Most massive halos with Gumbel Statistics

0.21 MB·

by  Olaf Davis

Mon.Not.R.Astron.Soc.000,1–6(2010) Printed18January2011 (MNLATEXstylefilev2.2)
Most massive halos with Gumbel statistics
O. Davis1⋆, J. Devriendt1, S. Colombi2, J. Silk1 & C. Pichon1,2
1Astrophysics, University of Oxford, DenysWilkinson Building, Keble Road, Oxford OX1 3RH, UK
2Institut d’Astrophysique de Paris, CNRSUMR 7095 & UPMC, 98bis bd Arago, F-75014 Paris, France
1
1
0 ABSTRACT
2 We present an analytical calculation of the extreme value statistics for dark matter
halos - that is, the probability distribution of the most massive halo within some
n
region of the universe of specified shape and size. Our calculation makes use of the
a
J counts-in-cells formalismfor the correlationfunctions, and the halo bias derived from
the Sheth-Tormen mass function.
4
1 We demonstrate the power of the method on spherical regions, comparing the
resultsto measurementsina largecosmologicaldarkmatter simulationandachieving

] good agreement. Particularly good fits are obtained for the most likely value of the
O
maximummassandforthe high-masstailofthe distribution,relevantinconstraining
C cosmologies by observations of most massive clusters.
.
h Key words: methods: analytical – methods: statistical – dark matter – large-scale
p structure of Universe
-
o
r
t
s
a 1 INTRODUCTION tersandthestudyofColes(1988)onthehottesthotspotsof
[ thecosmicmicrowavebackgroundtemperaturefluctuations.
1 Extreme value (or Gumbel) statistics are concerned with The past year or so, however, has shown a resurgence
the extrema of samples drawn from random distributions.
v of interest in the application of extreme value statistics
Ifalarge numberofsamples aredrawnfrom somedistribu-
6 to cosmology and questions of extreme structures, as re-
tion,theCentralLimitTheoremstatesthattheirrespective
9 vealedeitherintheclusteringofgalaxies(Antalet al.2009;
meanswill follow a distribution which tends,in thelimit of
8 Yamila Yaryuraet al.2010),theprevalenceofmassiveclus-
2 large sample size, to a member of the family of normal dis- ters (Holz & Perlmutter 2010; Cayo´n et al. 2010) or the
. tributions.Analogously,themaximum(orminimum)values
1 temperatureextrema of theCMB (Mikelsons et al. 2009).
uofthesampleswillhaveadistributionwhoselargesample
0 Inthispaper,weareinterestedinthedarkmatterhalos
size limit –wheresuch astablelimit exists1 –isamember
1 ofmassivegalaxyclusters.Thenumberdensityofextremely
ofthefamilyofGeneralisedExtremeValue(GEV)distribu-
1 massive clusters is indeed a sensitive probe of the effects of
tions as detailed byGumbel (1958):
:
v the underlying cosmological model and laws of physics on
i large scales (Mantzet al. 2010). These include for instance
X −lnP (y)=(1+γy)−1/γ, y=(u−α)/β. (1) the equation of state of dark energy (Mantz et al. 2008),
GEV
r the possibility of modified gravity (Rapettiet al. 2010) the
a Theshapeparameterγ issensitivetotheunderlyingdistri-
physical properties of neutrinos (Mantzet al. 2010) and
bution from which the maxima are drawn, while α and β
primordial non-Gaussian density fluctuations (Cayo´n et al.
are position and scale parameters.
2010). Although the majority of the previously mentioned
Despite their wide use in other fields, extreme
studies focus on the growth of massive clusters, simply
value statistics have historically seen very little applica-
knowing the mass of the most massive cluster in a sur-
tion in astrophysics; some exceptions are the work of
vey can already provide strong constraints on cosmology
Bhavsar & Barrow (1985) on the brightest galaxies in clus-
(Holz & Perlmutter 2010).
