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Paper text:
the inflationary models describe the dynamics of the first instants of the universe .
each model introduces a characteristic signature of non - gaussianity into the anisotropy distribution of the cosmic microwave background ( cmb ) radiation .
the level of non - gaussianity is described by the so - called _
bardeen s potential _
( ( * ? ? ?
* bardeen ( 1980 ) ) ) : @xmath2 + g_{nl}[\phi_l^3(\textbf{x})]\ ] ] where @xmath3 is the gravitational potential field of the universe and @xmath4 is its gaussian part .
the constants @xmath5 and @xmath0 , which parametrize the non - gaussian part , can be measured from the amplitude of the bispectrum and trispectrum of the field , i.e. the harmonic counterpart of the 3-point and 4-point correlation function of the field , respectively .    although there exist several optimal bispectrum estimators that allow to evaluate @xmath5 with high confidence ( see for instance ( * ? ? ?
* komatsu  ( 2005 ) ) or ( * ? ? ?
* planck coll .
( 2013 ) ) ) , the lack of an optimal trispectrum estimator has prevented the evaluation of strong constraints on @xmath0 so far .
spherical needlets ( ( * ? ? ?
* narcowich  ( 2006 ) ) , ( * ? ? ?
* baldi  ( 2009 ) ) ) are a wavelet system on the sphere defined by setting : @xmath6 where @xmath7 , @xmath8 are a set of cubature points and weights on the sphere , @xmath9 is a constant related to the width of the needlet and @xmath10 is a weight function satisfying the three following conditions : a ) compact support , @xmath11 if @xmath12 , 0 otherwise ; b ) partition of unity , for all @xmath13 , @xmath14 ; c ) smoothness , @xmath15 , i.e. , @xmath10 is @xmath16 times continuously differentiable , for some @xmath17 or @xmath18 .
the spherical needlet coefficients are provided by the analytical formula : @xmath19    as a consequence of the partition of unity property , the following well - known reconstruction formula holds : @xmath20      as argued earlier in ( * ? ? ?
* baldi  ( 2009 ) ) , spherical needlets enjoy an excellent localization property in real domain ( fig .
[ fig : needlet_shape ] ) : @xmath21 more explicitly , spherical needlets are then quasi - exponentially localized around any cubature point @xmath22 .
moreover , as extensively argued in the literature , the needlet coefficients evaluated on isotropic random fields are asymptotically uncorrelated , under mild regularity conditions .
it is then possible to derive analytically their statistical properties , and to understand the role and expressions for correction terms under realistic experimental conditions ( noise and masks , see donzelli et al . for the bispectrum case ) +
the sample trispectrum estimator is usually written using the spherical harmonics coefficients as : @xmath23 where @xmath24 is the gaunt integral defined for instance in ( * ? ? ?
* marinucci peccati ( 2011 ) ) .
this estimator is clearly unfeasible under realistic experimental conditions , due to the presence of missing data and anisotropic noise . as a consequence of the previous discussion , it is possible to exploit the needlet coefficients to derive an alternative , computationally feasible and statistically sound estimators of the trispectrum .
more precisely , exploiting wick theorem on higher order moments of gaussian variables we propose the following needlet trispectrum : @xmath25\\      \end{split}\ ] ]    heuristically , the needlet trispectrum is constructed combining quadruples of coefficients , evaluated at the scales of interest , and subtracting linear and quadratic components in order to cancel bias and minimize the variance .
further details , and a software which exploits the needlet trispectrum to evaluate @xmath0 on cmb maps , will be provided in the forthcoming paper ( * ? ? ? * troja   ( 2014 ) ) .
we propose a computationally feasible estimator for the needlet trispectrum , which develops earlier work on the bispectrum by ( * ? ? ? * donzelli  ( 2012 ) ) . 
 our proposal seems to enjoy a number of useful properties , in particular a ) the construction exploits the localization properties of the needlet system , and hence it automatically handles masked regions ; b ) the procedure incorporates a quadratic correction term to correct for the presence of instrumental noise and sky - cuts ; c ) it is possible to provide analytic results on its statistical properties , which can serve as a guidance for simulations . 
 the needlet trispectrum we present here provides the natural building blocks for the efficient estimation of nonlinearity parameters on cmb data , and in particular for the third order constants @xmath0 and @xmath1 .