Constraints on large-scale inhomogeneities from WMAP5 and SDSS: confrontation with recent observations

Mon. Not. R. Astron. Soc. 401, 547–558 (2010) doi:10.1111/j.1365-2966.2009.15670.x

Constraints on large-scale inhomogeneities from WMAP5 and SDSS:confrontation with recent observations

Paul Hunt1� and Subir Sarkar2�1Institute of Theoretical Physics, Warsaw University, ul Hoza 69, 00-681 Warsaw, Poland2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP

Accepted 2009 September 6. Received 2009 September 2; in original form 2009 February 4

ABSTRACTMeasurements of the Type Ia supernovae Hubble diagram which suggest that the Universeis accelerating due to the effect of dark energy may be biased because we are located in a200–300 Mpc underdense ‘void’ which is expanding 20–30 per cent faster than the averagerate. With the smaller global Hubble parameter, the Wilkinson Microwave Anisotropy Probe5 data on cosmic microwave background (CMB) anisotropies can be fitted without requiringdark energy if there is some excess power in the spectrum of primordial perturbations on100 Mpc scales. The Sloan Digital Sky Survey (SDSS) data on galaxy clustering can also befitted if there is a small component of hot dark matter in the form of 0.5 eV mass neutrinos.We show however that if the primordial fluctuations are Gaussian, the expected variance ofthe Hubble parameter and the matter density are far too small to allow such a large local void.Nevertheless, many such large voids have been identified in the SDSS Luminous Red Galaxysurvey in a search for the late integrated Sachs–Wolfe effect due to dark energy. The observedCMB temperature decrements imply that they are nearly empty, thus these real voids too arein gross conflict with the concordance � cold dark matter model. The recently observed highpeculiar velocity flow presents another challenge for the model. Therefore, whether a largelocal void actually exists must be tested through observations and cannot be dismissed a priori.

Key words: cosmic microwave background – cosmological parameters – cosmology: theory –dark matter – large-scale structure of Universe.

1 IN T RO D U C T I O N

The Einstein–de Sitter (E–deS) universe with �m = 1 is the simplestmodel consistent with the spatial flatness expectation of inflationarycosmology. However, Type Ia supernovae (SNe Ia) at redshift z �0.5 appear ∼25 per cent fainter than expected in an E–deS universe(Riess et al. 1998; Perlmutter et al. 1999). Together with measure-ments of galaxy clustering in the 2df survey (Efstathiou et al. 2002)and of cosmic microwave background (CMB) anisotropies by theWilkinson Microwave Anisotropy Probe (WMAP) (Spergel et al.2003), this has established an accelerating universe with a domi-nant cosmological constant term (or other form of ‘dark energy’)which presumably reflects the present microphysical vacuum state.This ‘concordance’ � cold dark matter (�CDM) cosmology (with�� 0.7, �m � 0.3, h � 0.7) has passed a number of cosmologicaltests, including baryonic acoustic oscillations (BAOs) (Eisensteinet al. 2005) and measurements of mass fluctuations from clustersand weak lensing (e.g. Contaldi, Hoekstra & Lewis 2003). Furtherobservations of both SNe Ia (Riess et al. 2004; Astier et al. 2006;

�E-mail: [email protected] (PH); [email protected] (SS)

Wood-Vasey et al. 2007) and the WMAP 3-year results (Spergelet al. 2007) have continued to firm up the model. However, thereis no physical basis for this model, in particular there are two fun-damental problems with the notion that the universe is dominatedby vacuum energy. The first is the notorious fine-tuning problemof vacuum fluctuations in quantum field theory – the energy scaleof the cosmological energy density is ∼10−12 GeV, many orders ofmagnitude below the energy scale of ∼102 GeV of the standardmodel of particle physics, not to mention the Planck scale of∼1019 GeV (see Weinberg 1999). The second is the equally acutecoincidence problem: since ρ�/ρm evolves as the cube of the cos-mic scalefactor a, there is no reason to expect it to be of O(1) today,yet this is apparently the case. In fact what is actually inferred fromobservations is not an energy density, just a value of O(H 2

0 ) for theotherwise unconstrained � term in the Friedmann equation. It hasbeen suggested that this may simply be an artefact of interpretingcosmological data in the (oversimplified) framework of a perfectlyhomogeneous universe in which H 0 ∼ 10−42 GeV ∼ (1028 cm)−1 isthe only scale in the problem (Sarkar 2008).

In fact the WMAP results alone do not require dark energy ifthe assumption of a scale-invariant primordial power spectrum isrelaxed. This assumption is worth examining given our present

C© 2009 The Authors. Journal compilation C© 2009 RAS

548 P. Hunt and S. Sarkar

ignorance of the physics behind inflation. We have demonstrated(Hunt & Sarkar 2007) that the temperature angular power spectrumof an E–deS universe with h � 0.44 matches the WMAP data well ifthe primordial power is enhanced by ∼30 per cent in the region of thesecond and third acoustic peaks (corresponding to spatial scales ofk ∼ 0.01–0.1 h Mpc−1). This alternative model with no dark energyactually has a slightly better χ 2 for the fit to WMAP3 data thanthe ‘concordance power-law �CDM model’ and, in spite of havingmore parameters, has an equal value of the Akaike informationcriterion (AIC) used in model selection. Other E–deS models witha broken power-law spectrum (Blanchard et al. 2003) have alsobeen shown to fit the WMAP data. Moreover, an E–deS universecan fit measurements of the galaxy power spectrum if it includesan ∼10 per cent component of hot dark matter (HDM) in the formof massive neutrinos of mass ∼0.5 eV (Blanchard et al. 2003; Hunt& Sarkar 2007). Clearly the main evidence for dark energy comesfrom the SNe Ia Hubble diagram.