Inthepresentwork,weoutlineananalyticalderivation
oftheextremalhalomassdistributioninstandardcosmolo-
⋆ E-mail:[email protected] gies with Gaussian initial conditions. Rather than taking a
1 Although it can be shown that where a stable limiting distri-
phenomenological approach, we aim to predict the distri-
bution exists it will take the form (1), certain pathological dis-
bution of the most massive halo in a region for any speci-
tributions give no such limit. For our purposes it is sufficient to
fied combination of power spectrum, cosmological parame-
notethatthelimitindeedexistsfordistributionswhichareofex-
ponential type, meaning the cumulative distribution function F ters and region size and shape. The paper is organized as
obeys limx→∞d/dx{(1−F(x))/F′(x)} = 0, and that this class follows. In § 2 we outline the basics of our method for ob-
includesallphysicaldistributionsreleventtoourapplications. taining an analytical expression of theGumbel distribution
(cid:13)c 2010RAS
2 O. Davis, J. Devriendt, S. Colombi, J. Silk & C. Pichon
of most massive clusters masses and make an explicit link and the typical number of halos above the threshold m in
with eq. (1). In § 3, the theoretical predictions are checked excess to theaverage in overdensepatches as:
against measurements in a very large N-body cosmological
N ≡nVξ¯h. (6)
simulation.Finally,§4follows withashortsummaryofthe c 2
main results and conclusions. In thehigh-m limit, thevoid probability can bewritten
P (m)=exp[−nVσ(N )], (7)
0 c
2 THEORY where thefunction σ(y)reads
Consideralargepatchoftheuniverse,whichcanbethought 1
σ(y)= 1+ θ e−θ, θeθ =y, (8)
of as representing the space covered by a volume-limited (cid:18) 2 (cid:19)
sampleofclusters,anddenotebym themassofthemost
max
(Bernardeau & Schaeffer1999).Notethat,aspointedoutby
massive dark matter halo in that patch. We wish to study
these authors, this expression for σ(y) follows from a spe-
analytically the Gumbel statistics, that is the probability
cific hierarchical behavior of higher-order correlation func-
distribution function p (m )dm of the values taken
G max max tions of very massive halos at large separations, ξ¯h ≃
by m if we sample a large number of such patches. Ob- N
max NN−2(ξ¯h)N−1.
viously, this distribution will depend on the size and shape 2
Wenowproceedtospecifythecumulativehalonumber
of the patch,as well as its redshift.
density and the average two-point correlation function of
halos in order to fully determinethe Gumbelstatistics.
2.1 General expression of the Gumbel statistics
Let usdefinethecumulative Gumbeldistribution by 2.2 Halo number density
m
P (m)≡Prob.(m 6m)≡ p (m )dm . (2) Thenumberdensityn(m,z)ofhalosatagivenmassmand
G max G max max
Z0 redshiftz,a.k.a.thehalomassfunction,weadoptistheone
Such a probability is also the probability P (m) that the calculatedbySheth& Tormen(1999).Itisbasedonamod-
0
patch is empty of halos of mass above the threshold m ification of the original model of Press & Schechter (1974),
(Colombi et al. 2010), hence which links the statistics of the initial matter density field
tothedistributionofvirializeddarkmatterhalosthrougha
dP
p (m)= 0. (3) spherical top hat description of their gravitational collapse.
G dm As a result, this mass function can be expressed as a uni-
Note that this assumes that there are no significant edge versal function of ν ≡ (δ /σ(m,z))2, where σ(m,z)2 is the
c
effects, i.e. that the boundaries of the catalog do not cross varianceoftheinitialdensityfieldsmoothedoverspheresof
too many clusters. This effect is negligible if the patch is radiusR(m)containinganaverageaveragemassmlinearly
large compared to thehalo size (and sufficiently compact). extrapolatedtoz,andδ isthecriticaloverdensitythreshold
c
Ifhalosareunclusteredthenthevoidprobabilityfollows needed to turn an initial spherical top hat density pertur-
simply from Poisson statistics, bation into a collapsed halo at redshift z. More specifically,
thenumberdensity of mass-m halos is given by:
P (m)=exp(−nV), (4)
0
where V is the volume of the patch and n=n(>m) m2n(m,z)dlogm =A 1+(aν)−p aν 1/2e−aν/2 (9)
the mean density of halos above mass m, with the ρ¯m dlogν (cid:0) (cid:1)(cid:16)2π(cid:17)
appropriate spatial average with redshift made implicit withρ¯m ≡Ωmρ¯theaveragedmatterdensityoftheUniverse.
n(>m)=hn[>m,z(x)]ix∈V. The shape of this mass function is parameterised by a and
WeexpectthePoisson limittobereachedforpatchsof p,and A is simply a normalisation factor.
