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ON THE LINEAR TERM CORRECTION FOR NEEDLETS/WAVELETS NON-GAUSSIANITY ESTIMATORS Simona Donzelli⋆,1 , Frode K. Hansen2,3 , Michele Liguori4,5 , Domenico Marinucci6 and Sabino Matarrese4,5

arXiv:1202.1478v1 [astro-ph.CO] 7 Feb 2012

Draft version September 15, 2017

ABSTRACT We derive the linear correction term for needlet and wavelet estimators of the bispectrum and the non-linearity parameter fNL on cosmic microwave background radiation data. We show that on masked W M AP -like data with anisotropic noise, the error bars improve by 10-20% and almost reach the optimal error bars obtained with the KSW estimator (Komatsu et al. 2005). In the limit of fullsky and isotropic noise, this term vanishes. We apply needlet and wavelet estimators to the W M AP 7-year data and obtain our best estimate fNL = 37.5 ± 21.8. Subject headings: cosmic microwave background — cosmology: observations — early universe — methods: data analysis — methods: statistical 1. INTRODUCTION

It is well known that most inflationary models predict the fluctuations in the Cosmic Microwave Background (CMB) to be close to but not exactly Gaussian. Non-Gaussian predictions are strongly model dependent, thus making primordial non-Gaussianity (NG) a powerful tool to discriminate among different Early Universe scenarios (see e.g Bartolo et al. (2004a); Chen (2010); Liguori et al. (2010) and references therein). In this paper we will focus on so called local nonGaussianity, which can be parametrized in the simple form: local Φ2L (x) − hΦ2L (x)i , (1) Φ(x) = ΦL (x) + fNL

where Φ(x) is the primordial curvature perturbation field at the end of inflation and ΦL (x) is the Gaussian part of the perturbation. The dimensionless paramelocal ter fNL describes the amplitude of non-Gaussianity8. Local non-Gaussianity is predicted to arise from standard single-field slow-roll inflation (Acquaviva et al. 2003; Maldacena 2003), although at a very tiny level, as well as from multi-field inflationary scenarios, like the curvaton (Moillerach 1990; Enqvist & Sloth 2002; Lyth & Wands 2002; Moroi & Takahashi 2001) or inhomogeneous (pre)reheating models (Dvali et al. 2004; Kolb et al. 2005, 2006)). Even alternatives to inflation, such as ekpyrotic and cyclic models (Koyama et al. 2007; Buchbinder et al. 2008) predict a local NG signature. The expected non-Gaussian amplitude fNL varies sig⋆ 1

[email protected] INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica Milano, Via E. Bassini 15, 20133 Milano, Italy 2 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway 3 Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway 4 Universit` a di Padova, Dipartimento di Fisica e Astronomia “G. Galilei”, Universit` a degli Studi di Padova , Via Marzolo 8, 35131 Padova, Italy 5 INFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy 6 Dipartimento di Matematica, Universit` a di Roma“Tor Vergata”, Via della Ricerca Scientifica 1, I-00133 Roma, Italy 8 For simplicity of notation in the following we will drop the superscript “local” and simply write fNL . No confusion can arise since in this context we are not dealing with other types of nonGaussianity

nificantly from model to model. For example, standard single-field slow-roll inflation predicts fNL ∼ 10−2 at the end of inflation (Acquaviva et al. 2003; Maldacena 2003) (and therefore a final value ∼ unity after general relativistic second-order perturbation effects are taken into account (Bartolo et al. 2004b,c)). Such a small value is not experimentally detectable and for this reason a detection of a primordial non-Gaussian signal in present and forthcoming CMB data will rule out single-field slow-roll inflation as a viable scenario. Motivated by these considerations many groups have attempted to measure fNL using CMB datasets, and Wilkinson Microwave Anisotropy Probe (W M AP ) data in particular. Consistently with theoretical findings, the most stringent NG bounds have been obtained using estimators of the CMB angular bispectrum (namely the three-point function of CMB fluctuations in harmonic space). While in the classical approach to fNL estimation (Komatsu et al. 2005; Creminelli et al. 2006; Yadav & Wandelt 2008; Yadav et al. 2007b) the starting point to build an optimal cubic statistic is a direct multipole expansion of the temperature field, alternative representations can be used as well, like e.g. the modal bispectrum expansion of Fergusson et al. (2009), or bispectra of wavelet and needlet coefficients. Since all these approaches are just based on expanding the same quantity (the angular bispectrum) in different bases (polynomial modes, wavelets, needlets etc.), they are also ultimately expected to yield very similar results when applied to data. This is indeed the case. For example, a recent estimate using an optimal bispectrum estimator has been made by Smith et al. (2009). They obtained the smallest error bars on fNL to date, finding fNL = 38 ± 21 on W M AP 5-year data. Consistent results, although with larger error bars, were found by Curto et al. (2009a) and Pietrobon et al. (2009), using parts of the bispectrum of Spherical Mexican Hat Wavelets (SMHW) (Mart´ınez-Gonz´ alez et al. 2002) and the skewness of needlet coefficients, and by Fergusson et al. (2010), using a modal bispectrum expansion. The most updated optimal result has been found by the W M AP team on the 7-year data with the estimate fNL = 32 ± 21 (Komatsu et al. 2011). Wavelets provide again a very similar result of fNL = 30 ± 23 (Fisher matrix bound