A mechanism that sets � = 0 is arguably more plausible than onewhich leads to the tiny energy density ρ� � 10−47 GeV4 associatedwith the concordance cosmology.1 If � is indeed zero, then perhapssome effect fools us into wrongly deducing the existence of darkenergy by mimicking a non-zero cosmological constant. It is naturalto connect this effect with inhomogeneities since cosmic accelera-tion and large-scale non-linear structure formation appear to havecommenced simultaneously. This approach offers the possibility ofsolving the cosmological constant problems within the frameworkof general relativity and keeps the introduction of new physics to aminimum.2 Several different ways in which inhomogeneities couldpotentially mimic dark energy have been considered in the litera-ture – for reviews see Celerier (2007), Buchert (2008) and Enqvist(2008). In an inhomogeneous universe, averaged quantities satisfymodified Friedmann equations which contain extra terms corre-sponding to ‘backreaction’ since the operations of spatial averagingand time evolution do not commute (Buchert 2000). The backreac-tion terms depend upon the variance of the local expansion rate, andhence increase as inhomogeneities develop. Whether backreactioncan indeed account for the apparent cosmological acceleration ishotly debated and remains an open question at present (Wetterich2003; Ishibashi & Wald 2006; Khosravi, Kourkchi & Mansouri2007; Vanderveld, Flanagan & Wasserman 2007; Wiltshire 2007;Behrend, Brown & Robbers 2008; Leith, Ng & Wiltshire 2008;Paranjpe & Singh 2008; Rasanen 2008).

Another possibility is that inhomogeneities affect light propa-gation on large scales and cause the luminosity distance-redshiftrelation to resemble that expected for an accelerating universe. Thishas been investigated for a ‘Swiss-cheese’ universe in which voidsmodelled by patches of Lemaıtre–Tolman–Bondi (LTB) space–timeare distributed throughout a homogenous background. However,the results seem to be model-dependent: some authors find thechange in light propagation to be negligible because of the cancel-lation effects (Biswas & Notari 2008; Brouzakis & Tetradis 2008;Brouzakis, Tetradis & Tzavara 2008), whereas Marra et al. (2007)claim it can partly mimic dark energy if the voids have radius

1 ‘Quintessence’ models, which attempt to address the coincidence problem,also assume that every other contribution to the vacuum energy cancels apartfrom that of the quintessence field.2 In models that seek to explain the observations through modifications ofgravity, the relevant scale of H−1

0 has to be introduced by hand, just as inquintessence models the quintessence field has to be given a mass of theorder of H0 – these are technically unnatural choices since this is an infraredscale for any microphysical theory.

250 Mpc (Marra, Kolb & Matarrese 2008). Mattsson (2009) hasnoted that observers may preferentially choose sky regions with un-derdense foregrounds when studying distant objects such as SNe Ia,so the expansion rate along the line of sight is then greater than aver-age; such a selection effect he argues can allow an inhomogeneousuniverse to fit the observations without dark energy.

In this paper, we are mainly interested in a ‘local void’ (some-times referred to as ‘Hubble bubble’) as an explanation for darkenergy; to prevent an excessive CMB dipole moment due to ourpeculiar velocity, we must be located near the centre of the void. Anunderdense void expands faster than its surroundings, thus youngersupernovae inside the void would be observed to be receding morerapidly than older supernovae outside the void. Under the assump-tion of homogeneity, this would lead to the mistaken conclusion thatthe expansion rate of the Universe is accelerating, although both thevoid and the global universe are actually decelerating. Henceforth,we use the ‘Hubble contrast’ δH ≡ (H in − H out)/H out to charac-terize the void expansion rate, where Hin and Hout are the Hubbleparameters inside and outside the void, respectively. (Other authorshave used the ‘jump’ J ≡ Hin/Hout = 1 + δH to characterizethe void.) The reduced Hubble parameter h is defined as usual byH out = 100 h km s −1 Mpc−1 throughout.

The local void scenario has been investigated by several au-thors using a variety of methods (Celerier 1999; Tomita 2000;Tomita 2001a,b,c; Iguchi, Nakamura & Nakao 2002; Tomita 2003;Mansouri 2005; Moffat 2005a,b, 2006; Alnes, Amarzguioui &Gron 2006; Chung & Romano 2006; Garfinkle 2006; Vanderveld,Flanagan & Wasserman 2006; Alexander et al. 2009; Alnes &Amarzguioui 2007; Biswas, Mansouri & Notari 2007; Caldwell& Stebbins 2008; Clarkson, Bassett & Lu 2008; Clifton, Ferreira &Land 2008; Garcia-Bellido & Haugboelle 2008a,b; Uzan, Clarkson& Ellis 2008). In a series of papers, Tomita modelled the void asan open Friedmann–Robertson–Walker (FRW) region joined by asingular mass shell to an FRW background, and found that a voidwith radius 200 Mpc and δH = 0.25 fits the supernova Hubblediagram without dark energy (Tomita 2001c). Alnes et al. (2006)showed that an LTB region which reduces to an E–deS cosmologywith h = 0.51 at a radius of 1.4 Gpc with δH = 0.27 can matchboth the supernova data and the location of the first acoustic peakin the CMB. Alexander et al. (2009) attempted to find the small-est possible void consistent with the current supernova results –their LTB-based ‘minimal void’ model has a radius of 350 Mpcand J � 1.2, i.e. δH � 0.2; a void of similar size but with δH =0.3 had been discussed earlier (Biswas et al. 2007). Unfortunately,since this model is equivalent to an E–deS universe with h = 0.44outside the void where the Sloan Digital Sky Survey (SDSS) lumi-nous red galaxies lie, as it stands it is unable to fit the measurementsof the BAO peak at z ∼ 0.35 (Blanchard et al. 2006). LTB modelsof much larger voids were considered by Garcia-Bellido & Haug-boelle (2008a) (with radii of 2.3 and 2.5 Gpc and Hubble contrastsof 0.18 and 0.30, respectively), and it was demonstrated that theycan fit the supernova data, BAO data and the location of the firstCMB peak. Clifton et al. (2008) found the best fit to the SNe Ia datafor a void of radius 1.3 ± 0.2 Gpc and an underdensity of about70 per cent at the centre, and Bolejko & Wyithe (2008) confirmedthat such a void provides an excellent fit to the latest ‘Union dataset’ (Kowalski et al. 2008). Moreover, Inoue & Silk (2006) haveshown that the unexpected alignment of the low multipoles in theCMB anisotropy can be attributed to the existence of a local void ofradius 300 h−1 Mpc. These authors also suggested that the anoma-lous ‘cold spot’ in the WMAP southern sky is due to a similar voidat z ∼ 1 and some evidence for this has emerged subsequently

C© 2009 The Authors. Journal compilation C© 2009 RAS, MNRAS 401, 547–558

Constraints on large voids from WMAP5 and SDSS 549

(Rudnick, Brown & Williams 2007). Recently, a large number ofvoids of varying sizes have been identified in the SDSS LuminousRed Galaxy (LRG) catalogue in a search for the late integratedSachs–Wolfe (ISW) effect due to dark energy (Granett, Neyrinck &Szapudi 2008b).

