lensing bias to cmb measurements of compensated...
TRANSCRIPT
Lensing Bias to CMB Measurements of Compensated Isocurvature Perturbations
Chen He Heinrich,1, 2, ⇤ Daniel Grin,2, 3 and Wayne Hu2, 3
1Department of Physics, University of Chicago, Chicago, IL 60637
2Kavli Institute for Cosmological Physics, Enrico Fermi Institute, Chicago, IL 60637
3Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637
Compensated isocurvature perturbations (CIPs) are modes in which the baryon and dark matterdensity fluctuations cancel. They arise in the curvaton scenario as well as some models of baryo-genesis. While they leave no observable e↵ects on the cosmic microwave background (CMB) atlinear order, they do spatially modulate two-point CMB statistics and can be reconstructed in amanner similar to gravitational lensing. Due to the similarity between the e↵ects of CMB lensingand CIPs, lensing contributes nearly Gaussian random noise to the CIP estimator that approxi-mately doubles the reconstruction noise power. Additionally, the cross correlation between lensingand the integrated Sachs-Wolfe (ISW) e↵ect generates a correlation between the CIP estimator andthe temperature field even in the absence of a correlated CIP signal. For cosmic-variance limitedtemperature measurements out to multipoles l 2500, subtracting a fixed lensing bias degrades thedetection threshold for CIPs by a factor of 1.3, whether or not they are correlated with the adiabaticmode.
PACS numbers: 95.35.+d, 98.80.Cq,98.70.Vc,98.80.-k
I. INTRODUCTION
Measurements of the cosmic microwave background(CMB) are consistent with primordial curvature fluctua-tions that are nearly scale-invariant, Gaussian, and adi-abatic [1–3]. They lend empirical support to the ideathat these perturbations were produced during inflation,an epoch of accelerated cosmic expansion, as quantumfluctuations of a single field (the inflaton) [4–8].
It is still possible, however, that two di↵erent fieldsdrive inflationary expansion and seed primordial fluctu-ations, leaving small but observable entropy or isocurva-ture fluctuations [9–14]. For example, in the curvaton
model a spectator field during inflation comes to dom-inate the density and hence produces curvature fluctu-ation after inflation. If the curvaton produces baryonnumber, lepton number, or cold dark matter (CDM) aswell, it could lead to a mixture of curvature and isocurva-ture fluctuations [6, 15–27]. Given their common sourcein the curvaton field fluctuations, these isocurvature andcurvature modes are typically correlated, which a↵ectstheir observability [17, 18].
Both correlated and uncorrelated isocurvature fluctu-ations between the non-relativistic matter (baryons andCDM) and photons are highly constrained by CMB ob-servations to be much smaller than curvature fluctuations[2, 3, 28–41]. In the curvaton scenario, these observationsimpose constraints to curvaton decay scenarios [3, 42–49].Nonetheless, there is an additional isocurvature mode,called a compensated isocurvature perturbation (CIP),that is allowed to be of order the curvature fluctuationor larger so long as the decay parameters lie in a rangeallowed by the matter isocurvature constraints [3, 46, 50–52]. These modes entirely evade linear theory constraints
from the CMB because the CDM and baryon isocurva-ture fluctuations are compensated so as to produce noearly-time gravitational potential or radiation pressureperturbation [50, 51, 53]. In addition to the curvatonmodel, CIPs could be produced in some models of baryo-genesis [54].
At higher order, the CIP modulation of the baryondensity leads to spatial fluctuations in the di↵usion-damping scale and acoustic horizon of the baryon-photonplasma [53, 55, 56]. As a result, the CIP field can be re-constructed using the induced o↵-diagonal correlationsbetween di↵erent multipole moments in CMB tempera-ture and polarization maps. CIPs also induce a secondorder change in CMB power spectra [57]. Although thephysics is di↵erent, these e↵ects are very similar to thoseencountered in the weak gravitational lensing of the CMB[53, 55].
Both e↵ects have been used to limit the amplitude ofCIP modes. Using WMAP data, limits from direct re-construction were imposed in Ref. [56], while limits fromCMB power spectra (using Planck data) were imposedin Ref. [57]. The latter work showed that the existenceof CIPs could even reduce internal tensions in CMB datasets (see, for example [2, 58, 59]) but only if the CIPfluctuations are orders of magnitude larger than the cur-vature fluctuations.
