Page 1
Mon. Not. R. Astron. Soc. 000, 1–9 (0000)Printed 20 September 2011(MN LATEX style file v2.2)
The integrated Sachs-Wolfe imprints of cosmic
superstructures: a problem for ΛCDM
Seshadri Nadathur,1Shaun Hotchkiss2and Subir Sarkar1
1Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK
2Department of Physics, University of Helsinki and Helsinki Institute of Physics, P.O. Box 64, FIN-00014 Helsinki, Finland
Accepted Xxxx. Received Xxxx; in original form Xxxx
ABSTRACT
A crucial diagnostic of the ΛCDM cosmological model is the integrated Sachs-Wolfe
(ISW) effect of large-scale structure on the cosmic microwave background (CMB).
The ISW imprint of superstructures of size ∼ 100 h−1Mpc at redshift z ∼ 0.5 has
been detected with > 4σ significance, however it has been noted that the signal is
much larger than expected. We revisit the calculation using linear theory predictions
in ΛCDM cosmology for the number density of superstructures and their radial density
profile, and take possible selection effects into account. While our expected signal is
larger than previous estimates, it is still inconsistent by > 3σ with the observation. If
the observed signal is indeed due to the ISW effect then huge, extremely underdense
voids are far more common in the observed universe than predicted by ΛCDM.
Key words:
theory, dark energy, large-scale structure of Universe
cosmic microwave background, cosmological parameters, cosmology:
1 INTRODUCTION
The standard ‘concordance’ ΛCDM cosmological model fits
many different observations, including the Type Ia super-
nova (SNe Ia) luminosity distance-redshift relation (e.g.
Hicken et al. 2009; Amanullah et al. 2010; Conley et al.
2011), the anisotropies in the CMB (Komatsu et al. 2011),
the locally measured Hubble parameter (e.g. Riess et al.
2011) and baryon acoustic oscillations (e.g. Percival et al.
2010, — but see Sylos-Labini et al. 2009; Kazin et al. 2010).
These observations, when interpreted assuming the homoge-
neous and isotropic Friedman-Robertson-Walker metric im-
ply that the expansion of the universe is accelerating, from
which it is inferred that the universe is presently dominated
by a cosmological constant (‘dark energy’) with negative
pressure. It is important to note that this evidence is purely
geometrical being based on interpreting measurements of
distances — made using ‘standard rulers’ (the sound horizon
at last scattering) and ‘standard candles’ (SNe Ia) — as due
to accelerated expansion. The same data can be equally well
fitted without dark energy if, e.g., the isotropic but radially
inhomogeneous Lema¨ ıtre-Tolman-Bondi metric is assumed
and other assumptions such as a power-law spectrum for the
primordial density perturbations are relaxed (e.g. Biswas,
Notari & Valkenburg 2010; Nadathur & Sarkar 2011).
Given that dark energy is a complete mystery from a
physical viewpoint, it is therefore imperative to establish if
there is any dynamical evidence for it. For example, the de-
cay of gravitational potentials after dark energy begins to
dominate (at redshift z ? 1) should lead to secondary CMB
anisotropies as the CMB photons traverse regions of over-
or under-density — the ISW effect (Sachs & Wolfe 1967). If
the universe is spatially flat, then detection of the ISW effect
through cross-correlation of the CMB with large-scale struc-
ture would provide direct evidence of dark energy’s negative
pressure, hence crucial confirmation of the ΛCDM model
(Crittenden & Turok 1996).1
To detect the ISW effect with 5σ significance in CMB-
galaxy cross-correlations requires z measurements for over
10 million galaxies (Afshordi 2004; Douspis et al. 2008). Such
datasets are not yet available but several authors (e.g., Fos-
alba, Gaztanaga & Castander 2003; Boughn & Crittenden
2004; Afshordi, Loh & Strauss 2004; Nolta et al. 2004; Pad-
manabhan et al. 2005; Giannantonio et al. 2006; Cabre et al.
2006; Raccanelli et al. 2008) have examined smaller source
catalogues and reported marginal detections with < 3σ sig-
nificance. On the other hand, Rassat et al. (2007) and Fran-
cis & Peacock (2010) were unable to reject the null hypoth-
esis (no ISW effect) and Sawangwit et al. (2010) even found
a slight anti-correlation thus rejecting ΛCDM at 2−3σ sig-
nificance. Some groups have combined different data sets to
try to push up the significance to 4σ (Ho et al. 2008; Gi-
annantonio et al. 2008) but it is difficult to then estimate
1The ISW effect should also boost low multipoles in the CMB
angular power spectrum, whereas these are in fact anomalously
low on the observed sky. However given the large ‘cosmic variance’
on these scales, the discrepancy with ΛCDM is not too significant.
c ? 0000 RAS
arXiv:1109.4126v1 [astro-ph.CO] 19 Sep 2011
Page 2
2 S. Nadathur et al.
errors reliably and it has been shown that the quoted errors
have in fact been underestimated (Lopez-Corredoira, Sylos
Labini & Betancourt-Rijo 2010).
Much of the uncertainty in full-sky studies arises from
the difficulty in reconstructing the underlying density field
from galaxy survey data, given Poisson noise in the galaxy
distribution. A different approach to this problem is followed
by Granett, Neyrinck & Szapudi (2008a,b), who study the
Sloan Digital Sky Survey (SDSS) Data Release 6 (DR6) lu-
minous red galaxies (LRGs). They use 3D galaxy informa-
tion rather than the projected 2D density, and select only
the most extreme density perturbations, which are unam-
biguously identified despite Poisson noise. Along the lines of
sight corresponding to these ‘superstructures’ they report a
4.4σ detection of the ISW effect. Aside from being the most
significant detection to date, this approach also provides in-
formation about the sizes and distribution of extreme struc-
tures in the universe so can be used to check the consistency
of the standard ΛCDM model of structure formation and can
constrain, e.g., primordial non-gaussianity.