size above a few hundred Mpc, where the matter distribu-
tion becomeshomogeneous. Below thispatchsize, however,
2.3 Halo correlation functions
halosaresignificantlyclustered.Inthatcase,thecalculation
of the void probability can be performed using a standard Atsufficientlylargeseparations,thetwo-pointcorrelationof
count-in-cell formalism if the connected N-point correla- halosofmassmcanberelatedtothatofthematterdensity
tionsfunctions,ξNh(x1,...,xN),ofhalosabovethethreshold through thebias function
areknown(e.g.,Szapudi& Szalay1993;Balian & Schaeffer
1989).Thesuperscripthinthepreviousexpressionindicates ξ2h(x1,x2,z)=b(m,z)2ξ2(x1,x2,z), (10)
halo correlation functions, while the naked ξ refer to cor-
N where ξ is the linear dark matter density autocorrela-
2
relations of the underlying matter density field. Since devi-
tion at redshift of interest. The function b(m,z) can be
ationsfromPoissonbehavioroccuronlyformoderatepatch
estimated analytically using the Press-Schechter formalism
sizes, the complex lightcone effects on the correlations in-
(Mo & White1996).Here,toremainconsistentwitheq.(9),
ducedbytheevolutionofclusteringwithredshiftinsidethe
weusetheexpressionforthebiasofSheth& Tormen(1999)
patch (e.g. Matsubara et al. 1997) can safely be neglected.
In particular, one can define the averaged correlations b(m,z)=1+ aν−1 + 2p/δc . (11)
overa patch of volume V: δc (1+aν)p
1 This result is valid in the regime where the separation
ξ¯h ≡ d3x ···d3x ξh(x ,··· ,x ), (5)
N VN Z 1 N N 1 N x = |x2 −x1| is large enough compared to the mass scale
V
(cid:13)c 2010RAS,MNRAS000,1–6
Most massive halos with Gumbel statistics 3
R(m).ThishasbeentestedsuccessfullyagainstN-bodysim- 3
-10
ulations by Mo et al. (1996) (see however e.g. Tinker et al.
2010,for possible improvementson eq.11). 3Mpc
m byWcealocbutlaatiningthtehebiwaseigohfthedaloasveerxacgeeeding mass threshold M) Sun-510
M / Sheth−Tormen, best fit
b(>m,z)= Rm∞b(∞mn′,(zm)n′,(zm)d′,mz)′dm′, (12) n / dlog(-710 + SShimeuthla−tTioonrmen, original
m d
R 9
and hence the averaged two-point correlation function for -10
halos abovethe threshold, uals 0.5
d
ξ¯2h(>m,z)=b(>m,z)2ξ¯2(z). (13) resi 0.0
Recallthatthisequationshouldbevalidintheregimewhere 1013 1014 1015
thepatch size is large compared to R(m), M / MSun
Figure 1. The upper panel shows the mass function of halos
3πm 1/3 inthe simulation (points), compared to the Sheth-Tormen mass
L≫R(m)= , (14)
(cid:18)4ρ¯ (cid:19) functionwith(p,a)equalto(0.3,0.707)and(−0.19,0.777)(solid
m
blueanddashedgreenlinesrespectively).Thelowerpanelshows
but small enough that light cone effects on the clustering theresidualsofthetwotheoretical curvescomparedtothedata;
inside it are negligible. i.e.(theory–data)/data.
3 NUMERICAL EXPERIMENT
2.4 Generalised Extreme Value parameterisation
To test our halo mass Gumbel distribution we compare the
The method outlined above allows us to compute the com- analytical result tomeasurements ontheHorizon 4ΠSimu-
plete distribution function of of the most massive clusters. lation (Teyssier et al. 2009),alargecosmological darkmat-
However, due to its complexity and the necessity of com- ter simulation performed using the RAMSES N-body code
putingsomeoftheintegralsnumerically,itdoesnotprovide (Teyssier 2002). The simulation followed the evolution of a
us with a neat analytic parameterisation of the distribu- cubic piece of the universe 2h−1Gpc on a side containing
tion.Therefore,inordertoachievesuchaparameterisation, 40963 particles,i.e.withaparticlemassof7.7h−1×109M⊙.
we turn to the general theory of extremes, eq. (1), using Initial conditions were based on the WMAP 3-year results
u = log m as the random variable. In order to calculate (Spergel et al.2007),withtheHubbleconstant,densityand
10
the parameters γ, α and β, we perform Taylor expansions characteristic parameters of the power spectrum given by
of the analytic PGEV, and P0 as computed by our method (h, ΩΛ, Ωm, Ωb, σ8, ns) = (0.73, 0.76, 0.24, 0.042, 0.77,
in thePoisson regime (eq.4).ThisTaylor expansion is per- 0.958). Halos in the simulation were identified at present
formedaboutthepeaksofthetwodistributionsdP /duand time, z = 0, using a ‘Friends-of-Friends’ algorithm (e.g.