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σF = 22.5) (Curto et al. 2011b). At this point one could reasonably ask why it is useful to implement estimators using many different bispectrum representations. After all in the end they are all expected to produce basically the same output in terms of fNL . The justification is two-fold. First of all, different expansions can provide information beyond fNL (like mode spectra and full bispectrum reconstruction (Fergusson et al. 2009, 2010)). Second of all, different expansions can present important practical advantages, such as computational rapidity or robustness to a number of contaminants and effects (masking, non-stationarity of the noise, foreground emission and so on). The W M AP 7-year results quoted above seem indeed to illustrate the latter point well. When we compare the bispectrum and the wavelet results we see that central values and error bars are both very similar. However, to achieve this result the bispectrum estimator needs to include a very important linear correction term. This additional term, originally introduced in Creminelli et al. (2006), subtracts from the measured three-point function a spurious fNL contribution due to the breaking of statistical isotropy introduced by masking and non-stationary noise. Without this contribution the error bars of the bispectrum estimator would be much larger9 than the quoted 21 (a factor at least 4 or 5 larger, as shown for example in Fig. 4 of Creminelli et al. (2006)). It turns out however that despite having an error bar only ∼ 10% larger than the bispectrum measurement, the W M AP 7-year wavelet result of fNL = 30.0 ± 23 (Fisher matrix bound σF = 22.5) (Curto et al. 2011b) was obtained without including any linear correction. It seems then that the wavelet expansion is much less affected by masking and anisotropic noise than the bispectrum estimator is. This raises two important issues, firstly pointed out in Fergusson and Shellard (2011). The first is why wavelets seem so much more efficient than a standard harmonic decomposition in dealing with breaking of isotropy in the data. This point was partly addressed in (Curto et al. 2011c), but we think that no definite and conclusive explanation has been provided to date (see also paragraph 2.3.3) . The other issue is whether it is possible to further reduce also the variance of waveletbased estimators through the introduction of the linear correction term. The aim of this work is to address both these questions: firstly we explicitly derive the linear correction for needlet and wavelet estimators of the bispectrum and of the non-linearity parameter fNL . We show how this linear term identically vanishes for full-sky maps with isotropic noise; we also explain why this term is in general smaller than for harmonic space based estimators, although non-negligible (Section 2). We then implement the linear term correction on a needlet bispectrum estimator and show that these results are confirmed by simulations; the procedures are then applied to W M AP 7-year data (Section 3). Conclusions are drawn in Section 4.

9 Note that the correction is very large for local NG, but much smaller for other types of NG, hence the reason to consider only local NG in this paper, which is entirely focused on issues related to the linear term

2. MOTIVATION FOR THE LINEAR TERM

CORRECTION 2.1. Some background results 2.1.1. Wick products We recall first some well-known background facts, to fix notation. Consider

Gaussian variables X1 , X2 , X3 , such that hXi i = 0, Xi2 = σi2 , hXi Xj i = σij . The Wick product of the three variables is defined as

: X1 , X2 , X3 := X1 X2 X3 −σ12 X3 −σ13 X2 −σ23 X1 . (2)

Example 1 For X1 = X2 = X3 = X : X, X, X := X 3 − 3σ 2 X .

For σ 2 = 1 this is the well-known Hermite polynomial of order 3, H3 (X) = X 3 − 3X. For the expected values of Wick products, the following Diagram Formula holds h: X11 , X12 , X13 :: X21 , X22 , X23 :i = hX11 X21 i hX12 X22 i hX13 X23 i + 5 permutations (3)

where the only permutations that are considered are those such that in each pair an element from the first triple {X11 , X12 , X13 } is coupled with an element from the second triple {X21 , X22 , X23 }. It is usually convenient to visualize the elements {X11 , X12 , X13 },{X21 , X22 , X23 } as vertices aligned on two different rows, and the pairs as edges connecting two different vertices; the above-mentioned diagram formula is then usually expressed by stating that “flat edges” are ruled out. Example 2 For the Hermite polynomial we have D E D 2 E 2 (H3 (X)) = X 3 − 3X =6.

We see that the expected value of the square of the third order WickD product is much smaller than the ex E 3 2 pected value of X = 15. In fact, given Gaussian random variables with unit variance a standard argument can be used to prove that Wick products yield the smallest variance among all other polynomials of the same order. This result is well-known and can be found in any monographs on related subjects, see for instance (Marinucci & Peccati 2011; Peccati & Taqqu 2011) for two recent references; we provide here a short proof for the case of cubic polynomials for the sake of completeness. Indeed consider Gaussian zero mean, unit variance random variables X1 , X2 , X3 , not necessarily independent, and form a generic polynomial P (X1 , X2 , X3 ) := X1 X2 X3 + c1 X2 X3 + c2 X1 X3 +c3 X1 X2 + c12 X3 + c13 X2 + c23 X1 + c123 , (4) where the c’s are arbitrary (fixed) real numbers. We can rewrite P (X1 , X2 , X3 ) =: X1 , X2 , X3 : +c1 X2 X3 + c2 X1 X3 + c3 X1 X2 +(c12 + hX1 X2 i)X3 + (c13 + hX1 X3 i)X2 +(c23 + hX2 X3 i)X1 + c123 , =: X1 , X2 , X3 : +Q(X1 , X2 , X3 ) ,

(5)

On the linear term correction for needlets/wavelets NG estimators where Q(X1 , X2 , X3 ) is a second order polynomial in (X1 , X2 , X3 ). Now, it is readily seen that : X1 , X2 , X3 : is uncorrelated with any polynomial of order 1 or 2 in the same variables; for instance h: X1 , X2 , X3 : X1 X2 i = hX1 X2 X3 X1 X2 i − hX1 X2 i hX3 X1 X2 i