However the magnitude of the temperature signal re-
ported by Granett, Neyrinck & Szapudi (2008a) (hereafter
G08a) is surprisingly large and has been argued (Hunt &
Sarkar 2010; Inoue, Sakai & Tomita 2010) to be quite incon-
sistent with ΛCDM. Papai & Szapudi (2010) have responded
by noting that the assumed profile of the superstructures
has a big effect on the signal, such that with a different
assumption than the ‘compensated top-hat’ profile adopted
by Hunt & Sarkar (2010) and Inoue, Sakai & Tomita (2010),
the discrepancy is only at the 2σ level.
Our aim in this paper is to clarify this important is-
sue. We calculate the expected temperature signal from
these superstructures making no a priori assumptions about
their nature except that they arose in a ΛCDM cosmol-
ogy with gaussian primordial density perturbations. We find
that while (Papai & Szapudi 2010) are right in that the ex-
pected signal does depend on the assumed density profile, its
value calculated using an exact treatment of (initially) gaus-
sian perturbations (Bardeen et al. 1986) is still discrepant
at > 3σ with the observations reported by G08a.
In Section 2 we briefly review the ISW effect and in Sec-
tion 3 we calculate the expected temperature signal of super-
structures in the standard ΛCDM model. In Section 4.1 we
describe the key features of the observation of G08a which
must be accounted for before making a comparison with the
theoretical calculation. In Section 4.2 we show that, even if
G08a had selectively picked out the regions in the survey
with the biggest ISW signal, there is still a significant dis-
crepancy. In the final Section 5 we discuss possible reasons
for this discrepancy and future observational tests.
2 THE ISW EFFECT
In a universe with matter density Ωm = 1 and no dark en-
ergy, density perturbations δ grow at exactly the same rate
as the scale factor of the universe a (≡ (1+z)−1), so at the
linear level there is no evolution of the gravitational poten-
tial Φ (∝ −δ/a). However, in a ΛCDM universe, a grows
faster than (linear) density perturbations, so perturbations
in Φ decay with time. For a CMB photon passing through
an overdense region the energy gained while falling in is not
cancelled by the energy lost in climbing out of the evolved,
shallower, potential well. Overdense regions (clusters) there-
fore appear as hot spots in the CMB; conversely, underdense
regions (voids) will appear as cold spots as the photon loses
more energy climbing the potential hill than it gains subse-
quently while descending.
The temperature fluctuation ∆T(ˆ n) induced along di-
rection ˆ n is (Sachs & Wolfe 1967):
∆T(ˆ n) =2
c3¯T0
ˆrL
0
˙Φ(r,z, ˆ n) a dr ,(1)
where ¯T0 is the mean CMB temperature, rL is the ra-
dial comoving distance to the last scattering surface (LSS),
˙Φ(r,z, ˆ n) is the time derivative of the gravitational potential
along the photon geodesic and c is the speed of light.
The Poisson equation relates Φ to the density contrast
δ ≡ (ρ − ¯ ρ)/¯ ρ (where ¯ ρ(t) is the mean density) through:
∇2Φ(x,t) = 4πG¯ ρ(t)a2δ(x,t).
This can be written in Fourier space as
?H0
where H0 is the current Hubble parameter. Taking the time
derivative of this equation yields:
?H0
We assume that linear theory holds on the large scales of
interest hence perturbations grow as δ(k,t) = D(t)δ(k,z =
0), where D(t) is the linear growth factor. A numerical sim-
ulation has shown that non-linear effects represent only a
10% correction at the low redshifts we are interested (Cai et
al. 2010).2In this approximation,
?H0
where β(z) ≡ dlnD/dlna is the linear growth rate. Hence
the time evolution is captured by the ISW linear growth
factor, G(z) = H(z)(1 − β(z))D(z)/a. For an Ωm = 1 uni-
verse, β(z) = 1 for all z so there is no ISW effect.
Given the density profile δ of any isolated superstruc-
ture, eqs. (1) and (5) can be used to calculate the tempera-
ture fluctuation it induces in the CMB. Assuming spherical
symmetry of the density profile, eq. (5) in real-space be-
comes:
˙Φ(r,z) =3
2ΩmH2
where
ˆr
0r
with δ(r?) evaluated at redshift z = 0. Thus F(r) contains
all information about the structure in question, while the
assumed cosmology enters through the prefactor and the
ISW growth factor G(z) in eq. (6).
(2)
Φ(k,t) = −3
2k
?2
Ωmδ(k,t)
a
, (3)
˙Φ(k,t) =3
2k
?2
Ωm
?
˙ a
a2δ(k,t) −
˙δ(k,t)
a
?
. (4)
˙Φ(k,z) =3
2k
?2
ΩmH(z)
a
[1 − β(z)]δ(k,z), (5)
0G(z)F(r), (6)
F(r) =
r?2
rδ(r?) dr?+
ˆ∞
r?δ(r?) dr?, (7)
2Both the linear and non-linear effects grow with time, however
at late times and large scales (∼ 100Mpc/h), the linear effect
dominates while at early times (when ΩΛ? 0), both effects are
smaller but the non-linear effect dominates.
c ? 0000 RAS, MNRAS 000, 1–9
Page 3
The ISW imprint of superstructures3
3 EXPECTED SIGNAL FROM
SUPERSTRUCTURES IN ΛCDM
‘Superstructures’ refer to density perturbations extending
over ? 100 h−1Mpc and should not be thought of as non-
linear collapsed structures in the usual sense, rather as
smooth hills and valleys in the density distribution. Col-
lapsed structures form only where the density perturbation
δ(r) exceeds unity, which happens on much smaller scales
than those of interest here.
