte Carlo simulations are shown by the shaded regions for 68%
(yellow/light-grey), 95% (magenta/grey) and 99% (red/dark-grey) levels.
100 150 200 250 300 350 400 450 500 550 600
PSfrag replacements
Wavelet half-width θhw (arcmin)
Figure 9. Correlation statistics computed for signed-intensity in the
matched intensity analysis, from the WMAP co-added map and the NVSS
map. Significance levels obtained from the 1000 Monte Carlo simulations
are shown by the shaded regions for 68% (yellow/light-grey), 95% (ma-
genta/grey) and 99% (red/dark-grey) levels.
tribute most significantly to the correlation detected, we again do
not do this since the ISW signal is expected to be weakly distributed
over the entire sky (as described in section 4.1). Instead, we display
in Fig. 7 the signed-intensity of matched features on the sky for the
scale (θhw ≃ 400′) corresponding to the most significance detection
of correlation using this analysis. Due to the strength of the cor-
relation, it is again possible to see it by eye between the WMAP
(Fig. 7 (b)) and NVSS data (Fig. 7 (c)).
100 150 200 250 300 350 400 450 500 550 600
PSfrag replacements
Wavelet half-width θhw (arcmin)
(a) Individual band maps
100 150 200 250 300 350 400 450 500 550 600
W−V−Q
Q1−Q2
V1−V2
W1−W2+W3−W4
PSfrag replacements
Wavelet half-width θhw (arcmin)
(b) Difference maps
Figure 10. Correlation statistics computed for the signed-intensity in the
matched intensity analysis, from the WMAP individual band and difference
maps and the NVSS map. Significance levels obtained from the 1000 Monte
Carlo simulations are shown by the shaded regions for 68% (yellow/light-
grey), 95% (magenta/grey) and 99% (red/dark-grey) levels.
5.2 Foregrounds and systematics
For this analysis procedure we also check whether the correlation
detected is perhaps due to foreground contamination or system-
atics. We follow the same procedure outlined in section 4.2. The
correlation signals computed for the matched intensity analysis us-
ing individual WMAP bands and differences maps in place of the
WMAP co-added data are plotted in Fig. 10. No frequency depen-
dence is observed between the signals computed for the individual
band maps (see Fig. 10 (a)), neither is any correlation observed
in the foreground contaminated W−V−Q map (see Fig. 10 (b)).
These findings suggest that foreground contributions are not re-
sponsible for the correlation detected in this analysis. Furthermore,
the same correlation signal is observed in each individual band (see
Fig. 10 (a)), but is not present in any of the difference maps for each
given band (see Fig. 10 (b)). This indicates that systematics are not
responsible for the correlation detected in this analysis. These re-
sults again strongly suggest that the ISW effect is responsible for
the correlation detected between the WMAP and NVSS data in this
second analysis based on the signed-intensity of matched features.
6 CONCLUSIONS
We have presented the first direct analysis of local morphological
measures on the sphere defined through a steerable wavelet analy-
sis. Using the wavelet on the sphere constructed from the second
derivative of a Gaussian, we are able to define three quantities to
characterise the morphology of local features: the signed-intensity,
c© 2007 RAS, MNRAS 000, 1–12
Probing dark energy with steerable wavelets 11
orientation and elongation. These local morphological measures
provide additional probes to search for correlation between the
CMB and LSS of the Universe that may have been induced by the
ISW effect.
Based on the morphological measures defined, we perform
two distinct analyses to search for correlation between the WMAP
and NVSS data. The first procedure, the local morphological anal-
ysis, correlates the morphological measures of local features ex-
tracted independently in the two data sets. This provides the most
general analysis and does not make any assumption regarding pos-
sible correlation in the data. The second procedure, the matched in-
tensity analysis, is based on the assumption that local features of the
LSS are somehow included in the local features of the CMB. Lo-
cal features are extracted from the WMAP data that are matched in
orientation to features in the NVSS data and the signed-intensities
of these features are correlated between the two data sets. Although
the two analysis procedures performed are obviously different con-
ceptually, we have shown that they differ practically also. Regard-
less of the procedure adopted, in the absence of an ISW effect the
WMAP and NVSS data should be independent and no correlation
should be present in any of the local morphological measures con-
sidered. However, strong detections are obtained in the correlation
of the signed-intensities of local features using both analysis proce-
dures. The most significant detection is made in the matched inten-
sity analysis at 99.9% significance, when all scales are considered
in aggregate. Moreover, in the local morphological analysis, mod-
erate detections are also made in the correlation of the orientation
and elongation of local features. In both analyses foreground con-
tamination and instrumental systematics have been ruled out as the
source of the correlation observed. This strongly suggests that the
correlation detected is indeed due to the ISW effect. Since the ISW
effect only exists in a universe with dark energy or a non-flat uni-
verse, and strong constraints have been placed on the flatness of
the Universe (Spergel et al. 2007), the detection of the ISW effect
made here can be inferred as direct and independent evidence for
dark energy.
In previous wavelet analyses to detect the ISW effect, theo-
retical predictions are derived for the wavelet correlation in or-
der to constrain dark energy parameters of cosmological models
(Vielva et al. 2006; McEwen et al. 2007b; Pietrobon et al. 2006).
As already discussed, our detections give a new insight on the na-
ture of the correlation between the CMB and the LSS, in terms of
the signed-intensity, orientation and elongation of local features.
This might in turn help to better understand the nature of dark en-
ergy through new constraints on dark energy parameters. However,
our steerable wavelet analysis is complicated by allowing the ori-
entation of wavelet coefficients to vary as a function of the data.
The derivation of the theoretical correlation for each local morpho-
logical measure is not easily tractable. We are currently exploring
this issue in more detail and leave any attempts to constrain stan-
dard dark energy parameters to a future work. We also intend to
explore the use of this, and other wavelet-based, ISW detections to
investigate perturbations in the dark energy fluid and to constrain
the sound speed of dark energy (see e.g. Weller & Lewis 2003;
Bean & Doré 2004).
ACKNOWLEDGEMENTS
We thank Patricio Vielva and Enrique Martı́nez-González for pro-
viding the preprocessed NVSS data. JDM is supported by PPARC.
YW acknowledges support of the Swiss National Science Founda-
tion (SNF) under contract No. 200021-107478/1. He is also post-
doctoral researcher of the Belgian National Science Foundation
(FNRS). Some of the results in this paper have been derived us-
ing the HEALPix package (Górski et al. 2005). We acknowledge the
use of the Legacy Archive for Microwave Background Data Anal-
ysis (LAMBDA). Support for LAMBDA is provided by the NASA
Office of Space Science.
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	Introduction
	ISW effect and steerable wavelets
	ISW effect
	Wavelets on the sphere
	Steerability and morphological measures
	Analysis procedures
	Data and simulations
	Generic procedure
	Local morphological analysis
	Matched intensity analysis
	Local morphological correlations
	Detections
	Foregrounds and systematics
	Matched intensity correlation
	Detections
	Foregrounds and systematics
	Conclusions