Future experiments (such as CMB-S4 [60]) that ap-proach the cosmic-variance limit for measurements of allCMB fields out to arcminute scales in principle can pro-vide a detection of correlated CIP modes through recon-struction, with magnitude of order 10 times the curva-ture fluctuations, and hence test curvaton decay scenar-ios [61]. The similarity between CIP reconstruction andgravitational lens reconstruction from quadratic combi-nations of CMB fields [62–64], however, suggests thatCIP estimators could be biased in the presence of CMBlensing. Past work has estimated this bias and showed
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that it does not alter the upper limits to CIPs fromWMAP [56]. Here we explore this issue further, andshow that lensing can significantly degrade the CIP de-tection threshold from CMB temperature reconstructionfor future experiments.
We begin in Sec. II by summarizing curvaton-modelpredictions for correlated CIPs, CIP reconstruction tech-niques, and the origin of lensing bias to CIP measure-ments. We describe the technique used to simulate lens-ing and CIP reconstruction as well as explore the biaslensing produces in the CIP auto and cross-temperaturespectra in Sec. III. We then perform a Fisher-matrix anal-ysis to evaluate the impact of CMB lensing on the sen-sitivity of a cosmic-variance limited experiment to corre-lated and uncorrelated CIPs in Sec. IV. We discuss theimplications and conclude in Sec V.
II. COMPENSATED ISOCURVATUREPERTURBATIONS
In this section we briefly review the origin of and ob-servable imprint in the CMB of compensated isocurva-ture perturbations. We refer the reader to Refs. [55, 61]for more details.
A. Modulated Observables
The primordial perturbations of the early Universe canbe decomposed into curvature fluctuations on constantdensity slicing ⇣, and entropy fluctuations in the relativenumber density fluctuations of the various species ni withrespect to the photons
Si� =�ni
ni� �n�
n�. (1)
Here i 2 {b, c, ⌫, �} with b for baryons, c for cold darkmatter (CDM), ⌫ for neutrinos, and � for photons. Thecompensated isocurvature mode � is a special combina-tion of entropy fluctuations for which the baryon-photonnumber density fluctuates but is exactly compensated bythe CDM in its energy density perturbations
Sb� = �, Sc� = �⇢b⇢c
�, S⌫� = 0. (2)
To linear order in the fluctuations, there are no observ-able e↵ects of this mode in the CMB. At higher order, thecurvature fluctuations and the acoustic waves they gener-ate propagate in a medium with spatially varying baryonto photon and baryon to CDM ratios. For CIP modesthat are larger than the sound horizon at recombination,subhorizon modes behave as if they were in a separateuniverse with perturbed cosmological parameters [55]
�⌦b = ⌦b�, �⌦c = �⌦b�. (3)
In this limit, we can take the usual calculation for theCMB temperature power spectrum given a curvaturepower spectrum P⇣⇣
C˜T ˜Tl =
2
⇡
Zk2dkT
˜Tl (k)T
˜Tl (k)P⇣⇣(k), (4)
and note that its dependence on the baryon and CDMbackground densities come solely through the radiationtransfer functions T
˜Tl (k). For a small �, we can Taylor
expand the power spectrum to extract its sensitivity toCIPs through the derivative
C˜T ,d ˜Tl =
2
⇡
Zk2dkT
˜Tl (k)
dT˜Tl
d�(k)P⇣⇣(k). (5)
Since this separate universe approximation involves aspatial modulation at recombination, the radiation trans-fer functions and C
˜T ˜Tl represent the power spectrum
in the absence of gravitational lensing. Formally, theyshould also omit post recombination e↵ects from reion-ization and the integrated Sachs-Wolfe e↵ect but sincewe are interested in the change in the subhorizon modeswe simply employ the usual radiation transfer functions.
The implied position-dependent power spectrum whenconsidered on the whole sky represents a squeezed bispec-trum in the CMB where the superhorizon mode is takento be much larger than the subhorizon modes. Quadraticcombinations of subhorizon modes can be used to recon-struct the superhorizon CIP modes in the same manneras in CMB gravitational lens reconstruction [55] as longas the CIP mode is larger than the sound horizon in pro-jection at recombination, i.e. for multipoles l . 100 [61].We explicitly construct this estimator of the CIP field �in Sec. III. To the extent that the CIP and lensing mod-ulations share the same structure, one will contaminatethe other.