G08a state that the most extreme structures in the
(500 h−1Mpc)3box of the Millennium N-body simula-
tion (Springel et al. 2005), when placed at z = 0, would
produce a signal of ∆T ∼ 4.2 µK. However when Hunt &
Sarkar (2010) calculated the signal distribution for super-
voids with the densities and sizes reported by G08a, they
obtained only ?∆T? = −0.42 µK. They assumed a com-
pensated top-hat profile for the gravitational potential mo-
tivated by the asymptotic final state of a void (Sheth &
van de Weygaert 2004). A similar profile was assumed by
Inoue, Sakai & Tomita (2010) who found a similar average
signal ?∆T? = −0.51 µK for the 50 most extreme density
perturbations of fixed radius r = 130 h−1Mpc expected in
a ΛCDM cosmology. Subsequently Papai & Szapudi (2010)
argued that this profile is not the appropriate choice for den-
sity perturbations on ? 100 h−1Mpc scales; Papai, Szapudi
& Granett (2011) chose instead an uncompensated gaussian
density profile to obtain significantly larger values of ∆T.
However, to arrive at this result they appear to have con-
sidered underdense regions with a physical density contrast
δ < −1 which is physically impossible!
It is thus necessary to revisit this issue. Using the statis-
tics of a homogeneous, isotropic, gaussian density field, we
now derive the expected mean density profiles of superstruc-
tures of all density contrasts and all sizes, as well as the
expected number density of such superstructures.
3.1 The number density of structures on different
scales
We identify superstructures of different sizes with extrema
of the linear density perturbation field δ(r) when smoothed
over different scales. Overdensities correspond to peaks
of the smoothed field and underdensities to troughs. In
the ΛCDM model, δ(r) is a homogeneous and isotropic,
gaussian-distributed random field and the statistical prop-
erties of the maxima and minima have been calculated
by Bardeen et al. (1986) (hereafter BBKS). We briefly review
below their key results and introduce necessary notation.
Let P(k,t) denote the matter power spectrum, defined
as the Fourier transform of the 2-point correlation function
ξ(r,t) of the density field at time t. Define a set of spectral
moments weighted by powers of k:
σ2
j(t) =
ˆ
k2dk
2π2W2(kRf)P(k,t)k2j, (8)
where W(kRf) is the window function appropriate to the
filter used to smooth the density field, and Rf is the (co-
moving) smoothing scale. Thus σ0 is just the standard RMS
fluctuation of the smoothed density field. Using a gaussian
filter, W(kRf) = exp(−k2R2
f/2), we define the parameters:
γ ≡
σ2
σ2σ0, R∗ ≡
1
√3σ1
σ2
. (9)
The (comoving) differential number density Nmax(ν) of max-
ima of height δ0 = νσ0 is then (Bardeen et al. 1986):
Nmax(ν)dν =
1
(2π)2R3
∗e−ν2/2G(γ,γν)dν .(10)
The function G(γ,γν) is given by eq.(A19) of BBKS; we use
a fitting form, accurate to better than 1%, given in their eqs.
(4.4) and (4.5). The density of minima is related simply to
that of maxima through: Nmin(ν) = Nmax(−ν).
3.2 Mean radial profiles
Having identified superstructures with the maxima or min-
ima in the smoothed density field, we wish to determine the
mean radial variation of the density field in the neighbour-
hood of these extrema. BBKS show that, given a maximum
δ = δ0 at r = 0, the mean shape in the vicinity of this point
after averaging over all possible orientations of the principal
axes as well as all values of the curvature at r = 0 is:
?
?
where ψ(r) ≡ ξ(r)/ξ(0) is the normalised density-density
correlation function, and x = −∇2δ/σ2. The expectation
value of x given a peak of height δ0at r = 0 is approximately
¯δ(r) =
δ0
(1 − γ2)
ψ +R2
∗
3∇2ψ
?
γ2ψ +R2
−
?x|δ0?σ0
γ(1 − γ2)×
∗
3∇2ψ
?
, (11)
?x|δ0? = γν + Θ(γ,γν) ,
where ν = δ0/σ0, and Θ(γ,γν) is given by the fitting func-
tion (6.14) of BBKS. It follows that eq. (11) can be rewritten
as (Lahav & Lilje 1991):
(12)
¯δ(r,t) =
1
σ0
ˆ∞
0
?ν − γ2ν − γΘ
k2
2π2
sin(kr)
kr
W2(kRf)P(k,t) ×
ΘR2
3γ(1 − γ2)1 − γ2
+
∗k2
?
dk. (13)
We use this for numerical evaluation of profiles. Note that
γ and R∗ depend on the smoothing scale Rf.
We make the simplifying approximation that the av-
erage ISW signal for a large number of superstructures is
the same as the signal due to a superstructure with the
mean profile (13) — this is justified because of the linear
relationship between ∆T, Φ and δ. (The actual distribution
of the size of structures depends on the distribution of x
values, which can be obtained from eq. (7.5) of BBKS.) In
Fig. 1 we plot some underdense profiles for selected values
of δ0 and Rf (these examples are chosen for clarity and are
not representative of the most likely actual underdensities).
Note that the size of the structures is much larger than the
smoothing scale Rf. Profiles calculated in this manner are
somewhat narrower than those obtained from the simpler
gaussian form used in Papai & Szapudi (2010) and Papai,
Szapudi & Granett (2011), and the profile also turns over
(δ?(r) changes sign) at large r, which the gaussian profile
does not. (This cannot however be seen in the figure be-
cause of the vertical scale used.)
