PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

7.1. FORMALISM
and the LSS studies.
7.1 Formalism
The post reionization matter overdensity field ∆(x) in Fourier space (∆ k ) can be related
to the primordial gravitational potential (Φ P ) on sub-Hubble scales as follows:
The function M(k,z) is given by [cf. Eq. (1.30)]
∆ k (z) = M(k,z)Φ P k . (7.2)
M(k,z) = − 3 5
k 2 T(k)
Ω m H 2 0
D + (z), (7.3)
where, as we had discussed, T(k) denotes the matter transfer function, while D + (z) represents
the growth factor associated with the density fluctuations. In our analysis below,
we shall make use of the conventional Bardeen-Bond-Kaiser-Szalay (BBKS) transfer
function [141] and the cosmological parameters obtained from an MCMC analysis of the
WMAP-7 data [85]. Actually, to obtain a more accurate result, a transfer function including
the baryonic acoustic oscillations should be made use of. But, since our main motivation
here is to arrive at bounds on the bi-spectrum and not to calculate exact numbers,
the approximate BBKS transfer function proves to be sufficient. The power spectrum of
the density field is defined as
〈∆ k ∆ k ′〉 = P(k)δ (3) (k+k ′ ). (7.4)
Clearly, the linear power spectrum of the density field is given byP(k) = M 2 (k,z)P P Φ (k),
wherePΦ P denotes the primordial power spectrum of the gravitational potential such that
PΦ P = P Φ G
+ O(f 2 ). [Note that the matter power spectrum P(k) above is the same as
NL
the quantity P M
(k) defined in Eq. (1.30). We shall drop the subscript in this chapter for
convenience.] The power spectrum P ΦG of the Gaussian field Φ G
shall be assumed to be
featureless and scale invariant.
Following the power spectrum which is the two point correlator of the density field,
the n point correlators can be defined as
〈∆ k1 ∆ k1 ...∆ kn 〉 =
and, using the definition above, we define the bi-spectrum as
n∏
M(k i )〈Φ P k 1
Φ P k 2
...Φ P k n
〉 (7.5)
i=1
〈∆ k1 ∆ k2 ∆ k3 〉 = B(k 1 ,k 2 ,k 3 )δ (3) (k 1 +k 2 +k 3 ). (7.6)
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