B. Curvaton CIPs
One scenario in which CIPs arise is the curvaton sce-nario, where a a spectator scalar field during inflationlater creates curvature perturbations. This field (the cur-vaton). It then decays into other particles, and depend-ing on whether it generates baryon number or CDM indoing so, isocurvature perturbations can arise and arecorrelated with curvature fluctuations. In particular, thepresence of CIP models distinguishes between possibledecay scenarios of the curvaton model [61].
If the curvaton gives rise to all of the curvature per-turbation ⇣, then any resulting CIPs would be fully cor-related with it
� = A⇣, (6)
where A determines the amplitude of the CIP. Thereare two decay scenarios that are particularly interestingbecause of their relatively large CIP amplitude: A ⇡
3
3⌦c/⌦b if baryon number is produced by the curvatondecay and CDM before curvaton decay, while A = �3 ifCDM is produced by curvaton decay, and baryon numberis produced before curvaton decay.
We can exploit the correlated nature of curvaton CIPsby measuring the cross spectra [61]
CXYl =
2
⇡
Zk2dkTX
l (k)TYl (k)P⇣⇣(k), (7)
with X, Y 2 T , � where the CIP transfer function
T�
l (k) = Ajl(kD⇤) (8)
represents a simple projection of � onto the shell at thedistance D⇤ to recombination in a spatially flat universe.For a low signal-to-noise reconstruction of CIP modes,the cross correlation enhances the ability of the CMB todetect the CIP modes (see Ref. [61] and Tab. I below).
Note that in practice we can only cross correlate thelensed CMB field to determine CT�
l though on the rel-evant l . 200 scales the di↵erence between the lensedand unlensed CMB is negligible. On the other hand, weshall see that the ISW-lensing cross correlation [65–69]provides a non-zero signal for this measurement even inthe absence of a true CIP mode.
III. SIMULATIONS
In this section, we simulate CIP reconstruction fromCMB temperature sky maps to assess its noise propertieswith and without non-Gaussian contributions from CMBlensing. We take a flat ⇤CDM cosmology consistent withthe Planck 2015 results [70]1 with baryon density ⌦bh2 =0.02225, cold dark matter density ⌦ch2 = 0.1198, Hubbleconstant h = 0.6727, scalar amplitude As = 2.207⇥10�9,spectral index ns = 0.9645, reionization optical depth⌧ = 0.079, neutrino mass of a single species contribut-ing to the total ⌦m = 0.3156, and T
cmb
= 2.726K. Thelensing simulations are performed using CAMB,2 Len-sPix,3 and HEALPix4 and CIP reconstructions employ amodified version of LensPix that we describe below.
A. CIP Reconstruction
To test the reconstruction pipeline, we begin with thecase for which the CIP quadratic estimator was designed[55], an otherwise Gaussian random CMB temperaturefield. In fact, we take the amplitude of the CIP signal
1 Specifically, we use results obtained with the TT, TE, EE + lowPlikelihood.
2 CAMB: http://camb.info3 LensPix: http://cosmologist.info/lenspix/4 HEALPix: http://healpix.sourceforge.net
-0.14 0.14
-0.014 0.014
FIG. 1. Top: Realization of CIP reconstruction noise assum-ing a Gaussian CMB temperature field. Bottom: target CIPsignal with A = 30 which is nearly scale-invariant shown witha factor of 10 smaller range than the nearly white noise. Mapshave been low pass filtered to l 30 to highlight modes wherein the curvaton scenario, the temperature-CIP cross correla-tion can be used to detect the signal.
to zero to simulate the noise properties of the estimatorand in this case the CMB temperature field is completelyGaussian by construction.
Using HEALPix, we draw nsim
= 4000 Gaussian ran-dom realizations of the dimensionless temperature fluc-tuation field Tlm from the CMB temperature spectrumCTT
l . Note that this is the power spectrum of the lensed
CMB but here the Tlm do not contain the non-Gaussiancorrelations of the properly lensed CMB sky. We do thisso that we can isolate the non-Gaussian aspects of lens-ing below. We take N
side
= 2048 and lmax
= 3900; wehave verified that these values yield su�cient accuracyto use modes out to l = 2500 in the estimator.