To identify superstructures in galaxy surveys (the
methodology of G08a is discussed in more detail in Sec-
c ? 0000 RAS, MNRAS 000, 1–9
Page 4
4 S. Nadathur et al.
0 20 40 6080 100 120 140160 180200
−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
r (h−1Mpc)
δ(r)
δ(r), δ0= −0.5, Rf= 20 h−1Mpc
δg(r), δ0= −0.5, Rf= 20 h−1Mpc
δ(r), δ0= −0.7 , Rf= 20 h−1Mpc
δ(r), δ0= −0.5, Rf= 30 h−1Mpc
Figure 1. Mean radial profiles for voids obtained from eq. (13)
for different values of central underdensity δ0 and the smooth-
ing scale Rf. The blue (solid) line and the green (dash-dot) line
are for the same Rf (= 20 h−1)Mpc but different values of δ0,
whereas the red (dotted) profiles have the same δ0 (= −0.5) as
the blue (solid) profiles, but a larger smoothing scale. The blue
(dashed) curve is the biased galaxy density contrast correspond-
ing to matter density contrast given by the blue (solid) line, with
bias factor b = 2.25 as is appropriate for LRGs.
tion 4.2) the galaxy density contrast δg is assumed to be
linearly biased with respect to the matter density: δg = bδ.
Denoting by ρsl the value of the density field at turnover,
and by ρ0the minimum density at the centre, a selection cut
is made on on w ≡ ρsl/ρ0, amounting to a lower bound on
the absolute value of δ0. This avoids false detections of over-
and under-dense regions that are just Poisson fluctuations
in galaxy number counts and ensures that only extreme su-
perstructures are included in the ensemble.
The definition of the radius Rvof any of the voids shown
in Fig. 1 is slightly ambiguous. We choose it to be the radius
of turnover in the density profile less Rf, since smoothing
necessarily increases the radius somewhat. This is a small
correction since in general Rv ? Rf.
3.3Temperature signal
The ISW signal of any individual superstructure will be too
small compared to the primordial CMB anisotropies to be
observable. Therefore, what is measured is the average tem-
perature fluctuation along the lines of sight of a selected
sample of either over- or under-densities. The primordial
anisotropies are uncorrelated with the large scale structue
and average out so, given a large enough sample, the corre-
lated ISW signal eventually dominates. Our calculation of
this averaged signal is done as follows.
At a given z, we use eq. (13) to calculate the matter
and galaxy density profiles about extrema of the density
field as functions of δ0 and Rf and obtain˙Φ along the line
of sight as discussed in Section 2. This enables us to cal-
culate ∆T(θ;δ0,Rf) where θ = 0◦is the line of sight pass-
ing through the centre of the superstructure. To compare
with the observations we first apply the selection criterion
on δ0 through the limit on w. Then, to calculate the expec-
tation value ?∆T? for the resulting ensemble, we weight the
results appropriately with the number density of extrema
(10). Hence for an ensemble of voids:
?∆T? =
˜W(θ)∆T(θ;δ0)Nminσ−1
πθ2
c
´Nminσ−1
where 0 ? θ ? θout; W(θ) is a filter chosen in order to match
that used in the actual observation and −1 ? δ0 ? δc
where δc
mum underdensity required to pass the selection criterion.
The choice of Rf determines the mean radial size of the voids
included in the ensemble; although structure-finding algo-
rithms may not have an explicit size dependence, in practice
there is obviously a lower limit on the size of the over- or
under-density that can be reliably found. As smaller struc-
tures are overwhelmingly more probable than larger ones,
it is important to capture this effect and we discuss this in
more detail in the next Section.
The expected signal from an ensemble of clusters follows
in an exactly analogous manner to eq. (14).
0
d2θdδ0
0
dδ0
, (14)
0,
0is the (mildly Rf-dependent) cutoff on the mini-
4 COMPARING THEORY TO OBSERVATION
4.1The measured ISW signal of superstructures
To compare our expectation for the ISW signal to the mea-
surement made by G08a, it is necessary to follow the same
selection procedure. They use a sample of 1.1 million LRGs
in the range 0.4 < z < 0.75 (mean z = 0.52) from the
SDSS DR6 (Adelman-McCarthy et al. 2008), which cov-
ers 7500 degree2on the sky and occupies a volume of
5 h−3Gpc3. They search for ‘supervoids’ and ‘superclusters’
using two publicly-available structure-finding algorithms:
ZOBOV (ZOnes Bordering On Voidness; Neyrinck 2008) for
supervoids, and VOBOZ (VOronoi BOund Zones; Neyrinck,
Gnedin & Hamilton 2005) for superclusters.
It is necessary to mimic the precise way in which these
algorithms select structures in choosing the ensemble for
which to calculate ?∆T? from eq. (14). ZOBOV uses a
parameter-free Voronoi tessellation to estimate the density
at each galaxy in the sample, based on the distance to its
nearest neighbours. Around each density minimum it then
finds the region of the density depression or supervoid. (Of
course large voids can contain multiple smaller voids, or even
isolated high-density regions.) The ‘significance’ of the de-
pression is estimated by comparing the density contrast, w
(defined as the ratio of the density at the lip of the void
to the density at its minimum) to an uniform Poisson point
sample. This yields the likelihood that a void of density con-
trast w could arise from Poisson noise, i.e. that it is a false
positive detection; a 3σ cut is then applied on the likeli-
hood which translates to requiring w > wc = 2.0 on the
density contrast (Neyrinck 2008). This procedure yields 50
supervoids, the properties of which are tabulated in Granett,
Neyrinck & Szapudi (2008b) (hereafter G08b). The VOBOZ
supercluster finder uses the same algorithm but applied to
the inverse of the density field, with density contrast de-
fined as the ratio of the peak density to the density at the
edge of the structure. However, now the 3σ cut on the likeli-
hood that an overdensity of given w could have arisen due to
c ? 0000 RAS, MNRAS 000, 1–9
Page 5
The ISW imprint of superstructures5
Poisson noise corresponds to w > 6.8 (Neyrinck, Gnedin &
Hamilton 2005). In fact G08a impose a tighter cut: w > 8.35
in order to obtain exactly 50 such superclusters; their prop-
erties are also tabulated in G08b.
G08a then search for the ISW signals of these super-
structures using an inverse-variance weighted combination of
the WMAP 5-year Q, V and W maps (Hinshaw et al. 2009)
with foreground subtracted and the KQ75 mask applied.