2 − hX1 X3 i X2 X1 − hX2 X3 i X12 X2 = 0 ,

(6)

because odd moments of Gaussian variables always vanish. Hence we have V ar {P (X1 , X2 , X3 )} = V ar {: X1 , X2 , X3 : +Q(X1 , X2 , X3 )} = V ar {: X1 , X2 , X3 :} + V ar {Q(X1 , X2 , X3 )} +2Cov {: X1 , X2 , X3 :, Q(X1 , X2 , X3 )} = V ar {: X1 , X2 , X3 :} + V ar {Q(X1 , X2 , X3 )} ≥ V ar {: X1 , X2 , X3 :} , (7) whence the result is established. In words, Wick polynomials of order 3 are uncorrelated by construction with any other polynomial of smaller order in the same random variables, and hence minimize the variance in the class of cubic polynomials of unit coefficient in the maximal term. This provides the heuristic rationale for the introduction of linear correction terms in standard and wavelet/needlet bispectrum estimators. 2.1.2. Needlets, Mexican needlets and SMHW Let b(t) be a weight function satisfying three conditions, namely • Compact support : b(t) is strictly larger than zero only for t ∈ [B −1 , B], some B > 1 • Smoothness: b(t) is C ∞ • Partition of unity: for all ℓ = 1, 2, ... we have ∞ X ℓ b2 =1. Bj j=0 Recipes to construct a function b(t) that satisfy these conditions are easy to find and are provided for instance by (Marinucci et al. 2008) and (Marinucci & Peccati 2011). Consider now a a grid of points {ξjk } on the sphere, e.g. the HEALPix10 centres (G´ orski et al. 2005); the needlet system is then defined by j+1

B X p ψjk (x) = λjk

ℓ X

b

ℓ=B j−1 m=−ℓ

ℓ Bj

(8) with the corresponding needlet coefficients provided by Z p βjk = λjk f (x)ψjk (x)dx S2

=

j+1 B X

ℓ X

ℓ=B j−1 m=−ℓ 10

b

ℓ Bj

http://healpix.jpl.nasa.gov

The coefficients {λjk } are proportional to the pixel area, see for instance (Baldi et al. 2009b) for more details. The needlet idea has been extended by (Geller & Mayeli 2009a,b) with the construction of so called Mexican needlets (see Scodeller et al. (2011) for numerical analysis and implementation in a cosmological framework). Loosely speaking, the idea is to replace the compactly supported kernel b(ℓ/B j ) by a smooth function of the form 2p ℓ ℓ ℓ2 b = (10) exp − 2j , Bj Bj B for some integer parameter p. Mexican needlets have extremely good localization properties in real space, and for p = 1 they provide at high frequencies a good approximation to the so-called Spherical Mexican Hat Wavelet (SMHW) construction. The latter is exploited for instance by Mart´ınez-Gonz´ alez et al. (2002); Curto et al. (2009a, 2011a,b), to which we refer for more discussion and definitions. In short, the SMHW coefficients at location n and scale R are provided by Z f (x)Ψ(x, n; R)dx , (11) w(n, R) = S2

where the wavelet filter is defined as y 2 y 2 2 2 1 1 ][2 − ]e−y /2R ; [1 + Ψ(x, n; R) = √ 2 R 2π N (R) (12) p here, N (R) = R 1 + R2 /2 + R4 /4 is a normalizing constant and y = 2 tan θ/2 represents the distance between x and n, evaluated on the stereographic projection on the tangent plane at n; θ is the corresponding angular distance, evaluated on the spherical surface.

2.2. The linear correction term for the KSW estimator For our arguments to follow, and to allow for a proper comparison with the results for needlet/wavelet based estimators below, we shall provide a brief heuristic argument to motivate the need for a linear term correction (Creminelli et al. 2006; Yadav et al. 2007a) in the wellknown KSW bispectrum estimator for the fNL parameter In particular, let us consider any three frequencies ℓ1 , ℓ2 , ℓ3 satisfying standard triangular conditions, and consider the angle-averaged bispectrum r (2ℓ1 + 1)(2ℓ2 + 1)(2ℓ3 + 1) ℓ1 ℓ2 ℓ3 Bℓ 1 ℓ 2 ℓ 3 = 0 0 0 4π X ℓ ℓ2 ℓ3 1 aℓ 1 m1 aℓ 2 m2 aℓ 3 m3 × m m m m1 m2 m3

Yℓm (x)Y ℓm (ξjk ) ,

aℓm Yℓm (ξjk ) .

(9)

3

=

Z

X

1

2

3

aℓ 1 m1 aℓ 2 m2 aℓ 3 m3

S 2 m1 m2 m3

× Yℓ1 m1 (x)Yℓ2 m2 (x)Yℓ3 m3 (x)dx Z = Tℓ1 (x)Tℓ2 (x)Tℓ3 (x)dx , S2 X Tℓ (x) = aℓm Yℓm (x) ,

(13)

m

where we have chosen a convenient (albeit non-standard) normalization for the bispectrum to make our argument

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notationally simpler - these normalizations do not affect by any means the substance of the argument. Now, the bispectrum should be more properly written as Z {: Tℓ1 (x), Tℓ2 (x), Tℓ3 (x) :} dx Bℓ 1 ℓ 2 ℓ 3 = S2 Z {Tℓ1 (x)Tℓ2 (x)Tℓ3 (x)} dx = 2 ZS {Γℓ1 ℓ2 (x)Tℓ3 (x) − S2

+ Γℓ1 ℓ3 Tℓ2 (x) + Γℓ2 ℓ3 Tℓ1 (x)}dx , Γℓu ℓv (x) = hTℓu (x)Tℓv (x)i , u, v = 1, 2, 3 ,

(14)

In the presence of full sky-maps with isotropic noise we have that Γℓu ℓv (x) ≡ Γℓu ℓv , i.e. it would be constant over pixels, whence for instance Z Z {Γℓ1 ℓ2 Tℓ3 (x)} dx {Γℓ1 ℓ2 (x)Tℓ3 (x)} dx = S2 S2 Z (15) {Tℓ3 (x)} dx = 0 , = Γℓ 1 ℓ 2 S2

because Z X

aℓm Yℓm (x)dx =

S2 m

X

aℓm

Z

S2

m

Yℓm (x)dx ≡ 0 .

(16) (assuming the monopole is zero). On the other hand, in the presence of anisotropic noise and/or masked maps the previous argument cannot hold, whence the linear term does not cancel and the variance of the Wick product (including the Wick product) is systematically smaller than any other cubic statistic; for instance, as before (Z ) 3 V ar Tℓ (x)dx S 2 \M

=6

Z

3

(S 2 \M)×(S 2 \M)

+9

Z

(S 2 \M)×(S 2 \M)

hTℓ (x)Tℓ (y)i dxdy

(17)

2 2 Tℓ (x) Tℓ (y) hTℓ (x)Tℓ (y)i dxdy .