0
dP /du.Equatingthefirsttwotermsintheseexpansions Zeldovich et al. 1982; Davis et al. 1985) with a standard
GEV
give usexpressions for the threeparameters: linkinglengthparametervaluegivenby0.2 timesthemean
interparticle distance.
(1+γ)(1+γ)
γ =n(>m )V −1, β= ,
0 n(m )Vm ln10
0 0
β 3.1 Fit of the mass function
α=log m − [(1+γ)−γ−1], (15)
10 0 γ
Any discrepancies between our derived Gumbel distribu-
where m is the mass at which the distribution dP /dz = tion and the true distribution can be thought of as arising
0 0
ln(10)mp (m)peaks–henceclosetothemostlikelyvalue from one of two sources: either inaccuracies in our chosen
G
of m – and is given implicitly by mass function, or inaccuracies due to the various assump-
tions made in proceeding from the mass function to p .
G
Aρ¯mV a e−aν0/2 1+(aν )−p = In order to quantify the respective contributions of each
m0 r2πν0 (cid:0) 0 (cid:1) of these two sources, we repeat our calculations with two
a 1 ap(aν )−(p+1) ν′′ sets of parameters for the mass function: (i) once taking
2 + 2ν + 1+(0aν )−p − ν′02, (16) the parameters used in Sheth& Tormen (1999), (p,a) =
0 0 0 (0.3,0.707),and,sincetheSheth&Tormenformisitsstan-
where ν , ν′ and ν′′ are ν and its derivatives with respect dard parametrisation is known to perform only approxi-
0 0 0
to m evaluated at m=m0. mately (e.g. Warren et al. 2006; Jenkins et al. 2001), (ii)
These equations, then, allow us to neatly summarise once with a best fit for a and p to the simulation’s mass
theinformation containedintheextremevaluedistribution function, leading to (p,a) = (−0.19,0.777). For this latter,
with thesingle parameterγ which describes itsshape. This we also weight bins by their mass, since it is the high-mass
statistic has the potential to be used as a tool to compare endofthedistributionwhichisofinteresttous.Fig.1shows
models with data or with each other, as Mikelsons et al. both these mass functions along with that measured in the
(2009) proposed for the CMB. simulation.
(cid:13)c 2010RAS,MNRAS000,1–6
4 O. Davis, J. Devriendt, S. Colombi, J. Silk & C. Pichon
5
2.
m) 2.0 1.5
g(
m) / d lo 01.5 1.0
P( 1.
d 0.5
5
0.
0 0
0. 0.
1015 1014 1015
m M m M
max Sun max Sun
(a) L=100/hMpc;γ=−0.11 (b) L=50/hMpc;γ=−0.14
2 Best fit with Poisson
1. 0.8 Best fit with bias
Original ST with bias
0
m) 1. 6 Asymptotic form
og( 0.8 0.
d l
m) / 0.6 0.4
dP( 0.4
2
2 0.
0.
0 0
0. 0.
1013 1014 1015 1012 1013 1014 1015
m M m M
max Sun max Sun
(c) L=20/hMpc;γ=−0.11 (d) L=8/hMpc;γ=−0.12
Figure 2. Distribution of largest cluster masses mmax. The y axis is probability density per log mass. Points are measurements from
the simulation with Poisson error bars for each mass bin. The solid blue line is the theoretical result using the best-fit mass function
(p=−0.19,a=0.777)andfullhaloclustering.GreendasheshaveinsteadtheoriginalSheth-Tormenparameters(p=0.3,a=0.707).Not
surprisingly,theydonotagreeaswellwiththemeasurementasthesolidblueline;reddotshavethebest-fitvaluesbutassumehalosare
Poissondistributed.Theorangedot-dashedlineistheGeneralisedExtremeValuedistribution,withparameterscalculatedasexplained
insection2.4andassumingPoissonstatistics.Allcalculationsusesphericalpatches,withradiusL=100,50,20,8h−1 Mpcrespectively.