They build stacked images by averaging the CMB tempera-
ture in the regions around the lines of sight passing through
the centres of the identified superstructures and use a com-
pensated top-hat filter of width θc in order to perform the
averaging. This corresponds to making the choice in eq. (14):
?
with θout =
of supervoids gives ?∆T? = −11.3±3.1 µK, and the sample
of superclusters ?∆T? = 7.9 ± 3.1 µK. When averaged to-
gether (with the negative of the supervoid image added to
the superclusters) this gives ?∆T? = −9.6 ± 2.2 µK, i.e. a
4.4σ detection.
W(θ) =
1,
−1,
√2θc. For a filter radius θc = 4◦,3the sample
0 ? θ ? θc;
θc < θ ? θout,
(15)
4.2Comparison to the theoretical expectation
We assume for simplicity that all the superstructures are lo-
cated at the mean redshift z = 0.52 and adopt the standard
ΛCDM cosmological model with Ωm = 0.29, ΩΛ = 0.71,
ns = 0.96 and σ8 = 0.83 (with h = 0.69 where required)
to obtain the matter power spectrum at z = 0.52 using
CAMB (Lewis, Challinor & Lasenby 2000).4The bias factor
for LRGs is taken to be b = 2.25.
Our first finding is that there are no overdense su-
perstructures within the linear regime (i.e. with δ0 < 1)
which meet the VOBOZ 3σ-significance selection criterion
that w > 6.82. Such a ratio of densities (between the lip
of an overdensity and its centre) can be achieved only for
non-linear collapsed structures. We conclude that Table 5 of
G08b does not list the most overdense large-scale linear per-
turbations, but the (mild) large-scale linear perturbations
that happen to contain the most overdense small-scale col-
lapsed structures. This means that the criteria used to select
the superstructures tabulated in G08b are being affected by
collapsed structures in a manner that our methodology is
unable to capture, hence we cannot estimate the expected
ISW effect. Note however that our calculations below for
the maximum possible amplitude of ISW signal from super-
structures holds equally well for over- and under-densities in
the linear regime. Any contamination of the regions selected
by VOBOZ will only reduce the expected signal, although
we are unable to estimate by how much.
Hence we concentrate only on the sample of underdense
regions (i.e. supervoids), which does not suffer from this
3This is based on the expectation that the CMB-galaxy cross-
correlation should peak at about 4◦(Padmanabhan et al. 2005).
G08a repeat the observation with a few other widths, 3◦? θc ?
5◦and obtain a maximum detection significance for θc= 4◦.
4These are the mean parameter values obtained from a fit to
WMAP 7-year (Komatsu et al. 2011) and SDSS DR7 (Abazajian
et al. 2009) data using COSMOMC (Lewis & Bridle 2002).
problem, and for which ?∆T?obs= −11.3±3.1 µK. To calcu-
late the expected ?∆T? using eq. (14), we must first choose
the smoothing scale Rf and determine the distribution of
void radii Rv in the ensemble using eq. (7.5) of BBKS. For
each Rf we calculate Rmin
v
such that 95% of all voids in the
ensemble have radius Rv > Rmin
way to characterise the ensemble. For Rf = 20 h−1Mpc, we
find Rmin
v
∼ 70 h−1Mpc, which is similar to the mean radius
of the voids in Table 4 of G08b;5at this scale, the number
of such voids that should satisfy the selection criterion is
Nv ∼ 104. This is to be compared with the Nv = 50 voids
that are actually tabulated in G08b. For the larger ensem-
ble, we find an expectation value ?∆T? = −0.3 ± 0.2 µK,
i.e. consistent with zero and indicative of enormous tension
with the observation.
Since G08a see only a small fraction of the total num-
ber of supervoids in the SDSS DR6 volume, strong selection
effects must be in operation. One of these is certainly our
neglect of shot noise, i.e., the assumption that the galaxy
distribution smoothly traces the total matter distribution;
while this has no effect on our ISW predictions, it will affect
the selection of structures in the LRG distribution. Other
selection effects that can enhance the expected signal are
a bias towards larger and deeper regions, i.e. if G08a did
not randomly select 50 of the ∼ 104expected supervoids
but chose some sample that is skewed towards regions with
larger ∆T values. We show below that the expected signal
from the 50 most extreme regions is indeed ∼ 6 times larger
than −0.3 µK; nevertheless the discrepancy with the G08a
observation is still > 3σ. Therefore, irrespective of how the
regions were selected by G08a, tension remains with the ex-
pectation in the standard ΛCDM model.
v
, this being a convenient
4.2.1Accounting for selection effects
We first consider the possibility that the void-finding al-
gorithm ZOBOV is sensitive only to the largest (and least
common) voids in the matter distribution which produce
the biggest ISW temperature signals. The average density
of LRGs in the SDSS DR6 sample is roughly 1 galaxy per
(15 h−1Mpc)3. In underdense regions LRGs will be even
more sparsely distributed so ZOBOV will certainly be less
able to identify smaller underdense structures, thus biasing
the sample towards larger voids.
In order to model the effect of such a selection bias,
we increase the value of Rf in eq. (14); this is equivalent to
including only the Nv largest voids with radius Rv ? Rmin
in the ensemble from which ?∆T? is calculated. In the left
panel of Fig. 2 we plot the expectation ?∆T? as a function of
Rmin
v
. The orange cross-hatched area shows the region that
is within 3σ of the observed value ?∆T?obs= −11.3±3.1 µK.