Here, we have used S 2 \M to denote integration over the sphere S minus the masked region M . In the full-sky case M = ∅, with isotropic noise, we have

2 2ℓ + 1 CMB (18) Cℓ + Cℓnoise Tℓ (x) = 4π whence the previous expression becomes 3 3 2ℓ + 1 CMB Cℓ + Cℓnoise =6 4π Z × Pℓ3 (x · y)dxdy

In the previous computations, we have written Pℓ for Legendre polynomials, x · y for scalar products, we have used the well-known Wigner’s 3j symbol arising from the (Gaunt) integral of the third power of Pℓ , and we exploited the well-known fact that Z Pℓ (x · y)dxdy = 0 , (20) S 2 ×S 2

whence the so-called flat edges terms vanish. This is not so in general, though. Note, however, that (Z ) 3

2 V ar Tℓ (x) − 3 Tℓ (x) Tℓ (x) dx S 2 \M

=6

Z

3

S 2 \M×S 2 \M

hTℓ (x)Tℓ (y)i dxdy ,

(21)

so that the flat terms are cancelled, even in the presence of anisotropic noise or masked regions. 2.3. Needlets/wavelets Non-Gaussianity estimators The situation for wavelet or needlet/like nonGaussianity estimators is to some extent analogous to the one for the KSW procedure. For instance, in (Curto et al. 2011a), Eqs. (14-16), the variance of the following statistic is considered Z 1 1 w(Ri , n1 )w(Rj , n1 )w(Rk , n1 )dn1 , (22) × 4π σi σj σk S2 where each w(Ri , n1 ) is the (random) SMHW coefficient at scale Ri and location n1 (Eq. (11)); this variance can clearly be written as 1 1 1 (4π)2 σi σj σk σr σs σt Z hw(Ri , n1 )w(Rj , n1 )w(Rk , n1 ) × S 2 ×S 2

w(Rr , n2 )w(Rs , n2 )w(Rt , n2 )idn1 dn2

(23)

For full-sky maps with isotropic noise, we would have that hw(Ri , n1 )w(Rj , n1 )w(Rk , n1 )w(Rr , n2 )w(Rs , n2 )w(Rt , n2 )i = hw(Ri , n1 )w(Rr , n2 )i hw(Rj , n1 )w(Rs , n2 )i × hw(Rk , n1 )w(Rt , n2 )i + 5 permutations, (24) e.g., the only terms that non-vanish are those where n1 and n2 appear in the same pair. However, as discussed above, in the presence of masked regions and/or anisotropic noise the terms with n1 , n1 or n2 , n2 do not vanish, and we should rather write

S 2 ×S 2

+9 Z × =6

2ℓ + 1 4π

2

(S 2 \M)×(S 2 \M) 3

2ℓ + 1 4π

hw(Ri , n1 )w(Rj , n1 )w(Rk , n1 )w(Rr , n2 )w(Rs , n2 )w(Rt , n2 )i = hw(Ri , n1 )w(Rr , n2 )i hw(Rj , n1 )w(Rs , n2 )i × hw(Rk , n1 )w(Rt , n2 )i + 14 permutations, (25)

2 CMB Cℓ + Cℓnoise Pℓ (x · y)dxdy

3 CMB Cℓ + Cℓnoise

ℓℓℓ 000

2

. (19)

i.e., there are 9 further “flat” permutations (those where two terms from the same row are coupled - there are three different way for each row to do this). These missing

On the linear term correction for needlets/wavelets NG estimators terms give to the integral a contribution of the form R 1 1 1 (4π)2 σi σj σk σr σs σt × S 2 ×S 2 hw(Ri , n1 )w(Rr , n1 )i

× hw(Rj , n2 )w(Rs , n2 )i hw(Rk , n1 )w(Rt , n2 )i dn1 dn2 R 1 1 1 hw(Ri , n1 )w(Rr , n1 )i = (4π) 2 σ σ σ σ σ σ S 2 ×S 2 i j k r s t × hw(Rj , n2 )w(Rs , n2 )i hw(Rk , n1 ) × w(Rt , n2 )i dn1 dn2

(26)

Now, for isotropic noise, hw(Ri , n1 )w(Rr , n1 )i ≃ const, whence the previous quantity is approximately proportional to R S 2 ×S 2 hw(Rk , n1 )w(Rt , n2 )i dn1 dn2 R

R = S 2 w(Rk , n1 )dn1 × S 2 w(Rt , n2 )dn2 = 0 , (27)

because

Z

w(Rk , n1 )dn1 = 0

(28)

S2

in the absence of masked regions. So when the celestial sphere is fully observed, it is equivalent to consider or not the extra, “flat” terms with the same indexes n1 , n2 in the pairs. As stated earlier, and as for the KSW estimators, these terms are no longer identically zero in the presence of masked regions or anisotropic noise. A linear correction term can therefore be needed; we discuss its derivation in the subsection below. 2.3.1. The linear correction term According to our previous argument, it is straightforward to see how, to decrease the variance of the wavelet cubic statistic, it is enough to change the cubic statistic from Z w(Ri , n)w(Rj , n)w(Rk , n)dn (29) S2

to Z

(: w(Ri , n), w(Rj , n), w(Rk , n) :) dn

(30)

S2

i.e. subtract a linear term of the form Z w(Ri , n)w(Rj , n)w(Rk , n)dn 2 ZS hw(Ri , n)w(Rj , n)i w(Rk , n)dn − 2 ZS hw(Ri , n)w(Rk , n)i w(Rj , n)dn − S2 Z hw(Rk , n)w(Rj , n)i w(Ri , n)dn . −

(31)

A similar situation exists for the needlets bispectrum (Lan & Marinucci 2008a; Rudjord et al. 2009), which we can implement as nX X 1 βj1 k βj2 k βj3 k − Γj1 j2 (k)βj3 k σj1 σj2 σj3 k k o X X − (32) Γj1 j3 (k)βj2 k − Γj2 j3 (k)βj1 k1 , k

where βjk are the usual coefficients for (standard or Mexican) needlets and Γj1 j2 (k) = hβj1 k βj2 k i .

