Impact of Secondary non-Gaussianities on the Search for Primordial Non-Gaussianity with CMB Maps

0.17 MB·English

by  Paolo Serra

Impact of Secondary non-Gaussianities on the Search for Primordial Non-Gaussianity
with CMB Maps
Paolo Serra and Asantha Cooray1
1Department of Physics and Astronomy, 4186 Frederick Reines Hall, University of California, Irvine, CA 92697
When constraining the primordial non-Gaussianity parameter fNL with cosmic microwave back-
groundanisotropymaps,thebiasresultingfromthecovariancebetweenprimordialnon-Gaussianity
and secondary non-Gaussianities to the estimator of fNL is generally assumed to be negligible. We
showthatthisassumptionmaynotholdwhenattemptingtomeasuretheprimordialnon-Gaussianity
out to angular scales below a few tens arcminutes with an experiment like Planck, especially if the
primordialnon-GaussianityparameterisaroundtheminimumdetectabilitylevelwithfNL between
5and10. In future,it will benecessary tojointly estimatethecombined primordial andsecondary
8 contributions to the CMB bispectrum and establish fNL by properly accounting for the confusion
0 from secondary non-Gaussianities.
0
2 PACSnumbers: 98.70.Vc,98.65.Dx,95.85.Sz,98.80.Cq,98.80.Es

n
a
J Introduction— The search for primordial non- of the non-Gaussianity with an overall normalization
1 Gaussianity with constraintsonthe non-Gaussianitypa- givenby bps andBlS1Zl2−l3κ andBlI1SlW2l3−κ are additionalfore-
2 rameterf usingcosmicmicrowavebackground(CMB) ground, secondary non-Gaussianities from the SZ and
NL
anisotropy maps is now an active topic in cosmol- ISWeffectscorrelatingwithCMBlensing[19,20]. These

]
ogy today [1, 2, 3]. The 3-year Wilkinson Microwave are certainly not all the non-Gaussiancontributions to a
h
p AnisotropyProbehasallowedthe constraintthat−54< CMB map. There are non-Gaussianities from ISW [21],
- f <114atthe95%confidencelevel[4],thoughamore kineticSZ/Ostriker-Vishniac[22],andtheSZeffectitself
o NL
recent study claims a non-zero detection of primordial [23]. We ignore ISW and kinetic SZ/OV related bispec-
r
t non-Gaussianity at the same 95% confidence level with tra as they are small compared to SZ generated non-
s
a 26.9 < f < 146.7 [5]. This result, if correct, has Gaussianities. TheSZ-SZ-SZbispectrumissignificantat
NL
[
significant cosmological implications since the expected arcminute angular scales, but given the power-law shot-

1 value under standard inflationary models is fNL . 1 noise behavior of the SZ bispectrum when l <1500, the
v [6,7, 8,9,10, 11,12],thoughalternativemodelsofinfla- SZ contribution to the bispectrum can be thought of as
6 tion, such as the ekpyrotic cosmology [14, 15], generally an additional correction to b . The shot-noise behavior
ps
7
predict a large primordial non-Gaussianity with f at ofthe SZ effect is especially applicable for the SZ contri-
2 NL
few tens. bution during reionization associated with hot electrons
3
. Moststudies thatconstrainfNL withCMBanisotropy in supernovae bubbles Compton-cooling off of the CMB
1
maps make use of an estimator for f of the form [16, [24]. Thus, we do not separately include the total SZ
0 NL
8 17, 18] bispectrum as a separate non-Gaussianity here.