The theoretical value of ?∆T? becomes marginally consistent
with the observed value when Rmin
the probability that within the SDSS survey volume there
are 50 supervoids of radius Rv ? 170 h−1Mpc which also
meet the ZOBOV selection criterion is negligibly small.
v
v
∼ 170 h−1Mpc. However
5At the radii reported in G08b, the density has not yet reached
the background level so their quoted values must be underesti-
mates of the void size relative to our criterion.
c ? 0000 RAS, MNRAS 000, 1–9
Page 6
6 S. Nadathur et al.
Figure 2. Left panel: The absolute value of ?∆T? for an ensemble of voids which satisfy the ZOBOV selection condition on density
(see text), as a function of the minimum radius of voids in the ensemble. The solid (blue) curve shows the mean value and the shaded
(lighter blue) contours the 1σ region. The (orange) cross-hatched area is the lower end of the 3σ range of the observed value ?∆T?obs=
−11.3±3.1 µK. Right panel: As above, but showing ?∆T? as a function of the number of voids in the ensemble from which the observed
sample of 50 voids is to be drawn, when only the Nv largest voids also meeting the ZOBOV selection condition on density are included.
In the right-hand panel of Fig. 2 we plot ?∆T? as a func-
tion of the size Nv of the ensemble of the largest supervoids
that should exist within the SDSS volume. It is from this en-
semble that the 50 observed supervoids should be regarded
as having been drawn. It can be seen that even under the
assumption that the VOBOZ algorithm selected exactly the
50 largest voids in the entire SDSS survey volume, the ex-
pected signal is only ?∆T? = −1.73 ± 0.18 µK which is still
discrepant by > 3σ with the observed value. We conclude
that the observed signal cannot be explained due to a simple
bias towards selecting only the largest voids.
It is interesting to note that the 50 largest supervoids
expected within the SDSS survey volume have Rmin
120 h−1Mpc. The largest void radius reported in G08b is
125 h−1Mpc and the mean is 70 h−1Mpc. We have argued
that these values somewhat underestimate the size of the
supervoids compared to our definition of Rv, yet it seems
unlikely that the difference could be so large that all the
voids tabulated in G08b should have Rv ? 120 h−1Mpc.
We consider next whether the ZOBOV algorithm is more
sensitive to deeper voids. In Table 4 of G08b, the edge of
most of the supervoids is defined at a radius where the
density contrast is still negative. This means ZOBOV sys-
tematically underestimates the value of w relative to our
definition (where δedge ? 0), i.e. the G08a observation ef-
fectively used a more stringent cut on w than the one we
have used. We can model this effect by varying δc
value determined by the stated algorithm. In the left-hand
panel of Fig. 3 we plot as examples ?∆T? as a function of
δc
v
∼ 70 h−1Mpc (the mean radius of the super-
voids in G08b) and Rmin
v
∼ 100 h−1Mpc. The right-hand
panel shows ?∆T? as a function of Nv, the number of voids
included in the ensemble when δc
For the smaller radius, the ensemble is dominated by small
v
?
0from the
0for Rmin
0 is varied in each case.
voids with a small ISW effect, so increasing Rmin
?∆T? at any δc
marginally consistent with the observation for δc
However, the probability of obtaining 50 supervoids with
Rv ? 100 h−1Mpc and δ0 ? −0.5 is negligibly small, as the
right-hand panel clearly demonstrates.
v
increases
0. With Rmin
v
∼ 100 h−1Mpc, ?∆T? becomes
0? −0.5.
4.2.2 Potential systematic errors
We now discuss the expected corrections due to the simpli-
fying assumptions we made in calculating the signal. Our
first approximation was to neglect non-linear effects and the
time evolution of the void density profile (Sheth & van de
Weygaert 2004; Colberg et al. 2005). On small scales voids
evolve towards a compensated top-hat profile through non-
linear evolution but this produces a smaller ISW signal due
to the effect of the overdense ridge at the boundary (Inoue,
Sakai & Tomita 2010; Papai & Szapudi 2010). Thus by using
the linear theory profile we are overestimating the expected
∆T, although on the large scales of interest ? 100 h−1Mpc,
the effects of non-linear evolution will be small in any case.
A more subtle assumption is that the real supervoid
profiles are adequately captured by our smoothing prescrip-
tion. The effect of the smoothing is to slightly broaden the
δ(r) profile which in turn leads to broadening of the ∆T(θ)
profile. This may become a problem if the ∆T(θ) profile is
significantly broadened so the compensated top-hat filter of
radius 4◦in eq. (14) then underestimates the real signal. If
this were the case then it would be more appropriate to use
a broader top-hat. To check this, we repeated our analysis
with a filter of radius 6◦, which is a generous overestimate
of the degree of broadening, and in the right panel of Fig.
3 we show the effect for the largest and deepest supervoids.
As expected, increasing the filter radius does increase ?∆T?,
c ? 0000 RAS, MNRAS 000, 1–9
Page 7
The ISW imprint of superstructures7
Figure 3. Left panel: The absolute value of ?∆T? for an ensemble of supervoids which satisfy δ0< δc
(green) curve shows the case when Rfis chosen such that voids with Rv ? 70 h−1Mpc are included in the ensemble; the solid (blue)
curve is for Rv ? 100 h−1Mpc. Shaded contours show the 1σ region about the mean and the orange cross-hatched area is as in Fig. 2.
Middle panel: As before, but showing ?∆T? as a function of the number of voids in the ensemble from which the observed sample of 50
supervoids is to be drawn, when only the Nv deepest supervoids are included. Right panel: As before, but for two different choices of the
radius θcof the compensating top-hat filter used in eq. (14). The solid (blue) curve is the mean value for θc= 4◦, and the broken (black)
curve for θc= 6◦. Shaded regions show the 1σ deviations from the mean.
0, as a function of δc
0. The dashed
but the effect is small. Even with a 6◦filter the expected sig-
nal remains > 3σ discrepant with observations for Nv = 50;
also as mentioned above, the actual effect on ?∆T? due to
the smoothing will be less than this extreme model.