(33)

Of course, under isotropic noise X ℓ ℓ 2ℓ + 1 b Cℓ . Γj1 j2 (k) = Γj1 j2 = b B j1 B j2 4π ℓ (34) Note that the linear terms have expected value zero always hΓj1 j2 (k)βj3 k i = Γj1 j2 (k) hβj3 k3 i = 0 ;

(35)

however the observed value of these terms over one realization of the sky is exactly equal to zero only if the sum (or the integral) is taken over the whole sphere and the noise is isotropic (assuming again zero monopole), i.e. for SMHW Z hw(Ri , n)w(Rj , n)i w(Rk , n)dn S2 Z w(Rk , n)dn = 0 , (36) = hw(Ri , n)w(Rj , n)i S2

and correspondigly for the needlets X X Γj1 j2 βj3 k3 = Γj1 j2 βj1 k1 = 0 , k3

(37)

k1

i.e. when there are no masked regions and the noise is isotropic. These are the assumptions under which the behavior of the needlet bispectrum was investigated by (Lan & Marinucci 2008a), where it was firstly introduced in the statistical literature. It should be noted, moreover, that in practical situations the contribution of the linear term for wavelet/needletlike bispectrum estimators will be typically smaller than for KSW. This can be explained as follows: consider X X X ℓ aℓm Yℓm (ξj3 k3 ) Γj1 j2 (k)βj3 k = Γj1 j2 (k) b B j3 k k ℓm X X ℓ Z = Γj1 j2 (k) b T (x)Y ℓm (x)Yℓm (ξj3 k )dx B j3 S2 k ℓm Z X ℓ T (x)Pℓ (x · y)dxdy . (38) ≃ Γj1 j2 (y) b B j3 S2 ℓ

S2

k

5

Now it is a consequence of needlet concentration in pixel space that Z X ℓ Z Γj1 j2 (y) b T (x)Pℓ (x · y)dxdy B j3 S2 S2 l # "Z Z X ℓ Pℓ (x · y)dy dx T (x) = Γj1 j2 (y) b B j3 S2 S2 ℓ "Z # Z X ℓ ≃ T (x) Γj1 j2 (y) b Pℓ (x · y)dy dx B j3 S2 Nε (x) ℓ

(39)

where Nε (x) is a small neighborhood of x; now assuming that noise is approximately constant over Bε (x), we can

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write X ℓ Pℓ (x · y)dy Γj1 j2 (y) b B j3 Bε (x) l "Z # X ℓ ≃ Γj1 j2 (y) Pℓ (x · y)dy ≃ 0 ,(40) b B j3 Bε (x) Z

better in cases where noise anisotropy is not extremely relevant, and less well otherwise. Also, for higher order polyspectra or alternative statistics (such as KSW) the equivalence between mean subtraction and linear term correction will no longer hold.

ℓ

because by localization Z X ℓ Pℓ (x · y)dy b B j3 Bε (x) ℓ Z X ℓ b ≃ Pℓ (x · y)dy ≃ 0 . B j3 S2

(41)

ℓ

2.3.2. The relationship with mean subtraction We shall now show how, in the presence of nearly isotropic noise, the behaviour of the linear term is wellapproximated by subtracting scale-by-scale the sky average of wavelets or needlets coefficients. Indeed, define 1 X βjk (42) βj = N k

then

1 X (βj1 k − β j1 )(βj2 k − β j2 )(βj3 k − β j3 ) N k ) ( 1 X 1 X = βj1 k βj2 k βj3 k − β j1 βj2 k βj3 k N N k k ) ) ( ( 1 X 1 X − β j2 βj1 k βj3 k − β j3 βj1 k βj2 k N N k

k

+ 2β j1 β j2 β j3 .

(43)

Now for nearly isotropic noise, e.g., if the covariance among coefficients Γj1 j2 (k) = hβj1 k βj2 k i ≃ Γj1 j2

(44)

is nearly constant over the sky, then, for high enough j, it is natural to expect that the cross-correlation among wavelet coefficients evaluated on a sky realization will be close to the ensemble average, ( ) 1 X Γj1 j2 ≃ (45) βj1 k βj2 k . N k

Moreover we also expect

β j1 β j2 β j3 ≃ 0 , whence (

)

(46)

1 X β j1 ≃ β j1 Γj2 j3 ≃ Γj2 j3 βjk , N k k (47) and similarly for the permutation terms, whence the linear term will be well-approximated by mean subtraction. At smaller frequencies j, this argument will not work, but at these scales noise is likely to be negligible. In general, it should be noted that the approximation will work 1 X βj1 k βj2 k N

2.3.3. A toy counterexample

In Curto et al. (2011c), Section 3.2 it is claimed that 3 the linear term is of order fNL and hence negligible. This is motivated on the basis of the asymptotic uncorrelation of the wavelet coefficients, implying the validity of the Central Limit Theorem (CLT) for the cubic (bispectrum statistic). We do agree on the uncorrelation of the coefficients and the Central Limit Theorem taking place; see for instance (Baldi et al. 2009a), (Lan & Marinucci 2008a,b) for analytic arguments in the needlets and Mexican needlets case. We fail to see, however, why this should necessarily imply that the resulting linear term should be negligible. As a toy counterexample, consider a sequence of independent Gaussian random variables Xi , with zero

mean hXi i and nonconstant (e.g. anisotropic) variance Xi2 = σi2 ; to fix ideas, we shall take σi2 = 1 for i = 1, 3, 5, (i.e. odd) σi2 = 3 for i = 2, 4, ..., (i.e. even). With these assumptions, we try to mimic the behavior of nearly independent wavelet coefficients with anisotropic noise; for simplicity, we neglect the effect of masked regions, but the argument would be analogous. Now consider the statistic n