iv:0 fˆNL = SˆprimN+Sˆlin , (1) pBˆeloW1cblts2rhlu3emn≈.feTNsthLiimBslapa1rtlli2limonl3wgwsfhaNenLne,setsiittmimaisatotiurnsfguoartlhflyeNLpartsihsmuroomrudegdihaltbhiast-
X
where
2
r Bprim
a Sˆprim =lX1l2l3 Bσlp21r(li2lm1l3,Blˆ2lo,1bll3s2l)3 , (2) Sˆprim =fNLlX1l2l3 σ(cid:16)2(ll11,l2ll23,(cid:17)l3), (3)
when Bprim is the primordial bispectrum with the as- with fˆNL =Sˆprim/N, where the normalization N is sim-
l1l2k3 ply the summed term. The above assumption that only
sumption that f = 1. Here, Sˆlin is a linear correction
NL theprimordialnon-Gaussianitycanbeconsideredisgen-
to account for issues related to the maps (such as the
erally motivated by the fact that the covariance term
mask) and N is an overallnormalizationfactor [18]. For associated with the mode overlap between Bprim and
the present discussion motivated from an analytical cal- l1l2l3
additional secondary contributions to Bobs via
culation, we can ignore the correction associated with l1l2l3
Sˆlin which involves imperfections in the data, such as BprimBi
due to the mask. We also assume all-sky data here. In Sˆprim,cov =XAi X σl21(l2ll3,l l,1ll2l)3 , (4)
equation (2), σ2 is the noise variance to the bispectrum i l1l2l3 1 2 3
[1, 19]. when A =(b ,A ,A ,...) is expected to be smaller
i ps SZ ISW
In general Bˆlo1bls2l3 = fNLBlp1rli2ml3 + bpsBlp1sl2l3 + than the dominant term from equation (3) [1]. Never-
A BSZ−κ+A BISW−κ+...,whereBps istheshape theless, an estimate of f only from equation (3) leads
SZ l1l2l3 ISW l1l2l3 l1l2l3 NL
2
103 τ = 0.089. We verified our calculations are consistent
102 PWrMimAoPrdial (fNL=1) with prior calculations in the literature. In Fig. 1, we
Planck show the the absolute value of the signal-to-noise square
101 Point Source Confusion bps=10−25 ratio for the primordial bispectrum (thick lines) and for
SZ−lensing Confusion
the covariances between primary and secondary bispec-
nl]3 100 ISW−lensing Confusion tra. The plotted quantity here involving d(S/N)2/dlnl3
2S/N)/dl10−1 roevseerml3blweshitlehekeeesptiinmgatthore sSˆignab(oigvneo,reinxgpetchte fsoigrntchheansugmes
d(10−2 lead to a higher bias as described in Ref. [25]). While
[
the primordial calculation involves (Bprim)2/σ2, the
10−3 P
“signal-to-noise” square of the covariance follows from
10−4 bps BprimBps/σ2,forexampleforthepoint-sourcecon-
P
fusion, and these confusions should not be interpreted
10−5
101 102 103 simplyasthesignal-to-noiseratiosquaretodetectanyof
l these secondary bispectra directly from the CMB maps.
3
Instead of squared signal-to-noise ratios, to highlight
FIG. 1: Absolute values of signal-to-noise ratio squared for
the bias introduced to f when the estimator ignores
thedetectionofprimordialbispectrum(blacklines)assuming NL
secondary non-Gaussianity covariances, we calculated
fNL =1asafunctionofl3. Thesignal-to-noise ratio squared
ftot = f + f where f is the bias that is gen-
forWMAPandPlanckareshownwithdashedanddot-dashed NL NL bias bias
line, respectively. The red, blue, and green lines show the erated artificially by the correlation of modes between
confusion resultingfromthecovariancebetweenprimaryand the primordial bispectrum and secondary bispectra. To
pointsource,primaryandSZ-lensing,andprimaryandISW- properly normalize the relative contribution from sec-
lensing bispectra, respectively. ondary non-Gaussianities, we assume normalizations for
thepoint-sourcebispectrumconsistentwithWMAPwith
b =3×10−25, consistent with Q+V+W residual fore-
ps
to a biased estimate because of the contributions from ground [4], and Planck with b = 5 × 10−27. The
ps
secondary anisotropies through equation (4).