From each realization we construct the inverse varianceand derivatively filtered fields
TV (n) =2500X
l=2
lX
m=�l
1
C ˜T ˜Tl
TlmYlm(n),
TD(n) =2500X
l=2
lX
m=�l
C˜T ,d ˜Tl
C ˜T ˜Tl
TlmYlm(n), (9)
where C˜T ˜Tl is the power spectrum of the unlensed CMB.
These filtered fields are then combined to form the min-imum variance quadratic estimator for cosmic-variancelimited temperature measurements out to l = 2500[55, 56]
�LM = ML
ZdnY ⇤
LM (n)TV (n)TD(n). (10)
4
101 102
L
10�7
10�6
10�5
10�4C
IPau
topow
erGaussian CMB
C��L (A = 30)
ML
Gaussian Noise
N��L distribution
FIG. 2. CIP estimator noise power N��L assuming Gaussian
CMB maps. Shown are the mean, 68% and 95% confidencebands (shaded) of 4000 realizations of the estimator in theabsence of a CIP signal. The mean matches closely the the-oretical expectation ML (dotted line) from Eq. (11). Theconfidence bands match the �2 expectation of Gaussian noisegiven the mean (solid lines). For reference we show a trueCIP signal with A = 30 (dashed line).
Here the normalization5
1
ML=
2500X
l,l0=2
(2l + 1)(2l0 + 1)
4⇡
✓l L l0
0 0 0
◆2
⇥⇥C
˜T ,d ˜Tl0 + C
˜T ,d ˜Tl
⇤2
2C ˜T ˜Tl C ˜T ˜T
l0
(11)
would return an unbiased estimator of the CIP field inthe separate universe approximation and in the absenceof lensing. Specifically, given that our realizations lack atrue CIP signal, the average over the ensemble of tem-perature realizations
h�LM i = 0. (12)
In Fig. 1, we show a single realization of the zero meannoise in the estimator. The noise of the estimator isnearly white. For reference we compare it to a realizationof a true CIP signal with A = 30 which is nearly scale-invariant and hence has relatively more power on largescales. We have also tested this estimator against CMBtemperature maps with a true CIP signal following theprocedure of Ref. [56] and find that the reconstruction isunbiased in the absence of lensing-induced o↵-diagonalcorrelations.
5 We correct a small (L + 1)ML ! LML error in the numericalresults on reconstruction noise and parameter errors obtainedfrom Ref. [61], Fig. 2.
101 102
L
�6
�4
�2
0
2
4
CIP
cros
spow
er
⇥10�8
Gaussian CMB
CT�L (A = 30)
Gaussian Noise
NT�L distribution
FIG. 3. CIP noise cross power NT�L assuming Gaussian
CMB maps. Shown are the mean, 68% and 95% confidencebands (shaded) of 4000 realizations of the estimator in theabsence of a CIP signal. The mean matches closely the the-oretical expectation NT�
L = 0 (dotted line). The confidencebands match the Wishart expectation of Gaussian noise (solidlines, see text). For reference we show a true correlated CIPsignal with A = 30 (dashed line).
.
From each realization we construct the all-sky estima-tor of the auto and cross power spectra of the CIP noise
NXYL =
1
2L + 1
X
M
X⇤LM YLM , (13)
for X,Y 2 T, �. Averaging the estimators over the en-semble of realizations yields the noise power spectra
NXYL = hNXY
L i. (14)
We compare these simulated spectra to the theoreticalexpectation N��
L = ML and NT�
L = 0 in Figs. 2 and 3and find good agreement. Given the quadratic natureof the � estimator, NT�
L is proportional to the tem-perature bispectrum which vanishes for a Gaussian tem-perature field. Note that this agreement also tests theslight mismatch in normalization versus noise due to theuse of CTT
l for the realizations whereas the filters are
built out of the unlensed C˜T ˜Tl . We have also tested that
NTTL = CTT
L as expected.Using the distribution of realizations NXY
L we can alsotest the approximation that the cosmic variance of thequadratic combinations of the temperature field producesnearly Gaussian random noise in the CIP estimator.While the product of Gaussian variates is not Gaussiandistributed, the estimator is formed out of many com-binations of temperature multipoles. The central limittheorem implies that the noise in the estimator shouldbe much closer to Gaussian than the product of any in-dividual pair.