1 X 3 B1n = √ X , n i=1 i n

n

3 1X 1 X Xi − X n , X n = B2n = √ Xi , n i=1 n i=1 n

n

1 X 1 X (: Xi , Xi , Xi :) = √ Xi3 − 3σi2 Xi , B3n = √ n i=1 n i=1 which correspond, in these circumstances, to the naive cubic statistics/bispectrum (without linear term), the cubic statistic with mean subtraction, and the proper bispectrum with Wick polynomials/linear term subtraction. Because the Xi are exactly independent, it is trivial to see that the CLT holds, and all three statistics are asymptotically Gaussian. Nevertheless, it is readily seen that n n 1X 1 X 6 V ar Xi3 = Xi n i=1 n i=1 6 n σ1 + σ26 15 X 6 σi → 15 = 210 , = n i=1 2

V ar {B1n } =

because Xi6 = 15σi6 by Wick’s Theorem / Diagram Formula Eq. (3) (which in this case must include the

On the linear term correction for needlets/wavelets NG estimators so-called flat edges). On the other hand V ar {B3n } =

1 n

1 = n

n X i=1 n X i=1

The difference between the variance with mean subtraction and the linear term is given by

V ar Xi3 − 3σi2 Xi

V ar {B2n } − V ar {B3n } →

9 9 − σ14 (σ22 − σ12 ) + σ24 (σ22 − σ12 ) 8 8 9 2 = (σ2 − σ12 )2 (σ22 + σ12 ) = 18 . 8

6 Xi − 6σi2 Xi4 + 9σi4 Xi2

n 1 X 15σi6 − 18σi6 + 9σi6 = n i=1 6 n σ1 + σ26 15 X 6 σ →6 = 84 . = n i=1 i 2

We see thus that the linear terms is indeed notnegligible (V ar 3σi2 Xi = 9σi6 , as compared to V ar Xi3 = 15σi6 ), and, in view of its negative correlation with the cubic statistics, it induces a major decrease in the variance. Concerning mean subtraction, simple computations show that n 3 1 X √ Xi − X n n i=1 n n n X X X 3 1 1 X3 − √ X2 Xj = √ n i=1 i n n i=1 i j=1 n

n

3 X 1 X 3 2 Xi X n − √ +√ X . n i=1 n i=1 n

Now the third and fourth term are easily seen to be converge to zero, from the law of large numbers. For the second summand, switching sums we obtain n n n n 3 X 1 X 2 3 1 X 2 X √ X Xj = √ Xi X , n n i=1 i j=1 n i=1 n j=1 j where, again by the law of large numbers we have the convergence (with probability one) n X σ 2 + σ22 1 Xj2 → 1 . n j=1 2 Hence, neglecting terms of order n−1 we have 3 n 1 X σ 2 + σ22 B2n ≃ √ Xi3 − 3 1 Xi ; n i=1 2

Using the Diagram Formula of Eq. (3) again, after some manipulations one obtains σ14 + σ24 + 2σ12 σ22 8 4 4 2 2 σ + σ2 + 2σ1 σ2 +9σ22 × 1 8 6 6 2 4 33σ1 + 33σ2 − 9σ1 σ2 − 9σ22 σ14 , 8 so that, for σ12 = 1, σ22 = 3 we have V ar {B2n } → 9σ12 ×

33 + 33 × 27 − 81 − 27 V ar {B2n } → = 102 . 8

7

Remark 3 Clearly the approximation of the linear term improves when the anisotropy decreases; in fact, the difference V ar {B2n } − V ar {B3n } → 0 as σ22 → σ12 . For instance, taking σ12 = 1, σ22 = 2 we have 6 σ1 + σ26 135 V ar {B1n } → 15 = = 67.5 , 2 2 6 σ1 + σ26 = 27 , V ar {B3n } → 6 2 V ar {B2n } − V ar {B3n } =

9 2 27 (σ2 − σ12 )2 (σ22 + σ12 ) = , 8 8

whence V ar {B2n } → 6

σ16 + σ26 2

= 30.375 .

3. APPLICATION TO W M AP DATA

3.1. The data In order to test the effect of the linear term correction we applied the needlet and SMHW estimators to the foreground reduced V + W maps of the W M AP 7-year data. For simplicity we coadded the two frequency band maps with a constant noise weight. We analyzed the maps at HEALPix resolution Nside = 512 and masking out galactic foregrounds and point sources with the extended temperature analysis mask, known as KQ75 (Gold et al. 2011). Where Gaussian simulations are necessary we used the parameters from the W M AP 7+BAO+H0 cosmological data to simulate the CMB sky (Komatsu et al. 2011), then applying the beam and noise properties supplied by the W M AP team. 3.2. The needlets/wavelets fNL estimator with the linear term The fNL estimator based on the needlets bispectrum has been developed and applied to W M AP data in in Rudjord et al. (2009, 2010). We recall the main features of this estimator. The needlets bispectrum can be expressed as X βj k βj k βj k 1 2 3 , (48) Ij1 j2 j3 = σj1 k σj2 k σj3 k k

where βjk is the needlet coefficient at scale j and direction, i.e. pixel, k, defined in Eq. (9), and σjk is the expected standard deviation of βjk . The needlets bispectrum is used to estimate fNL by a χ2 minimization procedure: χ2 (fNL ) = dT (fNL )C−1 d(fNL )

(49)

where the data vector is − fNL hIˆj1 j2 j3 i. d = Ijobs 1 j2 j3

(50)

8

Donzelli et al.

Here I obs is the bispectrum of the observed data and hIˆj1 j2 j3 i is the average first-order non-Gaussian bispectrum obtained from non-Gaussian simulations. In this analysis we used simulations with local-type nonGaussianity generated with the algorithm described in Elsner and Wandelt (2009). The covariance matrix C is obtained by means of Monte Carlo simulations: Cij = hdi dj i − hdi ihdj i.

