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5.2. CMB bispectrum from active models

Different cosmological models differ in their predictions for the statistical distribution of the anisotropies beyond the power spectrum. Future MAP and Planck satellite missions will provide high-precision data allowing definite estimates of non-Gaussian signals in the CMB. It is therefore important to know precisely which are the predictions of all candidate models for the statistical quantities that will be extracted from the new data and identify their specific signatures.

Of the available non-Gaussian statistics, the CMB bispectrum, or the three-point function of Fourier components of the temperature anisotropy, has been perhaps the one best studied in the literature [Gangui & Martin, 2000a]. There are a few cases where the bispectrum may be deduced analytically from the underlying model. The bispectrum can be estimated from simulated CMB sky maps; however, computing a large number of full-sky maps resulting from defects is a much more demanding task. Recently, a precise numerical code to compute it, not using CMB maps and similar to the CMBFAST code (17) for the power spectrum, was developed in [Gangui, Pogosian & Winitzki 2001b]. What follows below is an account of this work.

In a few words, given a suitable model, one can generate a statistical ensemble of realizations of defect matter perturbations. We used a modified Boltzmann code based on CMBFAST to compute the effect of these perturbations on the CMB and found the bispectrum estimator for a given realization of sources. We then performed statistical averaging over the ensemble of realizations to compute the expected CMB bispectrum. (The CMB power spectrum was also obtained as a byproduct.) As a first application, we then computed the expected CMB bispectrum from a model of simulated string networks first introduced by Albrecht et al. [1997] and further developed in [Pogosian & Vachaspati, 1999] and in [Gangui, Pogosian & Winitzki 2001].

We assume that, given a model of active perturbations, such as a string simulation, we can calculate the energy-momentum tensor Tµnu(x, tau) for a particular realization of the sources in a finite spatial volume V0. Here, x is a 3-dimensional coordinate and tau is the cosmic time. Many simulations are run to obtain an ensemble of random realizations of sources with statistical properties appropriate for the given model. The spatial Fourier decomposition of Tµnu can be written as

Equation 85 (85)

where k are discrete. If V0 is sufficiently large we can approximate the summation by the integral

Equation 86 (86)

and the corresponding inverse Fourier transform will be

Equation 87 (87)

Of course, the final results, such as the CMB power spectrum or bispectrum, do not depend on the choice of V0. To ensure this independence, we shall keep V0 in all expressions where it appears below.

It is conventional to expand the temperature fluctuations over the basis of spherical harmonics,

Equation 88 (88)

where nhat is a unit vector. The coefficients alm can be decomposed into Fourier modes,

Equation 89 (89)

Given the sources Thetaµnu(k, tau), the quantities Deltal(k) are found by solving linearized Einstein-Boltzmann equations and integrating along the line of sight, using a code similar to CMBFAST [Seljak & Zaldarriaga, 1996]. This standard procedure can be written symbolically as the action of a linear operator Bhatlµnu(k) on the source energy-momentum tensor, Deltal(k) = Bhatlµnu(k) Thetaµnu(k, tau), so the third moment of Deltal(k) is linearly related to the three-point correlator of Thetaµnu(k, tau). Below we consider the quantities Deltal(k), corresponding to a set of realizations of active sources, as given. The numerical procedure for computing Deltal(k) was developed in [Albrecht et al. 1997] and in [Pogosian & Vachaspati, 1999].

The third moment of alm, namely <al1 m1 al2 m2 al3 m3> , can be expressed as

Equation 90 (90)

A straightforward numerical evaluation of Eq. (90) from given sources Deltal(k) is prohibitively difficult, because it involves too many integrations of oscillating functions. However, we shall be able to reduce the computation to integrations over scalars [a similar method was employed in Komatsu & Spergel, 2001 and in Wang & Kamionkowski, 2000]. Due to homogeneity, the 3-point function vanishes unless the triangle constraint is satisfied,

Equation 91 (91)

We may write

Equation 92 (92)

where the three-point function Pl1l2l3(k1, k2, k3) is defined only for values of ki that satisfy Eq. (91). Given the scalar values k1, k2, k3, there is a unique (up to an overall rotation) triplet of directions bold k hati for which the RHS of Eq. (92) does not vanish. The quantity Pl1l2l3(k1, k2, k3) is invariant under an overall rotation of all three vectors ki and therefore may be equivalently represented by a function of scalar values k1, k2, k3, while preserving all angular information. Hence, we can rewrite Eq. (92) as

Equation 93 (93)

Then, using the simulation volume V0 explicitly, we have

Equation 94 (94)

Given an arbitrary direction bold k hat1 and the magnitudes k1, k2 and k3, the directions bold k hat2 and bold k hat3 are specified up to overall rotations by the triangle constraint. Therefore, both sides of Eq. (94) are functions of scalar ki only. The expression on the RHS of Eq. (94) is evaluated numerically by averaging over different realizations of the sources and over permissible directions bold k hati; below we shall give more details of the procedure.