How likely is the existence of such huge voids according tostandard theories of structure formation? Statistical measures ofthe void distribution such as the void probability function andunderdense probability function have been estimated from theTwo-degree Field Galaxy Redshift Survey (2dfGRS), SDSS andDeep Extragalactic Evolutionary Probe 2 (DEEP2) galaxy red-shift surveys (Croton et al. 2004; Hoyle & Vogeley 2004; Con-roy et al. 2005; Patiri et al. 2005; Tikhonov 2006; Tinker et al.2008; Tikhonov 2007; von Benda-Beckmann & Mueller 2007).Void probability statistics have also been examined theoreticallyusing analytical methods (Sheth & van de Weygaert 2004; Furlan-etto & Piran 2006; Shandarin et al. 2006) and N-body simulations(Little & Weinberg 1994, Schmidt et al. 2000; Arbabi-Bidgoli &Mueller 2002; Benson et al. 2003; Padilla, Ceccarelli & Lambas2005). However, such studies have been restricted to voids withradii of 10–30 Mpc. The scales of the large voids we are consid-ering lie in the linear regime where the variance of the Hubblecontrast is directly related to the matter power spectrum Pm(k).It has been noted (using results from Turner, Cen & Ostriker1992) that above 100 Mpc linear theory predictions agree wellwith N-body simulation results, although on smaller scales theHubble contrast is underestimated due to non-linear effects (Shi,Widrow & Dursi 1996). Applying the linear theory and using themeasured CMB dipole velocity, Wang, Spergel & Turner (1998)obtained the model-independent result 〈δH〉1/2

R < 10.5 h−1 Mpc/Rin a sphere of radius R. (This ought to be an acceptable procedureup to scales of the order of 800 h−1 Mpc – on larger scales, rela-tivistic corrections become increasingly important.) In this paper,we update these results by determining the probability distributionof δH and the density contrast on various scales using constraints onPm(k) from WMAP 5-year data (Komatsu et al. 2008) and the SDSSgalaxy power spectrum (Tegmark et al. 2004). We find that even the‘minimal local void’ is extremely unlikely if the primordial densityperturbation is indeed Gaussian as is usually assumed and the otherLTB model voids even less so. However by the same token, theISW effect due to the voids seen in the SDSS LRG survey (Granettet al. 2008b) appears to be too strong. Moreover, observed large-scale peculiar velocities appear to be much higher than expected(Kashlinsky et al. 2008; Watkins, Feldman & Hudson 2008). Itwould appear that the standard model of structure formation it-self needs re-examination, hence the existence of a large local voidcannot be dismissed on these grounds.

2 MO D E L S

We study variations of the Hubble parameter in the context of twodifferent cosmological models, both of which fit the WMAP andSDSS data but have different amounts of power on spatial scalesof O(100) Mpc. The intention is to examine whether previous con-clusions concerning the magnitude of such variations (Wang et al.1998) can be circumvented in an unorthodox model.

Our first model is the standard �CDM concordance model witha power-law primordial power spectrum. The spectral index andamplitude PR of the comoving curvature perturbation spectrum areevaluated at a pivot point of k = 0.05 Mpc−1. The second model isdubbed the ‘CHDM bump model’ since it has both cold and hot darkmatter and a ‘bump’ in the primordial spectrum. It was developed

by us (Hunt & Sarkar 2007) based upon the supergravity multipleinflation scenario in which ‘flat direction’ fields undergo gaugesymmetry-breaking phase transitions during inflation triggered bythe fall in temperature (Adams, Ross & Sarkar 1997; Hunt & Sarkar2004). Each flat direction ψ has a gravitational strength coupling tothe inflaton φ, giving a contribution to the potential of the form V ⊂12 λφ2ψ2. The flat directions are lifted by supergravity correctionsand non-renormalizable superpotential terms. Thus, when a phasetransition occurs the flat direction evolves rapidly from the originwhere it was trapped by thermal effects to the global minimum ofthe potential. Each phase transition changes the effective inflatonmass from m2

φ to m2φ − λ〈ψ〉2. Since the primordial power spectrum

is very sensitive to the inflaton mass, this can introduce features intothe spectrum. We showed that two flat directions ψ1 and ψ2, whichcause successive phase transitions about two e-folds apart and createa small bump in the power spectrum centred on k � 0.03 h Mpc−1,allow an E–deS model with h = 0.44 to fit the WMAP data (Hunt& Sarkar 2007). The effective scalar potential is

V (φ, ψ1, ψ2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

V0 − 12 m2φ2, t < t1,

V0 − 12 m2φ2 − 1

2 μ21ψ

21

+ 12 λ1φ

2ψ21 + γ1

Mn1−4P

ψn1 , t2 ≥ t ≥ t1,

V0 − 12 m2φ2 − 1

2 μ21ψ

21

+ 12 λ1φ

2ψ21 + γ1

Mn1−4P

ψn1

− 12 μ2

2ψ22 − 1

2 λ2φ2ψ2

2

+ γ2

Mn2−4P

ψn2 , t ≥ t2,

(1)

where t1 and t2 are the times at which the first and second phasetransitions begin, λ1 and λ2 are the couplings between φ and the flatdirections, γ 1 and γ 2 are the coefficients of the non-renormalizableterms of the order of n1 and n2, and V0 is a constant which dominatesthe potential. In the slow-roll approximation, the height of the bumpis PR

(1) and the amplitude of the primordial perturbation spectrumto the left and right of the bump is PR

(0) and PR(2), respectively,

where

PR(0) = 9H 6

4π2m4φ20

, (2)

PR(1) = P (0)

R(1 − �m2

1

)2 , (3)