(51)

Differentiating Eq. (49) yields the estimate fNL =

hIˆj1 j2 j3 iT C−1 Ijobs 1 j2 j3 T −1 ˆ ˆ hIj1 j2 j3 i C hIj1 j2 j3 i

(52)

Details of the estimation procedure can be found in Rudjord et al. (2009). We want now to adopt the linear term correction in order to decrease the variance of the estimator (52). Following Eq. (32), it is straightforward to subtract the linear term from the bispectrum in Eq. (48). The needlet in the data vector (50) can be exbispectrum Ijobs 1 j2 j3 pressed now as = Ijobs 1 j2 j3

X 1 {βj1 k βj2 k βj3 k σj1 σj2 σj3 k

− Γj1 j2 (k)βj3 k − Γj1 j3 (k)βj2 k − Γj2 j3 (k)βj1 k1 }, (53)

where

Γj1 j2 (k) = hβj1 k βj2 k i ,

(54)

and similarly for the SMHW where the βjk are replaced by the w(R, nk ). We note that the contribution of the linear term to the average hIˆj1 j2 j3 i in the data vector (50) is vanishing, as the expected linear term value is always zero (Eq. (35)). The terms Γj1 j2 (k) contains one contribution from the CMB and one from the noise. The former is calculated using Monte Carlo simulations, the latter is calculated analytically. We implemented slightly different algorithms of the fNL estimator described above. The cubic statistic in Eqs. (48, 53) has been obtained in three cases, namely with standard needlets, Mexican needlets and SMHW coefficients. 3.3. Analysis of the W M AP data In order to test the effect of the linear term correction, we applied the fNL estimator to the W M AP data described in Section 3.1, both before and after the addition of the linear term. The standard needlets coefficients has been obtained with two different bases: B = 1.781 with scales j = 1 − 13 and B = 1.34 with j = 3 − 22. For the Mexican needlets we chose B = 1.34 and p = 1 with j = 3 − 22. In all the cases in addition to the needlet scales we considered a “scale0”, i.e. the original map not convolved with the needlets. The scales selected for the SMHW are the same scales used in Curto et al. (2011a,b): R0 = 0 (the uncolvolved map), R1 = 2.9, R2 = 4.5, R3 = 6.9, R4 = 10.6, R5 = 16.3, R6 = 24.9, R7 = 38.3, R8 = 58.7, R9 = 90.1, R10 = 138.3, R11 = 212.3, R12 = 325.8, R13 = 500., R14 = 767.3 arc minutes. The SMHW coefficients have been analyzed in two ways: as they are or after subtracting the scaleby-scale average of the coefficients outside the applied

mask. Thereafter we tested the same “mean subtraction” procedure with standard and Mexican needlets. All the analysis are performed up to the multipole ℓ = 1500. With the exception of one case – where we analyzed the TABLE 1 Masked fraction of the sky with the extended KQ75 masks SMHW scale Ri % 0 (map) 29.4 1 29.4 2 29.4 3 29.4 4 29.4 5 29.4 6 29.4 7 29.6 8 30.6 9 33.3 10 37.8 11 44.5 12 54.0 13 67.4 14 83.4

Mexican needlets scale j % 1 66.2 2 60.9 3 55.8 4 51.0 5 45.9 6 41.8 7 39.1 8 36.9 9 35.3 10 34.2 11 33.4 12 32.7 13 32.3 14 32.0 15 31.6 16 31.3 17 31.0 18 30.7 19 30.4 20 30.2 21 30.1 22 30.0 scale0 29.4

full-sky with standard needlets B = 1.781 – we always applied the KQ75 mask. For the SMHW analysis it is necessary to properly extend the mask at each scale, because the pixels near the border are affected by the zero values of the cut. Given the similarity with the SMHW, for comparison we extended the mask also for one case with the Mexican needlets. Details of the mask extension procedure can be found in McEwen et al. (2005). The fractions of masked sky obtained with the extended masks at each scale are showed in Table 1. For each analyzed case we simulated a set of 51000 Gaussian maps with the same beam and noise properties of the W M AP coadded V + W cleaned map. 40800 simulations have been used to obtain the covariance matrix in Eq. (51). The standard deviation of the remaining 10200 simulations gives the error bar associated with the estimated fNL values. Moreover, in order to verify that the fNL estimator is unbiased, for each case we also analyzed a set of 700 non-Gaussian simulations with an input fNL = 30 (Elsner and Wandelt 2009). All the results are reported in Table 2. 3.4. Results The addition of the linear term achieves a decrease in the standard deviation in all the cases. The fullsky analysis shows only a little improvement of the error bar - 0.5% - indicating that the mask gives the major contribution to the correction operated by the linear term. Applying the KQ75 mask, all the fNL values estimated on the V + W foreground reduced maps

On the linear term correction for needlets/wavelets NG estimators TABLE 2 Results case

linear term

B = 1.781 fullsky B = 1.781 KQ75 B = 1.34 KQ75

no yes no yes no yes

standard needlets: // 18.4 // 18.3 63.5 25.4 41.5 22.2 43.6 24.9 39.3 22.1

B = 1.34, p = 1 ext. KQ75 B = 1.34, p = 1 KQ75

no yes no yes

Mexican needlets: 37.8 25.4 26.6 22.2 39.2 23.2 37.5 21.8

no mean subtr. ext. KQ75 mean subtraction ext. KQ75

no yes no yes

W M AP fNL a

SMHW: 77.8 33.1 37.5 34.4

Error Bar 1σb

27.3 21.9 22.3 22.0

∆σ %c

0.5 12.6 11.2

NG sims. hfNL i d

29.2 29.2 29.9 30.0 29.3 29.5

6.0

29.5 30.3 29.9 29.8

20.1 18.3e 1.3

29.7 30.1 29.5 29.7

12.6

needlets - mean subtraction: standard B = 1.34 KQ75 Mexican B = 1.34, p = 1 KQ75

no yes

33.0 37.1

22.5 22.3

9.6e 0.8

29.5 29.5

no yes

33.2 37.6

22.1 22.0

4.7e 0.4

30.0 29.8

(a)

local fNL on coadded V + W foreground reduced maps; standard deviation over 10200 Gaussian simulations; [σ(no linear term) - σ(with linear term)]/σ(no linear term); (d) average over 700 simulations with input f NL = 30; (e) comparison with the “no mean subtracted” σ. See the text for further details. (b)

