Large-Scale Structure of the Universe and Cosmological ...

Fig. 46. Projected leading order PT predictions (solid curves) and N-body results
(points with sampling errors) for the angular 3-point amplitude q3(α) at fixed
θ12 = θ13 = 2 deg for a survey with the APM selection function. N-body results
correspond to the average and variance of 5 realizations of the APM-like model
(top) and the SCDM model (bottom). The dashed lines show the corresponding PT
predictions for r12 = r13 = 15 Mpc/h projected with the hierarchical model.
model does not work well, as discussed above.
In the weakly nonlinear regime the third moment of smoothed angular fluctuations,
defined in (579), can be explicitly written in terms of the power
spectrum using PT. It is given by,
ω3 =6(2π) 2
+ 1
2
dχ χ 6 ψ 3
6
(χ)
7
k dk W 2 2D (k D θ) P(k)
kdkW 2 2 2D(k D θ) P(k)
k 2 dk Dθ W2D(k D θ) W ′
2D (k D θ) P(k)
(586)
where W ′ 2D is the derivative of the top -hat window W2D defined in Eq. (582).
Therefore, in case of a power-law spectrum P(k) ∼ k n , we have [48],
s3 = r3
36
7
3
− (n + 2) , (587)
2
with r3 given in general by Eq. (585). The coefficient r3 is found in practice
to be of order unity and to be very weakly dependent on the adopted shape
for the selection function.
It is worth to note that the hierarchical model in Sect. 7.2.3 yields a different
prediction for s3 than the above tree-level value. In the hierarchical case,
s3 ≃ r3S3 ([249,250]) with S3 = 34/7 − (n + 3). For example, for n ≃ −1,
the hierarchical model yields s3 ≃ 3.43 while the tree level prediction yields:
202