PR(2) = PR

(0)(1 − �m2

1 + �m22

)2 , (4)

where φ0 is the initial value of φ and

�m21 = λ1

m2

(μ2

1Mn1−4P

n1γ1

)2/(n1−2)

, (5)

�m22 = λ2

m2

(μ2

2Mn2−4P

n2γ2

)2/(n2−2)

(6)

are the fractional changes in the inflaton mass-squared due tothe phase transitions. The bump lies approximately between thewavenumbers k1 and k2, where k2 = k1eH (t2−t1). In this paper, weset γ 1 and γ 2 equal to unity, m2 = 0.005H 2, φ0 = 0.01MP, μ2

1 =μ2

2 = 3H 2 and λ1 = λ2 = H 2/M2P throughout as in our earlier

work (Hunt & Sarkar 2007). In fitting to the WMAP5 data, we alsoconsider continuous (non-integral) values of n1 and n2 to determine



whether a different shape of the ‘bump’ gives a better fit, keeping inmind that its physical origin may be different from multiple inflation(Chung et al. 2000; Lesgourgues 1999; Easther et al. 2001; Kaloper& Kaplinghat 2003; Gong 2005; Wang et al. 2005; Ashoorian &Krause 2006; Bean et al. 2008).

A pure cold dark matter (CDM) model exhibits excessive galaxyclustering on small scales. Therefore, it is necessary to include anHDM component which suppresses structure formation below thefree-streaming scale. We obtain a good match to the shape of theSDSS galaxy power spectrum with three neutrino species of mass∼0.5 eV. Hence, the CHDM bump model has �b � 0.1, �ν �0.1and �c � 0.8 (Hunt & Sarkar 2007).

3 THE DATA SETS

We fit to the WMAP 5-year (Nolta et al. 2008) temperature–temperature (TT), temperature–electric polarization (TE), andelectric–electric polarization spectra. Compared to the WMAP3 re-sults, the WMAP5 measurement of the TT spectrum is ∼2.5 per centhigher in the region of the acoustic peaks due to the revised beamtransfer functions, and the third acoustic peak is determined moreaccurately. Polarization measurements are improved by the use ofdata from an additional waveband.

We also fit the linear matter power spectrum Pm(k) to the mea-surement of the real-space galaxy power spectrum Pg(k) in theSDSS (Tegmark et al. 2004).

4 ME T H O D

The Hubble contrast δH smoothed over a sphere of radius R is (Shiet al. 1996)

δH(x)R =∫

d3 yv( y)

Hout· y − x| y − x|2 WR( y − x), (7)

where v is the peculiar velocity field and WR is the ‘top hat’ windowfunction,

WR(x) ={

3/(4πR3), |x| ≤ R,

0, |x| > R.(8)

Using the linear perturbation theory (Peebles 1993), it can be shownthat the variance of δH is related to the matter power spectrum as(Wang et al. 1998)

⟨δ2

H

⟩R

= f 2

2π2

∫ ∞

0dk k2Pm (k) W 2

H (kR) . (9)

Here, the window function WH is

WH (kR) = 3

k3R3

(sin kR −

∫ kR

0dy

sin y

y

), (10)

and the dimensionless linear growth rate f for a �CDM universecan be approximated by (Lahav et al. 1991; Hamilton 2001)3

f (�m, ��) � �4/7m + ��

70

(1 + �m

2

). (11)

Similarly, the variance of the density contrast δ ≡ (ρin − ρout) /

ρout in a sphere of radius R is

〈δ2〉R = 1

2π2

∫ ∞

0dk k2Pm (k) W 2 (kR) , (12)

3 Hamilton (2001) emphasized that the power-law exponent is 4/7 for a high-density universe, so we have corrected the previous formula from Lahav et al.(1991) accordingly.

where the window function

W (kR) = 3

k3R3(sin kR − kR cos kR) (13)

is the Fourier transform of WR.The variance of the peculiar velocity is given by

〈v2〉R = f 2H 2out

2π2

∫ ∞

0dkPm (k) W 2 (kR). (14)

Finally, we also consider �in = 8πGρ in/H2in, the ratio of the

matter density to the critical density as measured locally by anobserver inside the void (Wang et al. 1998). The variance of theperturbation δ� ≡ (�in − �m)/�m is then

〈δ2�〉R = 1

2π2

∫ ∞

0dk k2Pm (k) W 2

� (kR), (15)

where

W� (kR) = 3

k3R3

[(2f − 1) sin kR

+ kR cos kR + 2f

∫ kR

0dy

sin y

y

]. (16)

We use the Monte Carlo Markov Chain (MCMC) approach tocosmological parameter estimation, which is a method for drawingsamples from the posterior distribution P (� | data) of the cosmo-logical parameters � , given the data. For a discussion of the MCMClikelihood analysis, see appendix B of Hunt & Sarkar (2007). Givenn samples � (i) the best estimate for the distribution is

P (� |data) � 1

n

n∑i=1

δD[� − � (i)], (17)

where δD is the Dirac delta function. The CMB angular powerspectrum and the matter power spectrum [corrected for non-linearevolution using the ‘Halofit’ (Smith et al. 2002) procedure] of eachmodel are calculated using a modified version of the CAMB4 cosmo-logical Boltzmann code (Lewis, Challinor & Lasenby 2000) follow-ing the approach of Hunt & Sarkar (2007). While the temperatureof the CMB monopole would be affected if we are located nearthe centre of a spherically symmetric void, secondary anisotropiesdue to the void are expected to decay rapidly for higher multipoles(Alexander et al. 2009). We therefore neglect the possible effectsof the void on the angular power spectra since these are importantonly at low multipoles where the cosmic variance is large. Theintegrals for the variances in equations (9, 12–15) were evaluatednumerically. Care was taken to ensure that the precise values of theintegration limits did not affect the results. We compute f numeri-cally using the GROWλ software package (Hamilton 2001). We usea version of the COSMOMC5 package (Lewis & Bridle 2002) modi-fied to include 〈δ2

H〉1/2R , 〈δ2〉1/2

R , 〈v2〉1/2R and 〈δ2

�〉1/2R for eight values

of R as additional derived parameters determined from the baseMonte Carlo parameters. It follows from equation (17) that one-dimensional marginalized distributions of these quantities for eachR value are obtained by plotting histograms of the samples. Theprobability distribution P (δH|data)R of δH on the scale R given thedata can be written as

P (δH| data)R =∫

P (δH|� )RP (� |data) d� . (18)

4 http://camb.info5 http://cosmologist.info/cosmomc/



Table 1. The priors adopted on the base Monte Carlo parameters of the various models, as well as on thederived parameters: the Hubble constant and the age of the Universe.

