The Effect of Lensing on the Large-Scale CMB Anisotropy

0.25 MB·English

by  T. Shanks

Mon.Not.R.Astron.Soc.000,1–8(2006) Printed5February2008 (MNLATEXstylefilev2.2)
The Effect of Lensing on the Large-Scale CMB Anisotropy
T. Shanks⋆
Department of Physics, Durham University,South Road, Durham DH1 3LE, UK
7
Received;inoriginalform2006September
0
0
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ABSTRACT
n
We first compare the CMB lensing model of Seljak (1996) with the empirical model
a
of Lieu & Mittaz (2005) to determine if the latter approach implies a larger effect on
J
the CMBpower-spectrum.We findthatthe empiricalmodelgivessignificantlyhigher
0 results for the magnification dispersion, σ, at small scales (θ <30′) than that of Sel-
3
jak (1996),assumingstandardcosmologicalparameters.However,whenthe empirical

foreground model of Lieu & Mittaz is modelled via correlation functions and used in
2
v the Seljak formalism, the agreement is considerably improved at small scales. Thus
9 weconclude thatthe maindifference betweenthese resultsmaybe inthe differentas-
3 sumedforegroundmassdistributions.Wethenshowthataforegroundmassclustering
3 with ξ(r) ∝ r−3 gives an rms lensing magnification which is approximately constant
9 withangle,θ.InSeljak’sformalism,weshowthiscanleadtoasmoothingoftheCMB
0 power-spectrumwhichis proportionalto e−σ2l2/2 andwhichmaybe able to movethe
6
firstCMB acoustic peak to smaller l,ifthe mass clustering amplitude is highenough.
0
Evidence for a high amplitude of mass clustering comes from the QSO magnification
/
h resultsofMyersetal(2003,2005)whosuggestthatforegroundgalaxygroupsmayalso
p bemoremassivethanexpected,implyingthatΩ ≈1andthatthereisstronggalaxy
m
- anti-bias, b ≈ 0.2. We combine the above results to demonstrate the potential effect
o
r of lensing on the CMB power spectrum by showing that an inflationary model with
t neither CDM nor a cosmological constant and that predicts a primordial first peak
s
a at l = 330 might then fit at least the first acoustic peak of the WMAP data. This
: modelmaybe regardedassomewhatcontrivedsince the fitalsorequireshighredshift
v
reionisation at the upper limit of what is allowed by the WMAP polarisation results.
i
X Butgiventhe finely-tunednature ofthe standardΛCDMmodel, the contrivancemay
r be small in comparison and certainly the effect of lensing and other foregrounds may
a still have a considerable influence on the cosmologicalinterpretation of the CMB.
Key words: gravitationallensing, cosmic microwave background, dark matter
1 INTRODUCTION et al, 2004) and other experimental constraints when com-
binedwith theWMAPresult fortheCDMparticledensity,
The standard ΛCDM model gives an excellent fit to the
nowleaveonly asmall amountof ‘generic’ parameter space
CMB power spectrum from WMAP and other experiments
allowedbysomeofthebasicsupersymmetricmodelsforthe
(Hinshaw et al. 2003, 2006). The reduced chi-squared val-
neutralino(e.g.Ellisetal2003,Barenboim&Lykken2006);
uesare impressive for a model with 7basic parameters and
thisisknownasthe‘neutralinooverdensity’problem.There
fittingover40 approximately independentdatapoints.The
isalsowhat isknownasthe‘gravitinoproblem’.Iftheneu-
model also appears to give an excellent fit to independent
tralinoisthelightestsupersymmetricparticleandthegrav-
datasets such as the2dFGRSgalaxy power spectrum (Cole
itino is the next-to-lightest then if the neutralino is stable
et al. 2005).
as required for a CDM candidate then the gravitino may
Impressive though these fits are, the standard model
be also massive and unstable; if its lifetime is longer than
still appears to have a variety of more fundamental prob-
0.01 secs then the decay of these particles into other par-
lems (see Shanks, 2005 and references therein). For exam-
ticles such as baryons could upset the Big Bang prediction
ple,theCDMparticlestillhasitsownassociated problems.
of the light element abundances (e.g. Cyburt et al 2003).
The limits from theCryogenic Dark Matter Search (Akerib
Thustheusualagreement oflightelementabundanceswith
thebaryondensitymaynotbetakenforgrantediftheneu-
tralino is the CDM particle. In some sense, the invoking of
⋆ E-mail:[email protected]
2 T. Shanks
the CDM particle could appear to have over-specified the shall argue that a low H , Ω = 1 model may give an
0 baryon
model in terms of solving the baryon nucleosynthesis prob- excellent fit to the CMB TT power spectrum with no need
lem. to invokeeither a CDM particle or dark energy.