5
If the noise in the CIP estimator and the CMB tem-perature field are themselves Gaussian distributed, thepower spectra estimators NXY
L should be distributed ac-cording to a rank 2 Wishart distribution of 2L+1 degreesof freedom
P (NL|NL) = W2
✓NL
2L + 1, 2L + 1
◆. (15)
The Wishart distribution gives the joint probability den-sity of the set of power spectra NXY
L given NXYL which
form 2 ⇥ 2 symmetric matrices NL and NL with ele-ments X, Y 2 {T, �}. To obtain the marginal probabil-ity distribution for a single spectrum e.g. NT�
L one inte-grates the joint distribution over the other spectra (seee.g. Ref. [71]). The Wishart scale matrix NL/(2L + 1)reflects the fact that NL is defined as the average over the2L + 1 M -modes of the products of the Gaussian fieldsrather than the sum. For example, rescaling the statistic
2L + 1
NXXL
NXXL =
LX
M=�L
X⇤LM XLM
NXXL
(16)
brings its marginal distribution to a �2 of 2L+1 degreesof freedom.
In Figs. 2 and 3, we compare the 68% and 95% confi-dence regions of the N��
L and NT�
L distributions to theGaussian expectations of the marginal distributions de-scribed above. Again we find good agreement, indicatingthat the noise of the CIP estimator built from quadraticcombinations of a Gaussian CMB temperature field isitself nearly Gaussian.
B. Lensing Noise
Next we test the reconstruction pipeline with the prop-erly lensed and hence non-Gaussian CMB. In this casewe instead draw 4000 Gaussian random realizations Tlm
from the unlensed CMB power spectrum C˜T ˜Tl and 4000
lensing potentials �lm from C��l with correlations con-
sistent with the C˜T�l ISW-lensing cross power [66–69] as
supplied by CAMB.The angular positions of pixels in the unlensed map
are then remapped into the lensed map according to thegradient of the lensing potential [62–64]
T (n) = T (n + r�) (17)
using LensPix. As in the separate universe approxima-tion for CIPs, a large scale lens acts like a position de-pendent modulation of the CMB and so produces similare↵ects, in particular an o↵-diagonal two-point correlationof multipoles moments which biases CIP reconstruction.
For each realization, we repeat the steps in Eq. (9),(10), (13) and (14) but with these properly lensed tem-perature maps. In Fig. 4 (top) we show an example real-ization of the noise in the CIP estimator. In comparison
-0.14 0.14
-0.14 0.14
FIG. 4. Top: Realization of CIP reconstruction noise usinga lensed CMB temperature field. The noise is significantlylarger than the Gaussian CMB case of Fig. 1. Bottom: CIPreconstruction averaged over 40 realizations of the CMB skyat recombination lensed by the same potential. CIP recon-struction is biased by the statistical anisotropy of lensing con-tributing noise that is comparable in amplitude and spectrumto the Gaussian case.
101 102
L
10�7
10�6
10�5
10�4
CIP
auto
pow
er
Lensed CMB
C��L (A = 30)
ML
Gaussian Noise
N��L distribution
FIG. 5. CIP estimator noise power N��L using lensed CMB
maps. Shown are the mean, 68% and 95% confidence bands(shaded) of 4000 realizations of the estimator in the absenceof a CIP signal. The mean shows an excess of nearly a factorof two from the Gaussian CMB expectation ML (dotted line)from Eq. (11). The confidence bands still match the �2 ex-pectation of Gaussian noise with this enhanced mean (solidlines). For reference we show a true CIP signal with A = 30(dashed line).
6
101 102
L
�6
�4
�2
0
2
4C
IPcr
oss
pow
er⇥10�8
Lensed CMB
CT�L (A = 30)
Gaussian Noise
NT�L distribution
FIG. 6. CIP noise cross power NT�L using lensed CMB maps.