As a further test of the robustness of our calculation, we
compare our results with Fig. 1 of Cai et al. (2010) which
shows the ISW map from a cosmological N-body simula-
tion for a volume comparable to the SDSS DR6, but at
z = 0 rather than z = 0.52. The very largest density per-
turbations in this map yields a maximum ISW signal of
|∆T| ∼ 4 µK before applying a filter analysis. Taking into
account that ∆T is more pronounced at smaller redshift,
this tallies very well with our prediction from Fig. 3 that
the most extreme supervoid in the SDSS volume should pro-
duce ?∆T? ∼ −2 µK. Surprisingly (Cai et al. 2010) do not
emphasise that the observation of G08a is in stark contrast
to the expectation for a ΛCDM cosmology.
As seen in the full sky maps (Cai et al. 2010), there are
lines of sight along which the cumulative effect of density
perturbations between us and the LSS can lead to ISW ‘cold
spots’ with ∆T < −10 µK. Similarly, there will be several 4◦
circles on the sky for which the underlying CMB anisotropy
alone can give an average ∆T of similar magnitude — it is
just such fluctuations that generate the observational un-
certainty of ±3.1 µK in the ISW signal. However there is
no reason why such circles on the sky should be correlated
with large structures at z ∼ 0.5. Therefore, assuming only
that the lines of sight were not chosen a posteriori, the ob-
served signal can only result from rare (> 3σ) fluctuations
or anomalously large density perturbations at z ∼ 0.5.
It is interesting to note in this context that G08a re-
port a lower significance detection for both N = 30 and
N = 70 compared to N = 50. Fluctuations due to the un-
derlying CMB anisotropy dominate at small N and false
positive identifications of structures increase at large N so
the signal-to-noise ratio is expected to have a maximum at
some intermediate N, but a more detailed study is needed
to quantify where this should be.
G08a also study the variation of the signal-to-noise ratio
with the width of the compensating top-hat filter and report
a maximum at θc = 4◦for the combined sample of over-and
under-dense structures. However, for supervoids we find that
|?∆T?| increases as θc is raised from 4◦to 6◦(Fig. 3). This
is further evidence that the G08a sample of superstructures
cannot be modelled by linear structures in ΛCDM.
5 SUMMARY AND PROSPECTIVES
We have calculated the integrated Sachs-Wolfe effect ex-
pected in ΛCDM from ∼ 100 h−1Mpc size structures, using
the density profiles predicted by the linear theory of gaussian
perturbations (Bardeen et al. 1986). We find that the most
extreme superstructures in the SDSS volume will produce
an ISW signal of ∼ 2 µK. This matches well with ISW maps
generated from N-body simulations of the ΛCDM cosmology
(Cai et al. 2010).
Our result is about 4 times larger than earlier calcu-
lations which assumed compensated top-hat density pro-
files (Hunt & Sarkar 2010; Inoue, Sakai & Tomita 2010).
Such an assumption, while well-motivated for non-linear
structures that have formed at small scales (Sheth & van
de Weygaert 2004), should not apply on the large scales of
the superstructures considered here. Papai & Szapudi (2010)
c ? 0000 RAS, MNRAS 000, 1–9
Page 8
8 S. Nadathur et al.
have noted that the ISW signal from such structures should
therefore be larger and our results confirm this.
Nevertheless we have demonstrated that the ISW signal
claimed to have been detected by Granett, Neyrinck & Sza-
pudi (2008a) is still > 3σ larger than the signal expected in
ΛCDM. This tension persists even after allowing for likely se-
lection effects. In fact, even the most extreme underdensities
in the SDSS volume would still produce a signal discrepant
by > 3σ with the observed signal. Therefore the observed
signal cannot be due to a selection effect. We concur with
(Hunt & Sarkar 2010) that deep superstructures appear to
be far more numerous than expected in a ΛCDM cosmology.
This differs from the conclusion of Papai, Szapudi &
Granett (2011) who also used non-compensated density pro-
files. We believe that this is because those authors incor-
rectly applied a method that was calibrated at small density
contrasts to higher values where it necessarily breaks down
for voids (i.e., requires δ < −1!). Imposing the physical re-
striction δ > −1 will significantly decrease the amplitude of
the ISW signal calculated by these authors.
An interesting question is whether the expected signal
of the most extreme superstructures in a ΛCDM universe
is possible to detect in principle. The primordial CMB is
usually taken to be a gaussian signal with standard deviation
of ∼ 18 µK. Hence the standard deviation of the average of
N elements from such a distribution is (18/√N) µK, so
that for N ? 3000, the uncertainty on a measurement of
?∆T? is ∆Tnoise ? 0.33 µK. Thus a detection of |?∆T?| =
1 µK can be made with roughly 3σ significance, averaged
over an ensemble of 3000 superstructures. From Fig. 2 it
is seen that for the 3000 largest voids in the SDSS DR6
survey volume, |?∆T?| ∼ 1 µK which is of the right order
of magnitude but somewhat too small for detection at high
significance. However the SDSS window is not large enough
to contain 3000 independent 4◦patches on the sky so in any
case a larger survey would be needed in order to measure a
statistically significant signal and this would contain more
supervoids. This order-of-magnitude estimate indicates that
even if the G08a observation is a statistical anomaly, the
ISW imprint of superstructures in a ΛCDM cosmology may
be large enough to be detected in future surveys.
As noted earlier, the detection of the ISW imprint of
individual superstructures provides an important comple-
ment to full-sky CMB-galaxy cross-correlation studies. It
has the potential to provide information about the radii,
density contrasts and density profiles of specific structures
that lie in the extreme tail of the probability distribution
function. Our calculation demonstrates that the predicted
ISW signal from the most extreme superstructures is far too
small to explain the temperature fluctuations seen by G08a,
indicating a failing of the standard ΛCDM cosmology.