The cosmological constant ordark energy presentsfur- Hence, in Section 2 we review and modify the CMB
ther fundamental problems. Indeed, string theory prefers a lensing results of Lieu & Mittaz. In Section 3 we introduce
negativeΛratherthantheobservedpositiveΛ(e.g.Witten, theCMBFAST lensingformalism and comparewiththere-
2001). But it is the small size of the cosmological constant sults of Lieu & Mittaz. In Section 4 we use these results
that undoubtedly presents the biggest difficulty for anyone togetherwithcosmological parametersdeterminedfromthe
assessinghowsuccessfulthestandardmodelisinexplaining QSO lensing results of Myers et al (2003, 2005) to predict
thecurrentdata.Inthestandardmodel,just afterinflation a significantly increased effect of lensing on the CMB. In
the ratio of the energy density in radiation to the energy Section5wethenconsiderthebaryoniccosmological model
density in the vacuum was one part in 10120. This means thathasan intrinsicCMB first peakat l 330. Wediscuss
≈
that the standard model ends up potentially having more howhomogeneousreionisationathighredshiftandthenthe
fine-tuning than implied in the original ‘flatness’ problem effect of lensing can significantly improve agreement with
beforeitwassolvedbyinflation(Guth1981).Itisthisprob- the observed first peak at l 220. In Section 6 we discuss
≈
lemthatpresentsthedifficultyinassessingthesuccessofthe ourconclusions.
standardmodelwherethelowreducedchi-squared,fromfits
totheCMBpower-spectrumhastobebalancedagainstthe
huge degree of freedom allowed by the invocation of a very
2 LENSING MODEL OF LIEU AND MITTAZ
small cosmological constant.
(2005)
Hereweshall taketheviewthatthefundamentalcom-
plicationsofthepresentstandardmodelsuggestthatitmay We shall first consider the lensing model of Lieu & Mit-
still be worthwhile to explore if other models may explain taz (2005). These authors have suggested that the effects
thedata.Thismayimplytheintroductionoffurthermodel of lensing by foreground groups and clusters on the CMB
parameters which may locally increase the model’s compli- can be much larger than previously calculated. This area
cation,butiftheythenallowthedroppingofthehypothesis hasbeencontroversialinthepast withsomeauthorsclaim-
ofthecosmologicalconstantandeventheCDMparticle,the ing large effects of lensing on the CMB (Fukushige et al.,
model’s complication will be globally decreased. 1994, Ellis et al., 1998) while others find much smaller ef-
CMBforegroundswillformourmainescaperoutefrom fects (eg Bartelmann & Schneider, 2001). The lensing ef-
thetight constraints set bythe CMB power-spectrum. One fectsclaimed byLieu & Mittaz aresignificantly larger than
example will be the evidence for the effects of high red- those claimed by Bartelmann & Schneider and somewhat
shift (z 10) reionisation from the WMAP polarisation larger than claimed by Seljak. This may be because Lieu
results(K≈ogutetal2003,Pageetal2006).Thisalreadyde- & Mittaz use strong-lensing formulae appropriate for clus-
creases the height of the acoustic peaks by 20% relative ters modelled as isothermal spheres or NFW profiles rather
to the multipoles at larger scales even if the≈reionisation is than the weak-lensing approximations used by Seljak and
homogeneous. But galaxy clusters in the CMB foreground Bartelmann & Schneider, although all the above authors
at lower redshifts provide several other ways of contami- assume the weak-lensing Born approximation in terms of
nating the CMB signal. The hot gas in the richer clusters considering only small deviations from the original photon
scatter the CMB photons via the SZ effect and there are path. Another reason may be that they use empirically de-
claims from the cross-correlation analysis of WMAP data termined models for the foreground structure which may
with Abell clusters that the SZ decrements may extend up producelarger effects than standard ΛCDM models.
to0.5 degree scales (Myerset al2004). They can also grav- Lieu & Mittaz treat the demagnification in the empty
itationally affecttheCMBphotonsprimarilyviatheSachs- voids and the magnification in the clusters separately and
Wolfe effect and also by gravitationally lensing the CMB show that in terms of the average magnification η they
h i
photons.Althoughtheeffectsofforegroundlensingaregen- cancel,sothatthesame‘lightconservation’resultappliesin
erally claimed tobesmall, weshallstart byconsidering the thisinhomogeneouscaseasinthehomogeneouscasetreated
claims by Lieu & Mittaz (2005) that the effects of lensing e.g. by Weinberg (1976). They show that the same result
on the CMB power spectra may be bigger than previously applies when strong lensing is taken into account. Here we
′
estimated. Lieu & Mittaz analysed lensing by foreground followLieu&Mittazanddefineη=(θ θ)/θasthemagni-
−
non-linearstructuresbyusingcataloguedpropertiesofclus- fication ‘contrast’; the effect of the scattering is to increase
ters and groups rather than power-spectra, modelling the the average angular size θ of the source by this fractional
clusters as singular isothermal speheres. amount.
