# The integrated Sachs-Wolfe imprint of cosmic superstructures: a problem for ΛCDM

**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.

**0**Bookmarks

**·**

**71**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**We measure the average cold spot on the cosmic microwave background(CMB) produced by voids selected in the SDSS DR7 spectroscopic redshift galaxy catalog, spanning redshifts from 0 to 0.4. Our detection has a significance of ~3sigma based on the variance of random samples, and has an average amplitude of ~3 muK as viewed through a compensated top-hat filter scaled to the radius of each void. This signal, if interpreted as the late-time Integrated Sachs-Wolfe effect, serves as an evidence for the late-time acceleration of the Universe. The detection is achieved by applying the optimal filter size identified from N-body simulations. Two striking features are found by comparing ISW simulations with our detection, 1.) the void profiles traced by halos in our simulations are very similar to those in the data traced by galaxies. 2.) the same filter radius that gives the largest ISW signal in simulations also yields the largest detected signal in the observations. We model the expected ISW signal using voids from N-body simulations in LCDM, selected in the same way as in the observations. The detected signal, however, is many times larger than that from simulations, discrepant at the ~3sigma level. The large cosmic variance of large-scale modes in the gravitational potential can obscure an ISW measurement such as ours. However, we show how this cosmic variance can be effectively reduced by using a compensated top-hat filter for the detection. We test whether a few possible systematic effects could be producing the signal; we find no evidence that they do.The Astrophysical Journal 01/2013; · 6.28 Impact Factor - SourceAvailable from: Jeremy Butterfield
##### Article: On under-determination in cosmology

[Show abstract] [Hide abstract]

**ABSTRACT:**I discuss how modern cosmology illustrates under-determination of theoretical hypotheses by data, in ways that are different from most philosophical discussions. I emphasise cosmology's concern with what data could in principle be collected by a single observer (Section 2); and I give a broadly sceptical discussion of cosmology's appeal to the cosmological principle as a way of breaking the under-determination (Section 3). I confine most of the discussion to the history of the observable universe from about one second after the Big Bang, as described by the mainstream cosmological model: in effect, what cosmologists in the early 1970s dubbed the ‘standard model’, as elaborated since then. But in the closing Section 4, I broach some questions about times earlier than one second.Studies In History and Philosophy of Science Part B Studies In History and Philosophy of Modern Physics 01/2013; · 0.90 Impact Factor - SourceAvailable from: Carlos Hernández-Monteagudo
##### Article: On the signature of $z\sim 0.6$ superclusters and voids in the Integrated Sachs-Wolfe effect

[Show abstract] [Hide abstract]

**ABSTRACT:**Through a large ensemble of Gaussian realisations and a suite of large-volume N-body simulations, we show that in a standard LCDM scenario, supervoids and superclusters in the redshift range $z\in[0.4,0.7]$ should leave a {\em small} signature on the ISW effect of the order $\sim 2 \mu$K. We perform aperture photometry on WMAP data, centred on such superstructures identified from SDSS LRGs, and find amplitudes at the level of 8 -- 11$ \mu$K -- thus confirming the earlier work of Granett et al 2008. If we focus on apertures of the size $\sim3.6\degr$, then our realisations indicate that LCDM is discrepant at the level of $\sim4 \sigma$. If we combine all aperture scales considered, ranging from 1\degr--20\degr, then the discrepancy becomes $\sim2\sigma$, and it further lowers to $\sim 0.6 \sigma$ if only 30 superstructures are considered in the analysis (being compatible with no ISW signatures at $1.3\sigma$ in this case). Full-sky ISW maps generated from our N-body simulations show that this discrepancy cannot be alleviated by appealing to Rees-Sciama mechanisms, since their impact on the scales probed by our filters is negligible. We perform a series of tests on the WMAP data for systematics. We check for foreground contaminants and show that the signal does not display the correct dependence on the aperture size expected for a residual foreground tracing the density field. The signal also proves robust against rotation tests of the CMB maps, and seems to be spatially associated to the angular positions of the supervoids and superclusters. We explore whether the signal can be explained by the presence of primordial non-Gaussianities of the local type. We show that for models with $\FNL=\pm100$, whilst there is a change in the pattern of temperature anisotropies, all amplitude shifts are well below $<1\mu$K.Monthly Notices of the Royal Astronomical Society 12/2012; · 5.23 Impact Factor

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:

1INTRODUCTION

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

2S. 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.

2THE 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

3EXPECTED 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.

3.1The 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

4S. Nadathur et al.

020406080 100120 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.

3.3 Temperature 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-

4COMPARING 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

6S. 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.

4.2.2Potential 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

8S. 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

Granett B.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

SzapudiI., 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

#### View other sources

#### Hide other sources

- Available from Subir Sarkar · May 16, 2014
- Available from ArXiv