Parameter Model

�CDM power law CHDM bump with CHDM bump withn1 = 12, n2 = 13 n1, n2 continuous

Lower limit Upper limit Lower limit Upper limit Lower limit Upper limit

�bh2 0.005 0.1 0.005 0.1 0.005 0.1

�ch2 0.01 0.99

θ 0.5 10.0 0.5 10.0 0.5 10.0

τ 0.01 0.8 0.01 0.8 0.01 0.8

f ν 0.01 0.3 0.01 0.3

ns 0.5 1.5

104k1(Mpc −1) 0.01 600 0.01 600

104k2(Mpc −1) 0.01 800 0.01 1100

ln(1010PR

)2.7 4.0

ln[1010PR(0)] 2.0 6.0 2.0 6.0

ln[1010PR(1)] 2.0 6.0

ln[1010PR(2)] 2.0 6.0

h 0.4 1.0 0.1 1.0 0.1 1.0

Age (Gyr) 10.0 20.0 10.0 20.0 10.0 20.0

Using equation (17), this is approximated by

P (δH| data)R = 1

n

n∑i=1

P[δH|� (i)

]R

, (19)

where

P (δH|� )R = 1√2π

⟨δ2

H

⟩R

exp

(− δ2

H

2⟨δ2

H

⟩R

). (20)

We calculate the probability distribution P (δ|data)R in the sameway.

Flat priors are used on the parameters listed in Table 1. Here, θ

is the ratio of the sound horizon to the angular diameter distanceto last scattering (multiplied by 100), τ is the optical depth (dueto reionization) to the last scattering surface and f ν ≡ �ν/�d isthe fraction of dark matter in the form of neutrinos, where the totaldark matter density is �d ≡ �c + �ν . We assume the chains haveconverged when the Gelman–Rubin ‘R’ statistic falls below 1.02.We evaluate the sum in equation (19) when post-processing thechains.

5 R ESULTS

The mean values of the marginalized cosmological parameters to-gether with their 68 per cent confidence limits are listed in Table 2.As in our previous work (Hunt & Sarkar 2007), we also list the valueof the AIC relative to the �CDM power-law model. Recall that theAIC is defined as AIC ≡ −2 lnLmax + 2N (Akaike 1974), whereLmax is the maximum likelihood and N is the number of parameters.It is a commonly used guide for judging whether additional pa-rameters are warranted given the increased model complexity, andquantifies the compromise between improving the fit and addingextra parameters.

The CHDM ‘bump’ model with n1 = 12 and n2 = 13 has aχ 2 equal to the �CDM power-law model. Allowing n1 and n2 tovary freely further improves the fit to the data with the consequencethat the CHDM model with n1 and n2 continuous is favoured overthe �CDM model according to the AIC. The primordial powerspectrum of the models is shown in Fig. 1 together with the fit to theWMAP TT and TE spectra and the SDSS galaxy power spectrum.

The uncertainties of the derived parameters are smaller comparedto those derived from the WMAP 3-year results, as would be ex-pected for higher quality data. For example, the optical depth dueto reionization for the CHDM model with continuous n1 and n2 hasgone from τ = 0.075+0.012

−0.012 to τ = 0.0771+0.0073−0.0083 due to the more

accurate polarization measurements. The shape of the ‘bump’ inthe primordial power spectrum for the CHDM model with con-tinuous n1 and n2 is slightly changed by the new data. Althoughthe quantity ln[1010PR

(0)] is almost unaltered, ln[1010PR(1)] has

increased slightly from a 3-year value of 3.429+0.048−0.049 to a 5-year

value of 3.462+0.036−0.036 because of the increased amplitude of the TT

spectrum for multipoles � > 200. Due to the increased height of thethird acoustic peak, ln[1010PR

(2)] has increased from 3.091+0.071−0.067 to

3.183+0.043−0.041 and 104k2 fallen from 585+36

−82 to 500+21−54 Mpc−1. The in-

creased amplitude of the primordial power spectrum on small scaleshas raised σ 8 from a value of 0.662+0.063

−0.064 to 0.700+0.098−0.098.

The mean values of the variances 〈δ2H〉R, 〈δ2〉R, 〈v2〉R and 〈δ2

�〉R,together with their 1σ limits, are plotted in Fig. 2. The differentvariances in the two models can be understood with reference tothe matter power spectrum. From the relativistic Poisson equation,a given density perturbation leads to a larger curvature perturbationin a higher density universe. Since the amplitude of the primordialcurvature perturbation is similar in both models (as can be seen fromFig. 1), the density contrast during the early matter dominated erais greater in the �CDM universe than in the higher density CHDMuniverse. Although the growth of density perturbations at late timesis suppressed in a low-density universe, this means that the matter



Table 2. The marginalized cosmological parameters for the various models (with 1σ limits).The 12 parameters in the upper section of the table are varied by COSMOMC, while those in thelower section are derived quantities. The χ2 of the fit is given, as is the AIC relative to thepower-law �CDM model.