Thebasicrelationsarefortheexpectedvalueofηinthe
We shall then relate the above approach to the CMB
case where the source is far behind the lens from equation
lensingformalism ofSeljak(1996) asimplementedinCMB-
(18) of Lieu & Mittaz:-
FAST (Seljak & Zaldarriaga 1996). We shall show that the
standard formalism also allows models with a first peak at 3 xf (x x)x
WlarMgeArPwadvaetnau,mifbthere,ml,astshapnowtehrespsteacntdruamrdhmasodaesltrtoongfitantthie- hηi= 2ΩgroupsH02Z0 dx[1+z(x)] sx−s , (1)
biaswithrespecttothegalaxypowerspectrumassuggested and equation (31) of Lieu & Mittaz gives for δη
by the QSO lensing results of Myers et al (2003, 2005). In
8π3 3x 3x2 σ4
thisway weshallshowthatalarge rangeofmodelsmay be (δη)2= n x3 1 f + f p if(R ,b )(2)
able to reproduce the observed first peak. In particular, we 3 0 f(cid:18) − 2xs 5x2s(cid:19)Xi,j ijc4 j min
The effect of lensing on the CMB 3
where
R 8
f(R ,b )=ln j (3)
j min (cid:16)bmin(cid:17)− π2
and p is the probability of finding a group with velocity
ij
dispersion σ and radius R . 1
i j
In one case, they use a space density of groups which
correspondstothatobservedinthegroupsampleofRamella
etal(1999,2002)andassumingtheothergroupparameters
suchasthevelocitydispersion,σ.Thisproduces η 0.099
h i≈
andδη 0.093,assumingthegroupscontinuewiththesame
≈
comoving spacedensity out toz =1 in aΩ =0.27 model.
f m
Lieu & Mittaz note that this assumption may not be justi-
fiable for groups because of the effects of dynamical evolu-
tion.FortheirAbellclustermodeltheyfind η 0.099and
h i≈
δη=0.064.Lieu&Mittazarguethattheirassumptionofno
dynamical evolution of the cluster density may have more
empirical support in the case of clusters than groups. One
pointtonoteisthatLieu&Mittazusetheline-of-sightrms
velocity dispersion and the full 3-D rms velocity dispersion
shouldbe√3bigger.Although η wouldremainunchanged,
δη would then increase bya fachtoir of 3, implying δη=0.28 Figure 1.Lensingdispersions,δη=σ/θ,forthemodelsofLieu
& Mittaz and Seljak (1996). The LM clusters and groups mod-
for the Ramella et al groups and δη = 0.19 for the Abell
els are those of Lieu & Mittaz where the lenses are respectively
clusters.
Abell clusters and the groups of Ramella et al in a ΛCDM cos-
Thesevaluesforδηapplyinthecoherentregimeofscat-
mology (see Sect. 2). The results of Lieu & Mittaz have been
tering where a CMB patch has a mean angular size less multipliedby3×tobasetheresultsonthefull3-Drmsvelocity
thanα, thesize ofagroup at an average redshift.Atlarger dispersionsrather than line-of-sightdispersions.TheLM groups
scales, in the incoherent regime of scattering, the effective (Myers) model assumes each group contains the high mass im-
δη reduces according to δη 1/θ (see equations 33, 34 of pliedby theQSO lensingresultsof Myersetal. andassumes an
∝
Lieu & Mittaz). For the groups and clusters considered by EdScosmology(seeSect.4.1).TheΛCDMandSCDMmodelsare
Lieu&Mittaz,thedivisionbetweentheseregimesoccursat the results from CMBFAST including the non-linear extensions
10arcmin. The form of δη (= σ(θ)/θ in the terminology fromPeacock&Dodds(1994) andarecomparabletotheresults
≈ ofSeljak(1996). Themodelsbasedonthepower-lawcorrelation
ofSeljak,1996) withθ isgivenin Fig.1fortheabovemod-
functions alsouse Seljak’s formalism;the power-law shown with
icfioemdpgarroeutpheasnedtocltuhsetesrtamnoddaerldsCofMLBieFuAS&TMesitttimaza.tWeseofshtahlel rin0d=ex6-.30has−s1uMmpesc.r0T=he7c.4ohn−st1aMntpσca=nd(=th0e.0-015.8rpaodw.)erm-loadwealshsausmaens
magnification function in thenext section.