3.2 Results theoretical results as it would have seemed from the value
ofγ alone:weareclosinginontheso-called“scaleofhomo-
Fig.2showsthedistributionpG(mmax)calculatedasabove, geneity” above which the matter distribution is essentially
both for Poisson statistics (equation 4,red dots) and incor- unclustered. Reducing the patch size to 50h−1 Mpc (fig.
porating full clustering (equation 7, blue solid lines) using 2(b))causesthefullclusteringandPoissoncurvestodiverge.
our best-fitmass function. The full clustering calculation is As expected, only the calculations incorporating clustering
alsoshownfortheoriginalSheth-Tormenparameters(green remain a good match to the simulation on these smaller
dashes). Points show measurements from the Horizon 4Π scales.
simulation for comparison.
Decreasing L further (fig. 2(c), 2(d)) causes even the
Fitting a Gumbel distribution eq. (1) to the data pre- calculations including clustering to diverge from the data
sentedinthefourpanelsofthisfigureyieldsasinglevalueof asourapproximations–inparticulartheexpression forthe
γ around-0.21±00..0012 withreducedχ2 61.1,whereasthean- functionσ(y)insection2.1andthecondition(14)–failout-
alytic prediction presentedin section 2.4 gives −0.146γ 6 sidethelarge-Llimit.Forinstance,wefindR(1013M⊙)=3
−0.1. This lack of agreement has its root in the fact that γ Mpc/h, and R(1014M⊙) = 6.4 Mpc/h which is a signifi-
isverysensitivetothehigherorder(skewnessandkurtosis) cant fraction of the respective patch sizes of 8 Mpc/h and
momentsofthedatadistributionarounditspeak,andthese 20 Mpc/h. Despite these limitations, the description of the
arepoorly capturedbyassumingPoisson statistics, evenon dataisstillsignificantlybetterthanthatofthePoisson cal-
100h−1 Mpcscales. culation andremainsimpressiveat thehigh-massend.This
However,forapatch(inthiscaseasphereofradiusL) is excellent news as it is this high-mass tail of the distri-
of size L = 100h−1 Mpc, Fig. 2(a) shows that p (m ) bution which is of interest for assessing the significance of
G max
measured in the data is not as badly described by Poisson rareeventssuchassurprisinglymassiveclustersobservedin
(cid:13)c 2010RAS,MNRAS000,1–6
Most massive halos with Gumbel statistics 5
X-ray or redshift surveys. Indeed the lower-mass tail of the
−
distributionfor whichtheprediction fails most significantly −
−
lies at masses below ∼1013M⊙,which correspondstohalos −
containing one to a few galaxies rather than tens or hun- ax 1510 − −
dreds,and therefore are of limited interest in thesearch for m
m
themost massive cluster. y − −
alobgi1li0tMymodmrieasoxtrv-iebirsu,tfnaiooirtnleyftuahncaccttuitorhanete-pltoyhspaittrieoidsn,icotthfeetdhmbeyopsetthalkeiktoehflyetohvreaypluerveoebon-f most likel 1410
when the shape of the curve begins to diverge from that of − Best fit with Poisson
thedata.Fig.3shows thismost likelyvalue,log10mˆmax,as 1310 BSeimsut lfaitt iwointh bias
a function of L for both theoretical estimates and the sim-
ulation data (central line and points in the figure). Wealso
10 100 1000
showinthisfigurethe95%confidenceregion onlog m
10 max
L h/Mpc
(upperand lower lines and bars).This too is well fitby the
theory, particularly for the upper limit, and we emphasize
that this is a crucial test of the theory’s ability to give sig- Figure 3.Themostlikelyvalueoflog10mmax (middlelineand
points)andthe95%confidence limits(upper andlowerlineand
nificance values for observations of specific overly massive
bars). Dashed red and solid blue lines are analytic results for
clusters.
thePoissonlimitandfullclusteringcalculationsrespectively,and
InadditiontothefourvaluesofLshowninFig.2,Fig. pointsaresimulationvaluesforLcorrespondingtothefourpanels
3 has a final simulation point at L=500/h Mpc. Here the offig.2plusL=500/hMpc.
95%confidenceregionispoorlyfit,becausethepatchistoo
largecomparedtothesimulationboxtogetgoodstatistics.
However, the measured value of log m is still in good
10 max 4 CONCLUSION AND DISCUSSION
agreement with thetheory.