Parameter Model

�CDM power law CHDM bump with CHDM bump withn1 = 12, n2 = 13 n1, n2 continuous

�bh2 0.022 34+0.000 60

−0.000 61 0.016 74+0.000 41−0.000 47 0.017 62+0.000 95

−0.000 95

�ch2 0.1144+0.0046

−0.0046

θ 1.0397+0.0029−0.0031 1.0311+0.0039

−0.0039 1.0332+0.0048−0.0047

τ 0.0842+0.0077−0.0082 0.0721+0.0069

−0.0075 0.0771+0.0073−0.0083

f ν 0.114+0.015−0.012 0.085+0.015

−0.022

ns 0.961+0.014−0.014

104k1(Mpc −1) 81.7+8.5−8.3 87+11

−11

104k2(Mpc −1) 442+47−53 500+21

−54

ln(1010PR

)3.078+0.037

−0.037

ln[1010PR(0)] 3.294+0.031

−0.031 3.274+0.048−0.048

ln[1010PR(1)] 3.462+0.036

−0.036

ln[1010PR(2)] 3.183+0.043

−0.041

�ch2 0.1450+0.0079

−0.0077 0.156+0.012−0.013

�dh2 0.1634+0.0042

−0.0045 0.1702+0.0073−0.0074

h 0.695+0.021−0.021 0.4244+0.0052

−0.0055 0.4333+0.0093−0.0094

Age (Gyr) 13.78+0.14−0.14 15.36+0.20

−0.19 15.05+0.33−0.32

�m 0.284+0.025−0.025

�� 0.716+0.025−0.025

σ 8 0.817+0.027−0.027 0.617+0.059

−0.055 0.700+0.098−0.098

zreion 11.0+1.4−1.4 13.0+2.0

−2.0 13.4+2.1−2.0

�m21 0.074 95+0.000 46

−0.000 46 0.089+0.020−0.020

�m22 0.151 33+0.000 84

−0.000 84 0.136+0.015−0.016

H (t2 − t1) 1.68+0.12−0.13 1.73+0.14

−0.15

χ2 1339.9 1339.9 1330.2

�AIC 0 6.0 −3.7

power spectrum of the �CDM universe is larger on all scales thanthat of the CHDM universe, when measured in units of h−3 Mpc3.(This is not evident in Fig. 1 where the galaxy power spectrum isshown – the galaxies are more biased in the CHDM universe thanin �CDM so the matter power spectrum is lower.)

This also explains why, as seen in Fig. 2, 〈δ2〉R is uniformlygreater for the �CDM model. The linear growth factor f is smallerfor the �CDM universe, and the peak in the matter power spectrumoccurs at a larger scale. Thus, the quantity f 2Pm(k) which appearsin equation (9) is greater for the �CDM universe for wavenumbersbelow kcross � 0.01 h Mpc−1 but is greater for the CHDM uni-verse for wavenumbers above kcross. The window function WH (10)makes 〈δ2

H〉R sensitive to the value of f 2Pm(k) for the wavenum-ber k � π/R. Consequently, the 〈δ2

H〉R curves for the two modelscross at the scale π/kcross � 300 h−1 Mpc as seen in Fig. 2. The two〈v2〉R curves cross at a smaller scale of about 100 h−1 Mpc. This

is because the integral (14) for the variance in the peculiar veloc-ity 〈v2〉R is more strongly weighted towards small wavenumbersthan the corresponding expression equation (9) for the variancein the Hubble contrast 〈δ2

H〉R, which has an additional factor ofk2. Finally for 〈δ2

�〉R the situation is intermediate between that forthe variance in the density contrast and the variance in the Hub-ble contrast, since only some of the terms in W� contain factorsof f .

The scale dependence of 〈δ2H〉R is the reason that the P (δH|data)

distribution is broader for the �CDM power law and the CHDM‘bump’ models on scales above and below 300 h−1 Mpc, respec-tively, as shown in Fig. 3. Similarly, the P (δ|data)R distribu-tion is broader for the �CDM model on all scales, as seen inFig. 4.

To illustrate our findings, we calculate the probability of a fluc-tuation in the Hubble contrast greater than or equal to a given value



CHDM bump

ΛCDM power-law

k hMpc−1

P R( k

)10

−9

1 1010−4 10−3 10−2 10−1

2 0

2 8

2 4

3 2

CHDM bump

ΛCDM power-law

Multipole moment (l)

l (l+

1)C

l2π

μK

2

01 10 100 1000

2000

4000

6000

CHDM bump

ΛCDM power-law

Multipole moment (l)

(l+

1)C

l2π

μK

2

0

−1

1

−0 5

0 5

1 5

10 100 1000

CHDM bump

ΛCDM power-law

k hMpc−1

P(k

)M

pch−

13

104

10310−2 10−1

Figure 1. The top-left panel shows the primordial perturbation spectrum for the CHDM bump model (with n1 = 12 and n2 = 13) and for the �CDM power-lawmodel with ns � 0.96. The top-right and bottom-left panels show the best fits for both the models to the WMAP5 TT and TE spectra, while the bottom-rightpanel shows the best fits to the SDSS galaxy power spectrum.

CHDM bump

ΛCDM power-law

R h−1 Mpc

δ2 H1

2R

0001001

10−3

10−2

10−1

CHDM bump

ΛCDM power-law

R h−1 Mpc

δ21

2R

0001001

10−3

10−2

10−1

CHDM bump

ΛCDM power-law

R h−1 Mpc

δ2 Ω1

2R

000100110−3

10−2

10−1

CHDM bump

ΛCDM power-law

R h−1 Mpc

v2

12

R10

0km

s−1

1

0001001

5

Figure 2. The variation with increasing void radius of the variance of the Hubble parameter, the density contrast, the density parameter and the peculiarvelocity for the �CDM power-law and CHDM bump models, given the WMAP5 and SDSS data (with 1σ limits).



CHDM bump

ΛCDM power-lawR = 40 h−1 Mpc

δH

P(δ

Hdat

a)

−0 1 0.1 0.2−0 20

0

2

4

10

6

8

CHDM bump


δH

P( δ

Hdat

a)

0.05−0.05−0. 1.010

0

10

5

15

CHDM bump


δH

P( δ

Hdat

a)

0.04−0.040

0

10

5

15

20

25

−0. 080 .08

CHDM bump

ΛCDM power-lawR = 150h−1 Mpc

δHP

( δH

dat

a)0.02 0.04−0.02−0.04

00

10

20

30

40

CHDM bump


δH

P(δ

Hdat

a)

0.01 0.02−0.020

0

20

40

60

−0. 010 .03−0.03

CHDM bump


δH

P(δ

Hdat

a)

00

40

−0.012 −0.006 0.006 0.012

80

120

CHDM bump


δH

P(δ

Hdat

a)

00

100

200

300

−0. 0600 .006−0.003 0.003

CHDM bump


δH

P( δ

Hdat

a)

00

200

400

0.001−0.002 −0. 0100 .002

600

800

Figure 3. The probability distribution of the Hubble contrast (with 1σ limits), given the WMAP5 and SDSS data, for the �CDM power-law and CHDM bumpmodels, for spherical voids of radius R = (40, 70, 100, 150, 200, 300, 500, 800) × h−1 Mpc.