amplitude similarto that implied by the QSO lensing results of
Lieu&Mittaznotethatincaseswhereδη islargecom- Myersetal.
paredto η askeweddistributionforη maybeimplieddue
h i
to negative density fluctuations cutting off at ρ= 0 (Dyer-
Roeder, 1972, empty beam). Although this may also apply theΛCDMmodelliesbetweenthegroupandclusterresults
to the models above and elsewhere in this paper, here we of Lieu & Mittaz. However,at smaller scales, themodels of
shallsimply assumeasymmetricmagnification distribution Lieu & Mittaz are 2 3 higher than the ΛCDM model.
− ×
which is applied using equation A7 of Seljak (1996). Tests suggest that the neglect of linear theory evolution in
themodelsofLieu&Mittazdoesnotappeartoexplainthis
discrepancy.AnSCDMmodelisalso showninFig. 1.With
COBE normalisation, this model gives σ = 1.6 and lies
3 LENSING MODEL OF SELJAK (1996) 8
above theΛCDM model at all scales and above themodels
We next discuss the more usual lensing formalism of Sel- of Lieu & Mittaz at large scales.
jak (1996) as implemented in CMBFAST. This is based on The magnification rms dispersion, σ, is then used in
inferring the fractional magnification = σ(θ)/θ (= δη in Seljak’s equation A6 which, under the assumption that the
the terminology of Lieu & Mittaz) from the mass power isotropic term is dominant,becomes his equation A7:
spectrum using the variation of Limber’s formula given in
∞
equations(7)ofSeljak (1996). UsingtheCMBFAST source C˜(θ)=(2π)−1 ldle−σ2(θ)l2/2CJ (lθ). (4)
l 0
code, σ(θ)/θ can be generated. The ΛCDM model with Z
0
Ω =0.27, Ω =0.73 (as assumed by Lieu & Mittaz) with
m Λ where C is the CMB angular power spectrum and C˜(θ) is
the non-linear extension of Peacock & Dodds (1994) gives l
the lensed angular correlation function. This equation has
the result shown in Fig. 1. It can be seen that the σ(θ)/θ
the appearance of a Hankel transform but the dependence
drops like 1/θ at large scales as expected in the incoherent
ofσ onθ meansthatthisisonlytrueundertheassumption
regimewhileflatteningatsmaller scales.Atθ>100arcmin,
that theθ dependenceis slow.
When equation A6 is used with the standard model,
1 Note that equation (32) of Lieu & Mittaz (2005) contains a only small effects are seen on the CMB (see Seljak, Fig. 2).
typographicalerror;thenumericalvalueshouldread2.7×10−11. Indeed, we found that any σ(θ)/θ that flattens at scales
4 T. Shanks
smaller than 1 degree even with amplitudes as high as at 100 arcmin. The cut in thecluster mass profileat radius
σ(θ)/θ = 0.45, generally gave small lensing effects on the 2h−1Mpc assumed byLieu &Mittaz makesthecompari-
≈
first acoustic peak. Since the lensing dispersion is close to son less valid at larger scales. An 2.5 bigger correlation
≈ ×
that of Lieu & Mittaz, similar results arefound undertheir function scale length would be required to reach the con-
assumptions. stant σ model even at large scales ie r 27h−1Mpc. Also,
0
≈
A case of interest is the one where we start by assum- aswehaveseenthemodelimpliedbythe-1.8powerlawfor
ing that σ(θ) is a constant. This means that σ(θ)/θ 1/θ ξissignificantlyflatterthantheconstantσmodeldiscussed
∝
i.e. the usual large-scale behaviour extends to the smallest above and the SIS model with ǫ=2 and γ =1 as actually
scales. It also means that the combination e−σ2(θ)l2/2C is assumedbyLieu&Mittazwould giveanevenflatterslope.