b value for Planck is slightly higher than the values
ps
While the CMB map contains a large number of sec- routinely quoted in the literature for unresolved radio
ondary non-Gaussian signals, in terms of the covariance sources in Planck high resolution maps, but this is due
related to the fNL measurement, what is necessary is to the fact that we believe bps includes additional con-
not to account for all of these non-Gaussianities, but tributions such as from the SZ-SZ-SZ bispectrum from
toaccountfornon-Gaussianitieswithbispectrumshapes both clusters at low redshifts and supernovae halos dur-
Bl1l2l3 in (l1,l2,l3) moment space that align with the ing reionization with a power-law shot-noise spectrum
shape of the primary bispectrum. In this respect, pre- when l < 1500. For the ISW-lensing and SZ-lensing bis-
vious calculations have suggested that the point-source pectrum, we follow the calculation of Ref. [19] and gen-
bispectrum may be ignored[1], but the ISW-lensing bis- erate the SZ contribution and the SZ correlation with
pectrum must be accounted for the Planck analysis [25]. dark matter halos responsible for lensing of the CMB
Including the SZ-lensing bispectrum, we find that using the halo model [23]. To account for an overall un-
whiletheassumptionthatthecovariancefromsecondary certainty and the variation in SZ and ISW amplitudes
anisotropiescanbemostlyignoredforanexperimentlike we have introduced an overall amplitude A and A
SZ ISW
WMAP,itmaybenecessarytoaccountforcertaincovari- respectively. Finally, to illustrate our results, we assume
anceswhenestimating fNL fromahighresolutionexper- fNL consistentwithroughlytheminimumdetectablepri-
imentlikePlanck,especiallyiftheunderlyingprimordial mordial non-Gaussianity with WMAP and Planck with
non-GaussianityhasavaluearoundfNLbetween5and10 fNL =20 and 5, respectively. As we find later, the dom-
consistent with the minimum amplitude detectable with inant confusion is from lensing bispectra and not from
Planck. At the minimum detectability level of WMAP point sources.
withfNL ∼20,thesecondaryanisotropiesinvolvingboth We summarize our results in Fig. 2, where we plot
residual points sources and lensing correlations will bias ftot which can be thought of as the total primordial
NL
fNL by a factor between 1.2 and 1.5 if primordial non- non-Gaussianity parameter that one will extract with
Gaussianity estimate is performed out to angular scales the above estimator for f when no attempt has been
NL
corresponding to ℓ>700. made to separate out the confusion from secondary
To reach these conclusions, we first calculated Bprim anisotropies. For the most part, the bias is negligible
l1l2l3
followingRef.[1]withthefullradiationtransferfunction and becomes only important when l>500. For WMAP,
usingamodifiedcodeofCMBFAST[26]forthestandard shown with a dashed line in Fig. 2 with the assumption
flat ΛCDM cosmological model consistent with WMAP that f = 20 if non-Gaussianity measurements are at-
NL
with Ω = 0.042, Ω = 0.238, h = 0.732, n = 0.958, and tempted out to l > 700, capturing basically all informa-
b c
3
103 and K refers to the set of non-Gaussianity parameters:
β
(f ,b ,A ,A ). Thismethodassumesthatonehas
NL ps SZ ISW
WMAP (b =3x10−25)
Planck (bps=5x10−27) a goodmodel for (l1,l2,l3) dependence of secondary bis-
ps pectra Bβ . Even if the point source covariance is
l1l2l3
102 small,theamplitudeofthepointsourceconfusionisgen-
erally unknown. Moreover, at l < 1500 many secondary
bispectra such as the SZ effect has a power-law behav-
totfNL fNL=20 ior similar to the bispectrum of point sources. Thus, it
101 would be necessary to determine the amplitude bps from
a joint fit.
f =5 Our suggestion that an accounting of secondary
NL
anisotropies is necessary for primordial non-Gaussian
measurementis different from the generalassumption in
100
101 102 103 the literature that one can simply ignore the covariance
l between primordial and secondary non-Gaussianities.