δ0H in a sphere of radius R, given by

Probability(δH ≥ δ0

H

)R

=∫ ∞

δ0H

P (δH|data)R dδH. (21)

Since P (δH|data)R is symmetric, this is also equivalent to the prob-ability of a fluctuation being less than or equal to −δ0

H. As seen inFig. 5, the probability of a large excursion in δH is largest on smallscales, in accordance with physical intuition. Note that the proba-

bility on all scales tends to a value of half for small δ0H, because

the fluctuation has an equal probability of being positive or nega-tive. The probability is greater for the CHDM model than for the�CDM model on small scales because the P (δH|data)R distributionis broader for the CHDM model on these scales. Conversely sincethe distribution is broader on large scales for the �CDM model, theprobability is greater there for this model.

Similarly we calculate the probability of a fluctuation in the den-sity contrast less than or equal to a given value −δ0 in a sphere of



CHDM bump


δ

P(δ

dat

a)

−0 08 80.4−0 40

0

1

2

3

0 5

1 5

2 5

CHDM bump


δ

P(δ

dat

a)

0.2 0.4−0 2−0 40

0

1

2

3

4

5

6

CHDM bump


δ

P(δ

dat

a)

−0.1 0.1 0.2−0.20

0

2

4

10

6

8

CHDM bump


δP

(δdat

a)0.05−0.05−0. 1.01 0.15

00

10

5

15

20

−0.15

CHDM bump


δ

P(δ

dat

a)

0.06 0.09−0.06−0.090

0

10

20

30

0.03−0.03

CHDM bump


δ

P(δ

dat

a)

0.02 0.04−0.02−0.040

0

10

20

30

40

50

60

CHDM bump


δ

P(δ

dat

a)

0.01 0.02−0.020

0

100

150

50

−0.01

CHDM bump


δ

P(δ

dat

a)

00

100

200

300

400

−0. 0600 .006−0.003 0.003−0. 0900 .009

Figure 4. The probability distribution of the density contrast (with 1σ limits), given the WMAP5 and SDSS data, for the �CDM power-law and CHDM bumpmodels, for spherical voids of radius R = (40, 70, 100, 150, 200, 300, 500, 800) × h−1 Mpc.

radius R, which is given by

Probability(δ ≤ −δ0

)R

=∫ −δ0

−∞P (δ|data)R dδ. (22)

This probability is greater for the �CDM model on all scales asseen in Fig. 6, due to the broader P (δ|data)R distribution.

Moreover, we can determine the probability of one or more voidswith comoving volume V1 occurring within some larger comovingvolume V2. If the ratio V 2/V 1 is N to the nearest integer and p isthe probability of a void with volume V1, then the probability of

n voids within V2 is(

N

n

)pn(1 − p)N−n, where

(N

n

)is the binomial

coefficient. The expected number of voids within V2 is Np.

6 D ISCUSSION

A void with δH � 0.2–0.3 and a radius exceeding 100 h−1 Mpcis required to fit the supernova data without dark energy (Tomita2001c; Alexander et al. 2009; Biswas et al. 2007). The probabilitythat we are situated in such a void is less than 10−12 as can be seen



CHDM bump

δ0H

Pro

bab

ility

δ H≥

δ0 H

−1 Mpc = 800

1

100

10−12

10−10

10−8

10−6

10−4

10−4 10−3

10−2

10−2 10−1

150200

40

70300500

ΛCDM power-law

δ0H

Pro

bab

ility

δ H≥

δ0 H

−1 Mpc = 800

1

100

10−12

10−10

10−8

10−6

10−4

10−4 10−3

10−2

10−2 10−1

150200 4070300500

Figure 5. The probability of a fluctuation in the Hubble contrast greater than or equal to a given value δ0H in a sphere of radius R (with 1σ limits), given the

WMAP5 and SDSS data, for the �CDM power-law and CHDM bump models for R = (40, 70, 100, 150, 200, 300, 500, 800) × h−1 Mpc.

CHDM bump

δ0

Pro

bab

ility

δ≤

−δ0

−1 Mpc = 800

1

100

10−12

10−10

10−8

10−6

10−4

10−3

10−2

10−2 10−1

150200

40

70300500

ΛCDM power-law

δ0

Pro

bab

ility

δ≤

−δ0

−1 Mpc = 800

1

100

10−12

10−10

10−8

10−6

10−4

10−3

10−2

10−2 10−1

150200

40

70

300500

Figure 6. The probability of a fluctuation in the density contrast less than or equal to a given value δ0 in a sphere of radius R (with 1σ limits), given theWMAP5 and SDSS data, for the �CDM power-law and CHDM bump models for R = (40, 70, 100, 150, 200, 300, 500, 800) × h−1 Mpc.

from Fig. 5. The probability is exponentially smaller for the largervoids of Gpc size that have also been considered (Alnes et al. 2006;Clifton et al. 2008; Garcia-Bellido & Haugboelle 2008a).6

However, before we dismiss the possibility of a local void onthese grounds, we should also evaluate the probability of voidswhich have actually been claimed to exist elsewhere in the uni-verse. For example, it has been argued that a void with radius 200–300 h−1 Mpc and a density contrast of δ = −0.3 at z ∼ 1 can ac-count for the WMAP ‘cold spot’ in a �CDM universe (Inoue & Silk2006). Even if we conservatively take the radius to be 150 h−1 Mpc(and the same underdensity), the probability that one or more suchvoids lie within the volume out to z = 1 is only 1.05+5.24

−0.93 × 10−10.It has been argued that the WMAP cold spot may not be a local-

ized feature (Naselsky et al. 2007) and there may be no matchingvoid in the NRAO VLA Sky Survey (NVSS) radio source catalogue(Smith & Huterer 2008); however, an equally striking anomalyarises if we consider the large number of voids which have beenidentified in the SDSS LRG survey in a search for the late ISWeffect (Granett, Neyrinck & Szapudi 2008a; Granett et al. 2008b).These are of angular radius ∼40 corresponding to a (comoving)radius of ∼50 h−1 Mpc and are tabulated as having 1σ , 2σ or 3σ