l
the lensed power spectrum. By adjusting the size of σ it is We conclude that Lieu & Mittaz and Seljak appear
then possible to smooth small scale peaks away and even to obtain reasonably similar results for σ/θ when simi-
to partially smooth away and hence move the first peak. lar input models are assumed. Although the Lieu & Mit-
For the moment, we simply assume σ = 0.005 rad. in the taz group/cluster models show more power at small scales
case shown in Fig. 1 for θ > 7′. The question is whether than the ΛCDM model, when we assume mass correlation
this amplitude and form of σ(θ)/θ can be justified physi- functions which should be reasonably appropriate for the
cally. Equation 7 of Seljak(1996) implies that σ =constant Lieu & Mittaz cluster model in the Seljak formalism, im-
requiresthepowerspectrumofthedensity,P(k),( P k4 provedagreementisseenatsmallscales.Thereforeinterms
φ
∝ ×
whereP isthepowerspectrumofthepotential)tohavethe of the discussion (e.g. Lewis & Challinor 2006) of how the
φ
form P(k) k0 in the range of interest which implies that resultsofLieu&Mittazrelatetoearlierwork,theremaybe
ξ(r) r−3∼, since Fourier transforming the density power no serious disagreement, with their assumed higher ampli-
∼
spectrum, P(k) kn, to the correlation function implies tudeof foreground mass clustering probably being a bigger
ξ(r) r−(3+n) as∝ymptotically for 3< n<0. Thus it ap- factor in explaining any difference than their partial inclu-
∝ −
pears that the constant σ model will require a correlation sion of strong lensing in theirlensing formalism. But in the
function for the mass which is steeper than the observed wider context of affecting the first peak position, such dif-
galaxy correlation function, ξ(r) r−1.8. As can be seen, ferences between the models are academic because none of
∝
the amplitude of σ is also much higher than for the stan- the σ(θ)/θ models which flatten at small scales produces
dard model. But we shall see next that such an amplitude thesimplesmoothingoftheCMBfirstpeakoftheconstant
maybesuggestedbyconsiderationsofQSOlensingandthat σ model; models with more small-scale power and a larger
the σ =constant form shown in Fig. 1 then can be used to massclusteringamplitudearerequired.Wenextlooktosee
‘move’ thefirst acoustic peak. if there is any evidence that the size of the magnification
Wenowfurtherdrawthecomparisonwiththeresultsof dispersion may becurrently underestimated.
Lieu&Mittazbyexpressingtheirmodelinpower-spectrum
or correlation function terms. These authors assume a ran-
dom distribution of Abell clusters each with a singular
isothermal sphere for its mass distribution. According to 4 QSO MAGNIFICATION RESULTS
Peebles (1974) a model with randomly placed monolithic
clusters with density profile ρ(r) r−ǫ implies a correla- 4.1 Group masses from QSO lensing applied to
tion function with ξ(r) r−(2ǫ−3∝). ǫ = 2 as assumed by results of Lieu & Mittaz.
∝
Lieu & Mittaz gives γ =1.0 whereas ǫ=2.4 gives γ =1.8. We now consider the possible implications if galaxy group
Fourier transforming the correlation function to the power masses were as big as those found in the QSO magnifica-
spectrum(asabove)wefindthatisothermalspheresoflarge tion results of Myers et al (2003, 2005). Myers et al (2003)
radiusleadstoapowerspectrumofslope k−2 whichcanbe found that the QSO-group cross-correlation function fitted
assumed to apply in therange 0.01<k<10 h Mpc−1. SIS mass profiles with σ = 1125kms−1. This velocity dis-
Whenwetakethevalueforthelocalgalaxycorrelation persion is comparable to the values found for rich clusters,
functionscale length ofr =6h−1Mpcwefindthat,assum- although the sky density of these groups is 1deg−2 com-
0
ing ǫ = 2.4 for the Abell cluster model, σ(θ)/θ is approxi- pared to 0.1deg−2 for the Abell cluster≈s. Myers et al
≈
matelythesameatlargescalesasthelineartheoryΛmodel (2005)alsofoundthat2QZQSOlensingbyindividualgalax-
(see Fig. 1). Thus, as expected, in the Seljak formalism an iesagainproducedmoreanti-correlationthanexpectedfrom
approximately unbiased mass correlation function with a - thestandardmodel.Heretheyanalysedtheresultsusingthe
1.8 power-law gives approximately the same result for σ/θ galaxy power spectrum after Gaztanaga (2003) and found
as the ΛCDM model with non-linear extension. thatinthestandardcosmology ananti-biaswithb 0.1on
If we now assume a model in the spirit of Lieu & Mit- 1h−1Mpcscaleswouldberequiredtoexplainthe≈strength
≈
taz where all the mass is in galaxies and all the galaxies ofanti-correlation seen.Guimaraes,Myers&Shanks(2005)
are in randomly distributed Abell clusters, this will give a showed that these results were not due to selection effects
masscorrelationfunctionthatextendsasa-1.8power-lawto byshowing,forexample,thatthesameclusterfindingalgo-
>10h−1Mpc with a correlation length of r =10.6h−1Mpc rithms in the Hubble volume N-body simulation produced
0
(from equation 23 of Peebles, 1974). This then produces much smaller lensing effects.