Shown are the mean, 68% and 95% confidence bands (shaded)of 4000 realizations of the estimator in the absence of a CIPsignal. The mean carries a nonzero contamination due to thelensing-ISW bispectrum [66–69] with the Gaussian expecta-tion of zero shown for reference (dotted line). The confidencebands still match the Wishart expectation of Gaussian noisegiven the mean power spectra (solid lines). For reference weshow a true correlated CIP signal with A = 30 which has asimilar spectrum to the contamination (dashed line).
to Fig. 1, the noise is notably larger even though theCMB temperature field that it is constructed from hasthe same power spectrum CTT
l by construction.This additional noise contribution is from the non-
Gaussianity of the lensed CMB. In fact, if the CIP es-timator is averaged over the cosmic variance of the CMBat recombination with a fixed realization of the lensingpotential �, lensing produces a bias in the CIP map itself
h�LM i���
6= 0, (18)
due to the statistical anisotropy that it creates. In Fig. 4(bottom), we separately perform such an average over40 unlensed Tlm CMB realizations lensed by the samepotential. Notice that the fake CIP signal induced bythe fixed � has roughly the same amplitude and spectrumas the Gaussian CMB contribution to the CIP noise inFig. 1 (top).
Of course once averaged over random realization ofthe lens potential as in our main n
sim
realizations, thebias due to the statistical anisotropy of the lens becomesa non-Gaussian CMB source of zero mean noise. InFig. 5, we show that the non-Gaussian lensing contri-bution nearly doubles the noise power in N��
L . Fur-thermore, the lensing-ISW correlation [66–69] producesa nonzero NT�
L as well (see Fig. 6). This fake CIP-temperature correlation has a spectrum that is similarbut not identical to that predicted by the curvaton modeland so error propagation in its removal is important to
model.In spite of their origin in the non-Gaussianity of the
lensed CMB, these excess contributions to the noise inthe CIP estimator are nearly Gaussian distributed. InFig. 5 and 6, we compare the distributions of N��
L andNT�
L to the Gaussian noise expectation. Even the tailsof the distribution as displayed by the 95% confidenceregion are well modeled by a multivariate Gaussian.
For the CIP estimator noise to be Gaussian random,the covariance of the noise power must be diagonal in L,not just Wishart distributed in its variance.To test this,we construct the covariance matrix of N��
L :
Cij ⌘ hN��
LiN��
Lji � N��
LiN��
Lj. (19)
In Fig. 7, we show the correlation matrix
Rij =CijpCiiCjj
. (20)
The o↵-diagonal i 6= j elements are consistent with zero
up to the expected n�1/2sim
statistical fluctuations due tothe finite sample.
To quantify these bounds, we calculate the mean of theo↵-diagonal correlations
R =2
nL(nL � 1)
X
i,j>i
Rij , (21)
where nL is the number of multipoles L 2 [2, 200], andfind a negligible R = 0.00053 and the variance
�2
R =2
nL(nL � 1)
X
i,j>i
R2
ij � R2 (22)
which gives �R = 0.016. In Fig. 8, we further test thatthe scaling of �R with n
sim
by using the first nsim
of the
full 4000 set shows no sign of a deviation from n�1/2sim
thatwould indicate an underlying correlation.
Finally we test that binning the multipoles decreasesthe variance in the noise power in the manner expectedfor independent Gaussian random variables. The statis-tic
⌫VL1,L2 ⌘L2X
L=L1
(2L + 1)N��
L
N��
L
(23)
should be distributed as a �2 with ⌫ degrees of freedom(see Eq. 16) where
⌫ =L2X
L=L1
(2L + 1) = (L2
+ 1)2 � L2
1
. (24)
In Fig. 9, we show that the distribution of V31,40 in the
4000 simulations is in good agreement with the �2 dis-tribution. Other bins show the same level of agreement,which directly tests the approach taken in the next sec-tion of combining the information from each multipole asif they were independent.
7
0 50 100 150
i
0
50
100
150j
�0.10�0.08�0.06�0.04�0.020.000.020.040.060.080.10
FIG. 7. Correlation matrix Rij between multipoles Li and Lj
of the CIP noise power N��L (Eq. 20) using 4000 realizations
of the lensed CMB maps. The diagonal is removed to displaythe o↵-diagonal terms which fluctuate around a mean of R =0.00053 with an r.m.s. �R = 0.016 as consistent with the finitenumber of realizations.