This partly comes through, for example, the suggestion
FIG. 2: The maximum non-Gaussianity measured with an
that primordial and point-source bispectra are orthogo-
optimal estimator for the primordial bispectrum fNtoLt, which nalfollowingresultsfromanexercisethatinvolvedjointly
includesthetrueunderlyingprimordialnon-Gaussianitywith
measuring non-Gaussianamplitudes f and b using a
fNL as labeled on the figure and the bias correction coming NL ps
from the unaccounted secondary anisotropies. The bias is set of simulated maps in Ref. [3] to study if there are
generallysmallandnonexistingifprimordialnon-Gaussianity biases in the estimators. However,this study used simu-
measurements are limited out to l < 500, but depending on lated non-Gaussian maps that did not include any point
thevalueoffNL andtheresidualpointsourcecontamination, sources with b =0. This sets the covariance to be zero
ps
the correction is generally a factor of 1.5 to 2. If fNL . and we believe this may have led to the wrong conclu-
10, for Planck, it is necessary to account for secondary non-
sion that there is no bias in the optimal estimator for
Gaussianities properly.
f fromunresolvedpoint sources,thoughsucha bias is
NL
expected to be small, but non-negligible if f ∼1. Our
NL
tioninWMAPmaps,thenonefindsabiasbetweenafac- conclusions are consistent with some of the observations
tor of 1.5 to 2 if f ∼20. If f >30,then the relative in Ref. [25].
NL NL
contribution from secondary non-Gaussianities are sub- Herewehaveconsideredthe confusionfromsecondary
dominant compared to the primordial non-Gaussianity. non-Gaussianities such as point sources and those gen-
Alternatively, if WMAP data are used to constrain that erated by CMB lensing. Additional contributions to the
fNL < 30, then such a constraint must account for the bispectrum existwith correlationsbetween SZ,ISW and
covariancesfromsecondarynon-Gaussianities,especially pointsourcesastheyalltracethe samelarge-scalestruc-
those involving CMB lensing. ture at low redshifts. Previous studies using the halo
With Planck, non-Gaussianity estimates can be ex- modeltodescribethenon-lineardensityfieldhaveshown
tended to lmax ∼2000, but at such small angular scales, correlations such as between SZ and radio sources to be
one finds a bias higher by a factor of more than 2 rela- small[23],butsincethebispectrainthesecasesareofthe
tive to the lowest value of fNL that can be reached with formSZ-PS-PSnl, these bispectra may have a multipolar
Planck (dot-dashed line). In return, if Planck data were dependence in (l ,l ,l ) that is more aligned with the
1 2 3
to constrain fNL to be below ∼ 20, then such a con- CMB primary bispectrum. In an upcoming paper [27]
straint must account for the confusion from secondary we will discuss the impact of such foreground bispectra
anisotropiestothe“optimal”estimatoroffNL,sincelens- duetocorrelationsbetweenCMBsecondaryanisotropies
ing non-Gaussianities produce a correction to fNL with andpointsources. Whileourdiscussionhasconcentrated
fbias ∼10. onamomentumindependentnon-Gaussianityparameter
To account for secondary non-Gaussianities, one can f , or the so-called local type associated squeezed tri-
NL
modify existing“optimalestimators”for fNL andjointly angles, it is easy to generalize the calculation for more
fit for both the primordial non-Gaussianity and the sec- complex descriptions of f (k ,k ,k ) [28]. Due to dif-
NL 1 2 3
ondary non-Gaussianities through a series of estimators ferences in mode overlap, the exact momentum depen-
Sˆα where α denotes the non-Gaussianityof interestwith dence will change the covariance contributions and the
impact of secondary non-Gaussianities will be different
Sˆ =N K , (5)
α α,β β between attempts to measure local f and, for exam-
NL
where ple, equilateral fNL.
Based on our calculations on the covariance between
Bα Bβ
N = l1l2l3 l1l2l3 . (6) lensing and primary bispectra we have suggested a po-
α,β X σ2(l ,l ,l )
1 2 3 tentialconfusionforf measurementinPlanckdata. It
l1l2l3 NL
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