A likely explanation for this deviation is that the pri-
mordial perturbations are non-gaussian. This would influ-
ence both the abundance of these extreme regions (e.g., see
Matarrese, Verde & Jimenez 2000; Kamionkowski, Verde &
Jimenez 2009) as well as their density profile, thus chang-
ing their expected ISW signal. As pointed out by Enqvist,
Hotchkiss & Taanila (2011), a primordial skewness, param-
eterised by fNL, would not be able to enhance the abun-
dance of both over- and underdense regions simultaneously;
however a primordial kurtosis, parameterised by a positive
gNL, would indeed do so, and be less constrained by the
CMB. Note that non-gaussianity disproportionately affects
the tail of the distribution of density perturbations, which
is where most of the contribution to the ISW effect of in-
dividual superstructures comes from. Therefore, primordial
non-gaussianity may be able to explain this signal while pre-
serving the success of ΛCDM on other fronts. However, this
would then undermine the use of the ISW effect as an inde-
pendent test for Λ.
Another possible explanation might lie in a modifica-
tion of the growth rate of perturbations as can happen in
e.g. models based on scalar-tensor gravity (Nagata, Chiba &
Sugiyama 2004).The presence of large-scale inhomogeneities
can themselves alter the growth rate and this too deserves
further attention.
6 ACKNOWLEDGEMENTS
We thank Syksy R¨ as¨ anen, Ben Hoyle and Tom Shanks for
helpful comments and discussion. SH is supported by the
Academy of Finland grant 131454.
REFERENCES
Abazajian K.N. et al., 2009, ApJS, 182, 543
Adelman-McCarthy J.K. et al., 2008, ApJS, 175, 297
Amanullah R. et al., 2010, ApJ, 716, 712
Afshordi N., 2004 Phys.Rev.D, 70, 083536
Afshordi N., Loh Y.S., Strauss M.A., 2004 Phys.Rev.D, 69,
083524
Bardeen J.M., Bond J.R., Kaiser N., Szalay A.S, 1986, ApJ,
304, 15
Biswas T., Notari A. and Valkenburg W., 2010, JCAP,
1011, 030
Boughn S., Crittenden R., 2004, Nature, 427, 45
Cabre A., Gaztanaga E., Manera M., Fosalba P., Cas-
tander F., 2006, MNRAS, 372, L23
Cai Y.C., Cole S., Jenkins A., Frenk C.S., 2010, MNRAS,
407, 201
Colberg J.M., Sheth R.K., Diaferio A., Gao L., Yoshida N.,
2005, MNRAS, 360, 216
Conley A. et al., 2011, ApJS, 192, 1
Crittenden R.G., Turok N., 1995, PRL, 76, 575
Douspis M., Castro P.G., Caprini C., Aghanim N., A&A,
485, 395
Enqvist K., Hotchkiss S., Taanila O., JCAP, 1104, 017
Fosalba P., Gaztanaga E., Castander F., 2003, ApJ, 597,
L89
Francis C. L., Peacock J.A., 2010, MNRAS 406, 2
Giannantonio T. et al., 2006, Phys.Rev.D, 74, 063520
Giannantonio T., et al., 2008, Phys.Rev.D, 77, 123520
Granett B.R., Neyrinck M.C., Szapudi I., 2008a, ApJ, 683,
L99
GranettB.R.,Neyrinck M.C.,
arXiv:0805.2974 [astro-ph]
Hicken M. et al., 2009, ApJ, 700, 1097
Hinshaw G. et al., 2009, ApJS, 180, 225
Ho S., Hirata C., Padmanabhan N., Seljak U., Bahcall N.,
2008, Phys.Rev.D, 78, 043519
Hunt P., Sarkar S., 2010, MNRAS, 401, 547
Inoue K.T., Sakai N., Tomita K., 2010, ApJ, 724, 12
Szapudi I., 2008b,
c ? 0000 RAS, MNRAS 000, 1–9
Page 9
The ISW imprint of superstructures9
Kamionkowski M., Verde L., Jimenez R., 2009, JCAP,
0901, 010
Kazin E.A. et al., 2010, ApJ, 710, 1444
Komatsu E. et al., 2011, ApJS, 192, 18
Lahav O., Lilje P.B., 1991 ApJ, 374, 29
Lewis A., Bridle S., 2002, Phys. Rev. D, 66, 103511
Lewis A., Challinor A., Lasenby A., 2000, ApJ, 538, 473
Lopez-Corredoira M., Sylos Labini F., Betancort-Rijo J.,
2010, A&A, 513, A3
Matarrese S., Verde L., Jimenez R., 2000, ApJ, 541, 10
Nadathur A, Sarkar S., 2010, Phys.Rev.D, 83, 063506
Nagata R., Chiba T., Sugiyama N., 2004, Phys. Rev. D,
69, 083512
Neyrinck M.C., 2008, MNRAS, 386, 2101
Neyrinck M.C., Gnedin N.Y., Hamilton A.J.S., 2005, MN-
RAS, 356, 1222
Nolta M.R. et al., 2004, ApJ, 608, 10.
Padmanabhan N., et al., 2005, Phys.Rev.D, 72, 043525
Papai P., Szapudi I., 2010, ApJ, 725, 2078
Papai P., Szapudi I., Granett B.R., 2011, ApJ, 732, 27
Percival W.J. et al., 2010 MNRAS, 401, 2148
Raccanelli A., et al., 2008, MNRAS, 386, 2161
Rassat A., Land K., Lahav O., Abdalla F.B., 2007 MNRAS
377, 1085
Riess A.G. et al., 2011, ApJ, 730, 119
Sachs R.K., Wolfe A.M., 1967, ApJ, 147, 73
Sawangwit U., et al., 2010, MNRAS, 402, 2228
Sheth R.K., van de Weygaert R., 2004, MNRAS, 350, 517
Springel V. et al., 2005, Nature, 435, 629
Sylos Labini F., Vasilyev N. L., Baryshev Y. V., Lopez-
Corredoira M., 2009, A&A, 505, 981
c ? 0000 RAS, MNRAS 000, 1–9
Download full-text