a σ/θ result which would be 1.7 higher than that for Nowwediscussbelowotherobservationswhichdisagree
r = 6h−1Mpc shown in Fig. 1. T×his model gives a gen- with the results of Myers et al (2003, 2005). Nevertheless,
0
erally flatter result with θ than the LM cluster (or group) here we consider the possibility of whether galaxy groups
model,agreeing withtheLMclustermodelat 1arcmin,ly- may be dynamically young and the virialised assumption
ing a few times lower at 10 arcmin and a few times higher therefore unjustified. In this case the group lensing masses
The effect of lensing on the CMB 5
may be more correct, implying that the group masses are
bigger than expected and comparable to the mass of Abell
clusters. Ifthefullfactor of ten increase observedbyMyers
et al for QSO lensing over what is expected for the concor-
dancemodelistranslateddirectlyintotheresultsofSection
2 for CMB lensing, then 3 bigger values of δη will be
≈ ×
found, resulting in δη 0.6 for both groups and clusters in
≈
theabove model.
However, Myerset alpointed out that theirspaceden-
sity of high mass groups implies a Universe with a mass
density more consistent with that of an Einstein-de Sitter
model, rather than theΛCDM model. The space density of
groupsreportedbyMyersetalisn =3 1 10−4h3Mpc−3,
0
± ×
close to the value of Ramella et al as used by Lieu & Mit-
taz,(n =4.4 10−4h3Mpc−3).Takingaradiusof2h−1Mpc
0
with b = 7×h−1kpc and substituting these in the SIS re-
min
sult for δη, we find δη = 0.15. Here we have assumed that
the group and cluster distribution is now unevolved out to
z =0.5; our cluster and group densities are essentially de-
f
termined within this z range, reducing uncertainties dueto
evolution.WenotethatLieu&Mittazfindanaveragevalue Figure 2. The Cl for the Ωm = 1, h=0.35 baryonic model, as-
of the group velocity dispersion of 270kms−1 whereas the sumingreionisationwithopticaldepth,τ =0.4(dashedline).The
weighted sum in their equn (32) corresponds to an average lensed version of the Cl with τ = 0.4 is also shown (solid line)
valueof460kms−1.Clearly themoremassive groups,inthe assuming the constant σ (=0.005 rad.) model shown in Fig. 1.
ThesimpleSZmodeldescribedinSection 5isalsoshown(long-
tailofthedistribution,dominatethedeflectioneffect.Ifthe
dashed line). The data points are the 3-year WMAP results of
same effect applies for the groups of Myers et al, the rms
Hinshawetal(2006).
dispersion would rise to δη = 0.45 in the coherent regime,
asshownbythedottedlinemarked‘LMgroups(Myers)’in
Fig. 1. Althoughtheamplitudeshaverisentheformofthelens-
This is the extra dispersion in the CMB anisotropy ingdispersionsremainsthesame.Inparticular,theyflatten
caused by coherent scattering. Following Lieu & Mittaz we atsmallθmuchfasterthantheconstantσmodelpostulated
calculatethescaleabovewhichthescatteringoflightbythe in Section 3 above as having the appropriate form and am-
galaxy groups becomes incoherent rather than coherent. In plitude required to ‘move’ the first acoustic peak. The σ/θ
theEdSmodeltheangulardiameterdistanceattheaverage result for ξ(r)=(r/7.4h−1Mpc)−3 isshown inFig. 1where
group distance of z = 0.25 is dA = 500h−1 Mpc. Therefore the amplitude has been chosen to match the SCDM model
ourassumedgroupdiameterof4h−1Mpcsubtendsanangle atlargescales.Itcanbeseenthattoreachtheσ=constant
of 0.45deg at this redshift. At smaller θ, δη is constant, modelgiveninFig.1,ther−3masscorrelationfunctionmay
≈
andatlargeranglesδη 1/θ(seeFig.1).Withouttheeffect evenhavetosteepenfurthertoensureσ/θ maintainsits 1
of the tail, the results∝are similar to the Ωm = 1 results of slope down to 7′ scales, corresponding to 1h−1Mpc at−an
Seljak(1996)althoughtheδηextendstolargerscalesbefore averagegalaxydepthofz 0.15.Thestand≈ardCDMmodel
startingtodecrease.Withthetail,themodelgivesconsider- starts to flatten earlier at≈about 1 degree and the ΛCDM
ablyhighervaluesforδη(seeFig.1,dottedline)thanfound model at yet larger scales. This means that perhaps only a
bySeljakforthestandardΛCDMmodel(long-dashedline). baryonicmodelwith dissipation mayproducetherelatively
steep mass correlation function in ther<10h−1Mpc range
of interest for modifying the first acoustic CMB peak via
4.2 Galaxy masses from QSO lensing applied to
lensing.
results of Seljak (1996).