Substituting Eqs. (93) and (94) into (90), Fourier transforming the Dirac delta and using the Rayleigh identity, we can perform all angular integrations analytically and obtain a compact form for the third moment,

Equation 95 (95)

where, denoting the Wigner 3j-symbol by l1 l2 l3, we have

Equation 96 (96)

and where we have defined the auxiliary quantities bl1l2l3 using spherical Bessel functions jl,

Equation 97 (97)

The volume factor V03 contained in this expression is correct: as shown in the next section, each term Deltal includes a factor V0-2/3, while the average quantity Pl1l2l3(k1, k2, k3) propto V0-3 [cf. Eq. (94)], so that the arbitrary volume V0 of the simulation cancels.

Our proposed numerical procedure therefore consists of computing the RHS of Eq. (95) by evaluating the necessary integrals. For fixed {l1l2l3} , computation of the quantities bl1l2l3(r) is a triple integral over scalar ki defined by Eq. (97); it is followed by a fourth scalar integral over r [Eq. (95)]. We also need to average over many realizations of sources to obtain Pl1l2l3 (k1,k2,k3) . It was not feasible for us to precompute the values Pl1l2l3 (k1, k2, k3) on a grid before integration because of the large volume of data: for each set {l1l2l3} the grid must contain ~ 103 points for each ki. Instead, we precompute Deltal (k) from one realization of sources and evaluate the RHS of Eq. (94) on that data as an estimator of Pl1l2l3 (k1,k2,k3) , averaging over allowed directions of bold k
hati. The result is used for integration in Eq. (97).

Because of isotropy and since the allowed sets of directions bold k hati are planar, it is enough to restrict the numerical calculation to directions bold k hati within a fixed two-dimensional plane. This significantly reduces the amount of computations and data storage, since Deltal (k) only needs to be stored on a two-dimensional grid of k.

In estimating Pl1l2l3 (k1, k2, k3) from Eq. (94), averaging over directions of bold k hati plays a similar role to ensemble averaging over source realizations. Therefore if the number of directions is large enough (we used 720 for cosmic strings), only a moderate number of different source realizations is needed. The main numerical difficulty is the highly oscillating nature of the function bl1l2l3(r). The calculation of the bispectrum for cosmic strings presented in the next Section requires about 20 days of a single-CPU workstation time per realization.

We note that this method is specific for the bispectrum and cannot be applied to compute higher-order correlations. The reason is that higher-order correlations involve configurations of vectors ki that are not described by scalar values ki and not restricted to a plane. For instance, a computation of a 4-point function would involve integration of highly oscillating functions over four vectors ki which is computationally infeasible.

From Eq. (95) we derive the CMB angular bispectrum curlyCl1l2l3, defined as [Gangui & Martin, 2000b]

Equation 98 (98)

The presence of the 3 j-symbol guarantees that the third moment vanishes unless m1 + m2 + m3 = 0 and the li indices satisfy the triangle rule |li - lj| leq lk leq li + lj. Invariance under spatial inversions of the three-point correlation function implies the additional `selection rule' l1 + l2 + l3 = even, in order for the third moment not to vanish. Finally, from this last relation and using standard properties of the 3 j-symbols, it follows that the angular bispectrum curlyCl1l2l3 is left unchanged under any arbitrary permutation of the indices li.

In what follows we will restrict our calculations to the angular bispectrum Cl1l2l3 in the `diagonal' case, i.e. l1 = l2 = l3 = l. This is a representative case and, in fact, the one most frequently considered in the literature. Plots of the power spectrum are usually done in terms of l(l + 1)Cl which, apart from constant factors, is the contribution to the mean squared anisotropy of temperature fluctuations per unit logarithmic interval of l. In full analogy with this, the relevant quantity to work with in the case of the bispectrum is

Equation 99 (99)

For large values of the multipole index l, Glll propto l3/2 Clll. Note also what happens with the 3 j-symbols appearing in the definition of the coefficients curlyHl1l2l3m1m2m3: the symbol l1 l2 l3 is absent from the definition of Cl1l2l3, while in Eq. (99) the symbol l l l is squared. Hence, there are no remnant oscillations due to the alternating sign of l l l.

However, even more important than the value of Clll itself is the relation between the bispectrum and the cosmic variance associated with it. In fact, it is their comparison that tells us about the observability `in principle' of the non-Gaussian signal. The cosmic variance constitutes a theoretical uncertainty for all observable quantities and comes about due to the fact of having just one realization of the stochastic process, in our case, the CMB sky [Scaramella & Vittorio, 1991].

The way to proceed is to employ an estimator Chatl1l2l3 for the bispectrum and compute the variance from it. By choosing an unbiased estimator we ensure it satisfies Cl1l2l3 = <Chatl1l2l3>. However, this condition does not isolate a unique estimator. The proper way to select the best unbiased estimator is to compute the variances of all candidates and choose the one with the smallest value. The estimator with this property was computed in [Gangui & Martin, 2000b] and is

Equation 100 (100)

The variance of this estimator, assuming a mildly non-Gaussian distribution, can be expressed in terms of the angular power spectrum Cl as follows

Equation 101 (101)

The theoretical signal-to-noise ratio for the bispectrum is then given by

Equation 102 (102)

In turn, for the diagonal case l1 = l2 = l3 = l we have

Equation 103 (103)

Incorporating all the specifics of the particular experiment, such as sky coverage, angular resolution, etc., will allow us to give an estimate of the particular non-Gaussian signature associated with a given active source and, if observable, indicate the appropriate range of multipole l's where it is best to look for it.

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