We have presented an analytic prediction of the the prob-
ability distribution of m , the most massive dark mat-
max
terhalo/galaxy cluster in aspecified region of theuniverse,
making use of the counts-in-cells formalism. Our calcula-
3.2.1 Senstivity to the mass function
tion, valid for Gaussian initial conditions, is numerically
Worthy of note is the close similiarity of the curves for the consistent with that proposed byHolz & Perlmutter(2010)
two sets of parameters in the Sheth-Tormen mass function when performed assuming such massive halos are Poisson
(solid blue and dashed green lines in Fig. 3). Although the distributed spatially. However, the work presented in this
massfunctionsthemselvesdifferbyarelativelylargeamount paper improves on the calculation performed by these au-
comparedtothebest-fitfunction’sagreementwiththesim- thors in two aspects: (i) our results are given in a fully ex-
ulation points (Fig. 1, lower panel), this translates to only plicitanalyticformand(ii)theyincludethecontributionof
amodestchangeinthedistributionofmmax.Similarcalcu- clustering of halos.
lations performed with the mass functions of Tinker et al. Wealso compared ouranalyticpredictionstomeasure-
(2008)andJenkins et al.(2001)leadustoconcludethatthe ments from a large (2000h−1 Mpc on a side) and well re-
extreme value distribution is fairly robust to the choice of solved(particlemass7.7h−1×109M⊙)cosmologicalN-body
any reasonable analytic fit to the mass function. simulation at zero redshift. We achieve remarkable agree-
ment with the simulation in the area of parameter space in
which our formalism is expected to be valid, namely patch
radii above a few tens of h−1 Mpc. More surprisingly, even
outsidethisrangeofscalesthehigh-masstailofthedistribu-
3.2.2 Redshift variation
tion is well fit byour ”fully clustered” theoretical estimate,
WhiletheabovecalculationsusepatchsizesLsmallenough as is the most likely value of log m .
10 max
that the redshift evolution within the patch is negligible, it Thisunexpectedsuccessoverawiderangeofscaleswar-
isalsointerestingtocalculatem foralargerregionwith rantstheapplicationoftheformalismtoquantifythestatis-
max
significant ∆z. As noted in section 2.1, redshift variation tical significance of individual clusters observed in surveys.
can be taken into account by a weighted spatial average By applying our method to a patch of shape, size and red-
providedthePoissonapproximationholds,whichisthecase shift equivalent to a real survey we can obtain the Gumbel
for such large patches. In particular, averaging the number distribution and hence a likelihood for the observed value
densityof halos overtherange z=0to ∞ givesa valuefor ofm .Moreover, we arequiteconfidentthat ourmethod
max
the expected largest mass cluster in the entire observable can be extendedto non-standard cosmologies such as those
universe. including initial non-Gaussianities. It could therefore pro-
Weperformedthiscalculation assumingPoisson statis- vide a measure of the evidence for such cosmologies from
tics,andfoundmmax=4.6±10..26×1015M⊙ at the1−σ con- existingsurveysofthemostmassiveclustersasadvocatedin
fidencelevel.Wenotethatthisisinfairagreementwiththe Cayo´n et al.(2010).Weplantotacklethisexcitingprospect
similar calculation performed byHolz & Perlmutter(2010), in the verynear future.
who obtained m =3.8±0.6×1015 using a WMAP7 cos- In addition to the full likelihood curve of m we are
max 0.5 max
mology and themass function of Tinker et al. (2008). able, via the analytic Generalised Extreme Value formal-
(cid:13)c 2010RAS,MNRAS000,1–6
6 O. Davis, J. Devriendt, S. Colombi, J. Silk & C. Pichon
ism, to produce a summary of the distribution in the form SzapudiI., Szalay A. S., 1993, ApJ, 408, 43
of the three parameters a, b and γ. The latter in particular Teyssier R.,2002, A&A,385, 337
hasbeenproposed previouslyasastatistic forusein model TeyssierR.,PiresS.,PrunetS.,etal.2009,A&A,497,335
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described in this paper uses a single cosmology and power ren M., Yepes G., Gottl¨ober S., Holz D. E., 2008, ApJ,
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sensitivetotheshapeofthepowerspectrum.Iftheeffecton Warren M. S., Yepes G., Gottlober S., 2010, ArXiv e-
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5 ACKNOWLEDGEMENTS ThispaperhasbeentypesetfromaTEX/LATEXfileprepared
bythe author.
The authors are grateful to an anonymous reviewer whose
comments led to improvementsin this paper.
JD’sresearchissupportedbytheOxfordMartinSchool,
Adrian Beecroft and STFC.
CP acknowledges support from a Leverhulme visiting
professorshipattheAstrophysicsdepartmentoftheUniver-
sity of Oxford.
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