We next apply the galaxy lensing results of Myers et al
(2005)totheresultsofSeljak(1996).ForthestandardΛcos-
5 APPLICATION TO THE MODEL OF
mology,Myersetalfoundvaluesofthebiasb 0.05 0.15at
≈ − SHANKS (1985)
sub-Mpcscales, dependingon themethodology. IntheEdS
cosmology, they found values in the range b 0.15 0.3. Anexampleofamodelwhichinitially hasthefirstacoustic
≈ −
Littleevidenceofscale dependencewasseen in thedata,so peakatl=330ratherthanl=220istheEinstein-deSitter,
theseresultsmay beassumed toalso applyat larger scales. baryon dominated, low H model of Shanks(1985); Shanks
0
Scaling σ/θ from the unbiased results of CMBFAST in the et al. (1991); Shanks (2005). The C from this model also
l
ΛandEdScosmologies accordingtotheabovederivedanti- producestoohighapeakheight,sotoobtainafittothedata
bias, the results for σ/θ clearly rise by 1/b i.e. by 10 we need to assume that the optical depth to reionisation is
≈ ×
in the Λ case and 5 in the EdS case. Clearly these re- τ =0.4whichisatthehighendofwhathasbeensuggested
≈ ×
sultsareonlyapproximate.Inafuturepaperweshallrelate by the large-scale WMAP polarisation results (Kogut et al
the QSO galaxy lensing result of Myers et al (2005) to the 2003, Page et al 2006). In fact as shown by CMBFAST,
CMB result more directly via equations (A14) of Myers et as the τ goes up, the polarisation power spectra quickly
al (2005) and (7) of Seljak (1996). saturateanditbecomesmoredifficulttorejecthighervalues
6 T. Shanks
ofτ.TheresultingmodelC iscomparedtotheWMAPdata bytheabovemodel(Kogutetal2003,Pageetal2006).For
l
in Fig. 2. example, it is known that lensing will produce B modes as
Wethenusetheσ=constantmodelfromFig.1.There well as E modes in the polarisation signal (Zaldarriaga &
we haveseen that thismodel has a high amplitudebut one Seljak 1998) anditwill beinterestingtoseeif thisprovides
whichmaybemotivatedbytheQSOmagnificationresultsof a constraint on the current approach.
Myersetal(2003,2005).Theexponentiallysharpincreasein For the moment, we conclude that using high-redshift
smoothingaslincreasesthatthenresultsfromequationA7 reionisationtoreducethefirsttwoacousticpeaks,thenlens-
ofSeljakisthesort ofbehavourwhichmightberequiredto ing to move the first peak to larger scales and finally SZ
smooth thebaryonmodelintotheWMAPdata.Theresult to recreate the second peak, it appears possible to achieve
of applying this lensing smoothing function is shown as the an approximate fit to the WMAP CMB fluctuation power
solid line in Fig. 2 and it can be seen to give an improved spectrum.Althoughthisapproachmaybecriticisedasfine-
fit to the first acoustic peak. (Note that we are assuming tuningCMB foreground effects to achieve a fit to the data,
that l2C (θ) << 1 in equations A5 and A6 of Seljak in it may also be viewed as being much less fine-tuned than
gl,2
applying his equations A6 and A7 in this σ =constant case the approach taken via exotic particles and dark energy, as
where σ(θ)/θ 1 at θ <20′.) The question of whether the employed to construct the standard ΛCDM model.
≈
rise in σ towards the smallest scales is self-consistent with
the baryon model also arises. Certainly the linear theory
baryon modeldoes not producesuch arise, indeedshowing
6 CONCLUSIONS
less small scale power than the standard model. However,
the non-linear power spectrum of the baryon model after Wefirstcomparedmodelsfortheeffectofforegroundgalaxy
top-down collapse is not easy to calculate and this leaves clusters and groups on the CMB power-spectrum. We have
open the possibility that the baryon mass power spectrum shown that the lensing effects considered by Lieu & Mit-
maystillbeassteepasrequiredtoproducetheσ=constant taz may have bigger effects than they suggested, if the full
model.Aswehaveseen,amasscorrelationfunctionassteep 3-D velocity dispersion is used rather than the 1-D disper-
asr−3approximatesthebehaviourneeded,comparedtothe sionsassumedbytheseauthors.Thischangeproduces 3
observedgalaxycorrelationfunction’sr−1.8behaviour.How- bigger results for δη(= σ(θ)/θ) than they suggested, w≈hic×h
ever, galaxy correlation functions as steep as r−2.2 are seen will increase theeffect on theCMB peaks.