(c)

are consistent with the W M AP 7-year best estimate of fNL = 32 ± 21 (Komatsu et al. 2011). We began the analysis choosing the same needlet base B = 1.781 as in Rudjord et al. (2010). With respect to the previous result - fNL = 73 ± 31 - we already obtained a 18% smaller error bar analyzing the 7-year data in place of the 5-year release and using more scales. But a further improvement of 12.6% is given by the linear term, demonstrating that the correction is non-negligible. The average hfNL i over the 700 non-Gaussian simulations shows that the estimator is unbiased. We can note also that the fNL value estimated with the correction is closer to the W M AP result than without the linear term. We then chose a smaller needlet base B = 1.34, and therefore we constructed the cubic statistic of Eqs. (48, 53) with more needlet scales. We indeed note a smaller standard deviation of 24.9 even without the linear correction with respect to B = 1.781. The error bar is decreased to 22.1, i.e. another 11.2%, adding the linear term. Moving to the Mexican needlets case, we started with a conservative analysis applying an extended KQ75 mask as for the SMHW case. Despite the reduced sky coverage (Table 1) we found the same 1σ error bars obtained with standard needlets - B = 1.781 (but fNL values closer to

9

W M AP ). Motivated by the results of Scodeller et al. (2011) that has shown negligible neighbour bias effects of the mask, we repeated the Mexican needlets analysis with the original KQ75 mask. Looking at the results on the non-Gaussian simulations, we notice that the hfNL i are indeed unbiased. In this case the linear term correction leads to a 6% improvement of the error bar, sufficient to achieve our best estimate fNL = 37.5 ± 21.8. This standard deviation is very close to the W M AP result with the optimal KSW estimator, where σ = 21. With the addition of the linear term correction the needlets fNL estimator is almost optimal. We believe that further standard deviation reductions can be achieved with an optimal V + W coadding and with a different choice of the needlet base B and scales j. Anyway these exploitations are behind the primary scope of this work. Furthermore we consider the case with the SMHW coefficients. The extensions of the KQ75 mask returns sky coverage percentages close to Curto et al. (2011a,b). Without subtracting the scale-by-scale coefficients average, the linear term correction leads to a reduction of the error bar from 27.3 to 21.9, equal to 20.1%. With a very similar SMHW analysis on coadded V + W 7-yr cleaned maps, Curto et al. (2011b) found fNL = 32.5 ± 23 (Fisher matrix bound σF = 22.5), where here we considered the result uncorrected from point-source contribution. Therefore the linear term addition rather improves their result. However the analysis of Curto et al. (2011b) is performed after the scale-by-scale mean subtraction. Following the same procedure, we indeed achieved a reduction of the error bar without the linear term correction: from 27.3 to 22.3, equal to 18.3%. The σ found subtracting the mean is indeed closer to the one of Curto et al. (2011b). Anyway we note that also in this case the addition of the linear term can still slightly improve the error bar (1.3%), and the σ found with this correction - subtracting or not subtracting the mean - is almost identical. Motivated by the SMHW analysis, we tested the scaleby-scale mean subtraction procedure with both the standard and the Mexican needlets. We found that also in the needlets case this procedure well approximates the linear term correction, providing an error bar reduction of 9.6% (i.e. from 24.9 to 22.5) and 4.7% (i.e. from 23.2 to 22.1) for standard and Mexican needlets respectively. But again the addition of linear term provides even smaller error bars: 0.8% and 0.4% respectively. Moreover we obtained the smallest σ with the linear term but without subtracting the mean. We indeed got 22.1 against 22.3 for the standard needlets, and 21.8 against 22.0 for the Mexican needlets. 4. CONCLUSIONS In this paper, we have derived for the first time the linear correction term for wavelets/needlet fNL estimators. As expected, under ideal experimental circumstances (isotropic noise, no masked regions) this term turns out to be identically zero. Under a realistic experimental set-up, the term is non-negligible, although smaller than for the KSW estimators; this is due to the localization properties of wavelets/needlets statistics, which soften to some extent the effects of unobserved regions and anisotropic noise. We have also argued that the linear correction term is well-approximated by scale-byscale mean subtraction, thus providing an explanations

10

Donzelli et al.

for recent results from (Curto et al. 2011a,c), where numerical estimates on the variance of wavelet estimators where shown to be very close to the KSW bound; in fact, we have shown that mean subtraction can cover approximately 85-90% of the linear term effect under W M AP -like circumstances. The procedures we advocate are applied to W M AP 7-year V + W foreground cleaned data, confirming that the corrections achieved are non-negligible. Applying the linear term correction we obtained the best estimate fNL = 37.5 ± 21.8. The error bar is very close to the optimal standard deviation σ = 21 found by the W M AP team (Komatsu et al. 2011). In view of these results, we argue that wavelets/needlets statistics can provide a statistically sound and computa-

tionally convenient technique for non-Gaussianity analysis on CMB data. FKH acknowledges an OYI grant from the Norwegian Research Council; research by DM is supported by the European Research Council grant n.277742 (Pascal ). This research has been partially supported by the ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2 and by the PRIN 2009 project ”La Ricerca di non-Gaussianit` a Primordiale”. Super computers from NOTUR (The Norwegian metacenter for computational science) have been used in this work. We acknowledge the use of the HEALPix software package (G´orski et al. 2005) and the Legacy Archive for Microwave Background Data Analysis (LAMBDA) to retrieve the W M AP data set.

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