6 There is a further constraint on Gpc scale voids from the observed ab-sence of a ‘y-distortion’ in the spectrum of the CMB (Caldwell & Stebbins2008) and from the ‘kinetic Sunyaev–Zeldovich’ effect observed for X-ray-emitting galaxy clusters (Garcia-Bellido & Haugboelle 2008b). However,this has no impact on smaller voids.

underdensities. These numbers relate to the detection significance(the likelihood of detecting the void by chance out of a Poisson dis-tribution) rather than the likelihood of finding such underdensitiesin a Gaussian field which we have computed in this paper (Granett,private communication). Moreover, the observed LRGs are biasedwith regard to the dark matter, hence the underdensities in darkmatter are likely to be smaller than the quoted values.

However, if Granett et al. (2008a,b) have indeed detected thelate ISW effect as they assert, we can simply circumvent theseuncertainties by requiring that the voids be large enough and/orunderdense enough to yield the observed CMB temperature decre-ments. To calculate the late ISW effect, we consider the propagationof CMB photons to us from the last scattering surface through an in-tervening void. The photon temperature change caused by the voidis

�T

T= − 2

c2

∫ anear

afar

d�

dada, (23)

where afar is the scalefactor when the photon crossed the far side ofthe void and anear is the scalefactor when the photon crossed the nearside of the void. The gravitational potential of a void with properradius r is

� = 4πG

3r2ρbδ (a) , (24)

where the background density is given by ρb = 3H 20�m/8πGa3

and the density perturbation is given by δ(a) = D(a)δ(a0) where D



-80 -60 -40 -20

Temperature (μK)

Num

ber

00

10

−100

5

15

20

25

30

Num

ber

Probability of void in SDSS LRG volume

0.1

1

10

10−200 10−5010−10010−15010−250 1

Figure 7. The left-hand panel shows the ISW signals of the 50 voids detected by Granett et al. (2008a), calculated using equation (25); in order to match theobserved average ISW signal of −11.3 μK, it has been necessary to increase the void radii by a factor of 1.75 and the underdensities by a factor of 5. Theright-hand panel shows the probability of such voids occurring in the SDSS LRG survey volume according to the concordance �CDM model.

is the linear growth factor. Hence,

�T

T= �m

(R

c/H0

)2 [D(afar)

afar− D(anear)

anear

]δ. (25)

Using this we calculate the expected ISW signal for the 50 highestsignificance voids in table 4 of Granett et al. (2008a), employing theconcordance �CDM cosmology to determine afar and anear for eachvoid from the void redshift measurements. The ISW signal is foundto be only −0.42 μK on average if the dark matter underdensitiesare smaller than the observed underdensities in the LRG counts bythe bias factor of 2.2 (taking σ 8 = 0.8). This is in contrast to thedetected mean signal of −11.3 μK which is over 20 times bigger!We must therefore conclude that the void radii and/or underdensitieshave been significantly underestimated. The void radii can at mostbe increased by a factor of 1.75 within the quoted uncertaintiesso the observed signal of −11.3 μK can be matched only if theunderdensities are increased by a factor of 5 (implying a bias factorof 0.2). The CMB temperature decrements of such model voidscalculated using equation (25) are shown in Fig. 7 and are (byconstruction) similar to the actual measurements shown in Fig. 2 ofGranett et al. (2008b). While such an underbias for the observedLRGs may seem implausible, we emphasize that this is the onlyway in which the temperature decrements observed by Granett et al.(2008a,b) can be accounted for as being due to the late ISW effect.

Fig. 7 displays a histogram of the probabilities for finding suchvoids in the SDSS LRG survey volume (5 h−3 Gpc3 in the redshiftrange 0.4 < z < 0.75). The most improbable void is at z = 0.672 – inorder to yield the observed average CMB temperature decrement itmust have a density contrast of −0.72 (quoted galaxy underdensityof −0.316 multiplied by 5/2.2) and a radius of 230 h−1 Mpc (radiusderived from the quoted volume of 107 h−3 Mpc3 and multiplied by1.75). The probability of such a void is 1.9 × 10−247 according to ourcalculations. Although the linear theory may not be applicable forsuch a deep void, it is clear that its existence is in gross conflict withthe standard theory of structure formation from Gaussian primordialdensity perturbations.

This conclusion is strengthened by the recent detection of verylarge peculiar velocities on large scales. As seen in Fig. 2, theexpected variance of the peculiar velocity as calculated by equa-tion (14) is about 200 km s−1 on a scale of 100 h−1 Mpc, whereasthe measured value is at least five times higher, and the discrepancyis even bigger on larger scales up to 300 Mpc (Kashlinsky et al.2008).

It is also seen from Fig. 5 that if a determination of the Hub-ble constant is required with say 1 per cent accuracy, then mea-surements extending out to at least 150 h−1 Mpc must be made toovercome local fluctuations. A similar estimate was made by Li,Seikel & Schwarz (2008) who noted that the observed variance inmeasurements of h is in accord, thus consistent with the assump-tion of a Gaussian density field. However, the voids observed in theSDSS LRG survey (Granett et al. 2008b) call this assumption intoquestion. In particular whether there is a large local void is thenan issue that must be addressed observationally and not dismissedon the grounds that it is inconsistent with Gaussian perturbations.The Hubble flow is presently poorly measured in the redshift range0.1 � z � 0.3 – just where the effects of such a local void would bemost apparent (Alexander et al. 2009). Given that dark energy maywell be an artefact of such a void, this issue needs urgent attention.

The question of how such voids can have been generated withoutconflicting with the CMB observations is beyond the scope of thepresent work. Some suggestions have been made in the context ofmultifield inflationary models (Occhionero et al. 1997; DiMarco &Notari 2006; Itzhaki 2008).

AC K N OW L E D G M E N T S

This work was supported by an STFC Senior Fellowship award toSS (PPA/C506205/1) and by the EU Marie Curie Network ‘Uni-verseNet’ (HPRN-CT-2006-035863).

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