insamplesofearly-typegalaxiesonr<1h−1Mpcscales(Ze- WefindthattheempiricalmodelofLieu&Mittazthen
havietal2005,Rossetal2006).Theamplitudewouldhave gives significantly higher results for the magnification dis-
to be of order r0 25h−1Mpc implying strong anti-bias of persionatsmall scales(θ<30′)thanthoseofSeljak(1996)
≈
thetypesuggested bytheQSO magnification results. assuming standard cosmological parameters. We have also
Note that we have only postulated the σ = constant shown that under the same foreground assumptions, now
model in Fig. 1and not derivedit from equation (7) of Sel- modelledinmasscorrelationfunctionterms,thelensingfor-
jak (1996). Therefore, although the model has σ/θ > 1 for malism of Lieu & Mittaz produces approximately the same
θ < 20 arcmin, we have not violated the condition that results as thethat of Seljak (1996). But both formalisms in
equation (7) only applies in the case σ/θ<1 because of its the case of the standard ΛCDM foreground or in the em-
assumption of the Born approximation. The main assump- pirical cases considered by Lieu & Mittaz tend to produce
tions we therefore have made in deriving the lensed Cl in σ/θ relations which are too flat towards small θ and have
Fig. 2 is that l2Cgl,2(θ) <<1 in the application of Seljak’s too small amplitudes to modify significantly the first peak
equation A7as notedabove. Nevertheless,thelensed result of theCMB power spectrum.
in Fig.2 shouldstill beregarded asonly an approximation. In conjunction with a model that assumed lensing dis-
It has also not been possible to fit the second peak. persion σ(θ) = constant we then used the QSO magnifi-
To address this issue, we could consider using the SZ ef- cation results of Myers et al (2003, 2005) to show that, if
fect caused by the known presence of hot gas in the rich galaxygroupshavebiggerthanexpectedmasses,therecould
Abell clusters to re-create the second peak after its era- besignificanteffectsofgravitationallensingoneventhefirst
sure by the lensing. Myers at al (2004) have statistically peakoftheCMB.AlthoughtheresultsofMyersetalremain
detected SZ decrements around Abell clusters in WMAP controversial (Scranton et al. 2005, Mountrichas & Shanks
data out to radii of 30 arcmin. These decrements simply 2007), it is clear that at least in principle, lensing can af-
add power to the power spectrum on cluster-sized scales. fect theinterpretation of theCMB powerspectrum. But to
Assuming a random distribution of circular ‘clusters’, with haveasignificanteffectonthefirstpeak,themassclustering
a sky density of 3deg−2, radius 12 arcmin and with ‘decre- power spectrum would then need to havebeen significantly
ment’∆T/T = 3 10−5,wecanproduceaquitesignificant underestimated in previous studies with anti-bias, b 0.2,
− × ≈
‘second peak’ (see Fig. 1). Although the match to the sec- being required.
ondpeakisonlyapproximate,thissimplemodelwithjusta There are many pieces of evidence which favour the
‘top-hat’SZprofileatleastservestoindicatetheprincipleof standard model bias value of b 1. For example, redshift
this approach. The observed spectral index of the WMAP space distortion analyses imply≈β = Ω0.6/b 0.5 0.1
m ≈ ±
CMB fluctuations which effectively constrain the SZ con- (e.g. Ratcliffe et al 1996, Hawkins et al 2003.) But such
tribution to the first acoustic peak (Huffenberger & Seljak analyses tend to assume that the galaxies trace the mass
2004)aremuchlesseffectiveconstraintsagainstasignificant with linear bias and in the case where groups contain as
SZ contribution tothe second peak. muchmassasclustersthisassumptiondoesnotapply.Even
FurtherworkisneededtocheckiftheCMBEEandTE here we note that the infall velocity patterns around the
polarisation power spectrafrom WMAPcan bereproduced rich environments of passive galaxies in 2dFGRS are very
The effect of lensing on the CMB 7
similar to those around the poorer environments of active and is based on known physics, without recourse to unde-
galaxies (Madgwick et al. 2003), suggestive of the evidence tected neutralinos or postulated dark energy.
from theQSOsthat groups dominated by spirals may have
as much associated mass as the richer environments of the
early-types.
ACKNOWLEDGMENTS
But still, the most direct previous constraints on the
mass power spectrum come from estimates of cosmic shear. Richard Lieu is gratefully acknowledged for useful discus-
The QSO lensing results are again in clear contradiction sions. The referee is also thankedfor comments which have
with these since they are mainly consistent with b=1. For significantly improved theclarity of thepaper.
example,Seljak(1996,Fig.1)suggeststhatthecosmicshear
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