102 103
nsim
10�1
�R
n�1/2sim
FIG. 8. The r.m.s fluctuations in the o↵-diagonal correlations�R vs. the number of simulations nsim for multipoles L 2[2, 200]. The dispersion �R (solid) scales as n�1/2
sim (dotted),as expected for statistical fluctuations around an insignificantcorrelation.
IV. FORECASTS
In the previous section we have shown that the non-Gaussianity of the lensed CMB nearly doubles the noisepower of the CIP estimator for cosmic-variance limitedtemperature measurements out to l = 2500. This has asubstantial impact on the detectability of CIPs that weaddress in this section.
Even when the significant non-Gaussian e↵ects of lens-ing are included, the total noise in the CIP estimator is
0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
L = 31 � 40 Noise Power
0
2
4
6
8
10
�2/�
V31,40
FIG. 9. Binned and normalized noise power V31,40 distribu-tion for the multipole band L = 31 � 40. The distribution iswell modeled by the Gaussian noise expectation that ⌫V31,40
is distributed as a �2 with ⌫ = 720 degrees of freedom.
TABLE I. 2�A detection threshold for A given various noiseassumptions for CIPs that are correlated with CMB tem-perature fluctuations and uncorrelated. Here CIPs are re-constructed to multipole LCIP = 100 from quadratic estima-tors with cosmic-variance limited CMB temperature measure-ments out to l = 2500.
2�A
Noise Corr. CIP Uncorr. CIP
Gaussian CMB Analytic 24 45Gaussian CMB Sim. 25 47Lensed CMB Sim. 31 58
nearly Gaussian distributed (see Sec. III B). This meansthat we can treat the total noise like detector noise orcosmic variance in the data analysis. To estimate theimpact of the lensing contributions on the detection ofthe CIP amplitude A, we use the Fisher matrix. Wefocus on CIP reconstruction using cosmic-variance lim-ited measurements of the CMB temperature field out tol = 2500 as in the simulations of the previous section.We also simplify the analysis by assuming that all cos-mological parameters except for A are fixed.
Under these assumptions the Fisher matrix has a sin-gle entry whose inverse is the estimate of the varianceof measurements of A, �2
A. It combines the informationfrom the various CIP power spectra and multipole mo-ments
��2
A =LCIPX
L=2
X
XY,X0Y 0
@CXYL
@A
�C�1
L
�XY,X0Y 0
@CX0Y 0
L
@A,
(25)
8
where as before X, Y 2 {T, �} and CL is the covariancematrix
CXY,X0Y 0
L =CXX0
L CY Y 0
L + CXY 0
L CX0YL
2L + 1. (26)
We only include L < LCIP
= 100 in order to remain inthe regime where the separate-universe assumption madein constructing the estimator is valid [61].
The di↵erence between this covariance and that im-plied by the Wishart distribution of the previous sectionis that we now include the sample variance of a real CIPsignal
C��
L = C��
L + N��
L ,
CT�
L = CT�
L + NT�
L ,
CTTL = CTT
L = NTTL . (27)
The sample variance depends on the assumed value of Aand so we take its value to be the detection thresholdA = 2�A for the various simulated and analytic noisespectra NXY
L from the previous section (see Tab. I).To establish the baseline, we start with the analytic
form for the noise given a Gaussian CMB in the absenceof CIPs and lensing as in Ref. [61],
N��
L = ML, NT�
L = 0, (28)
for which 2�A = 24 with correlated CIPs. Employing in-stead the Gaussian CMB simulations for the noise spectrawe obtain a consistent 2�A = 25. Finally with the lensedCMB simulations the threshold is raised to 2�A = 31, afactor of ⇠1.3 higher compared to the case of analyticalestimate of the noise for Gaussian CMB.
Given that these detection thresholds are larger thaneven the largest curvaton motivated value, it is also inter-esting to consider the case of uncorrelated CIPs CT�
L = 0with the same C��
L . We find that the lensing noise con-tributes here as well by raising the 2� threshold a factorof ⇠ 1.3 from 45 to 58.
If we take LCIP
= 200, the detection thresholds areslightly lower in the correlated case but the relativedegradation from lensing is similar. In the uncorrelatedcase, the results are insensitive to the exact value of L
CIP
since most of the signal-to-noise comes from large scalesfor a nearly scale invariant auto-spectrum.
V. CONCLUSIONS
A large scale CIP field modulates the two point statis-tics of small scale CMB anisotropies, much like gravita-tional lensing. The quadratic reconstruction of the for-mer from the latter can therefore be contaminated bylensing. We have shown, using simulations, that non-Gaussian modulation by lensing provides an additionalcontribution to the noise in the CIP estimator that iscomparable to that of the cosmic variance of the smallscale CMB modes themselves. Moreover, the lensing-ISW bispectrum (present in the absence of real CIPs)
[65–69] provides a false signal that resembles the CIP-temperature cross correlation in the curvaton model;such contamination similarly a✏icts estimators of theamplitude f
NL
of primordial local-type non-Gaussianity[3, 65–68].
Despite these non-Gaussian e↵ects in the CMB, theresultant noise in the CIP estimator is nearly Gaussian byvirtue of the central limit theorem. We find that the noisepower is uncorrelated to good approximation – no morethan 0.1% correlations averaged over all multipole pairswith pair fluctuations that are consistent with our finitesample of simulations and less than 2%. Furthermore,binning of multipoles reduces the variance of the noisepower in the same manner as a Gaussian random field.
Treating the lensing contamination as excess Gaussianrandom noise, we estimate its impact on the detectionof CIPs using the Fisher information matrix. Assum-ing cosmic-variance limited measurements of the CMBtemperature anisotropies out to l = 2500, we find thatthe detection thresholds for the CIP amplitude param-eter A are raised by a factor of 1.3 when the lensingbias is included for both the uncorrelated and correlatedCIPs. Here we have employed CIP reconstruction up tothe separate-universe limit L
CIP
= 100 and assumed allother cosmological parameters are fixed.
Our methodology, which employs direct simulation oflensing, is also straightforward to generalize to caseswhere measurement noise, systematic e↵ects and othercosmological parameters are included. In these cases thelensing contamination is treated as an additional signalthat is jointly modeled along with the CIP contributionsand prior information in a realistic data pipeline. Cos-mological parameter uncertainties can make the lensingcontamination even more important given parameter de-generacies. For example the ISW-lensing contamination[66–69], depends on the cosmic acceleration model andtakes a similar form to the CIP cross correlation. Fortu-nately, the lensing contamination to the CIP auto corre-lation takes a very di↵erent form which can be used tobreak degeneracies.
Likewise, CMB polarization information can also as-sist in distinguishing lensing from CIPs as well as reducethe overall noise in the CIP estimator. In particularquadratic reconstruction from the E and B modes hasthe highest signal-to-noise for lensing reconstruction [72]and the lowest for CIP reconstruction [61]. More gener-ally, these additional fields provide consistency tests forthe specific type of modulation expected by CIPs in theauto and cross power spectra of their quadratic estima-tors.
Finally, there are consistency tests that are internalto just the temperature based CIP estimator. The esti-mator is constructed out of many pairs of temperaturemultipoles whose weights were chosen to be optimal inthe absence of lensing. Whereas the impact of CIP mod-ulation does not depend strongly on the orientation ofthe modes, lensing does since its e↵ect vanishes if thedeflection is in a direction orthogonal to the modes. In
9
principle, the pairs can be reweighted to de-emphasizethe most contaminated modes at the expense of compu-tational e�ciency of the estimator. We leave these topicsfor future study.
ACKNOWLEDGMENTS
We thank Cora Dvorkin and Duncan Hanson for stim-ulating discussions. CH and WH were supported by
U.S. Dept. of Energy contract DE-FG02-13ER41958 andNASA ATP NNX15AK22G. DG is funded at the Uni-versity of Chicago by a National Science Foundation As-tronomy and Astrophysics Postdoctoral Fellowship underAward AST-1302856. This work was supported in partby the Kavli Institute for Cosmological Physics at theUniversity of Chicago through grant NSF PHY-1125897and an endowment from the Kavli Foundation and itsfounder Fred Kavli. Computing resources were providedby the University of Chicago Research Computing Cen-ter.
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