PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 7. IMPRINTS OF PRIMORDIAL NON-GAUSSIANITY IN THE LY-ALPHA FOREST
Ly-α forest flux fluctuations δ F are given by
P F (k) = b 2 1P(k),
B F123 = b 3 1 B 123 +b 2 1 b 2 [P(k 1 )P(k 2 )+permutations]. (7.15)
The bi-spectrum of Ly-α flux is hence completely modeled using the three parameters
(f NL
,b 1 ,b 2 ).
We shall now set up the Fisher matrix for constrainingf NL
using the Ly-α bi-spectrum.
Following the formulation described in Ref. [136], we shall define the bi-spectrum estimator
as
ˆB F123 = V f
V 123
,
∫k 1
d 3 q 1
∫k 2
d 3 q 2
∫
k 3
d 3 q 3 δ D
(q 123 ) ∆ o F (k 1)∆ o F (k 2)∆ o F (k 3). (7.16)
Here q 123 = q 1 + q 2 + q 3 , and the integrals are performed over the q i -intervals
(k i −δk/2,k i +δk/2). Also, V f = (2π) 3 /V , where V is the survey volume. The survey
volume is given by
V 123 =
∫k 1
d 3 q 1
∫k 2
d 3 q 2
∫
k 3
d 3 q 3 δ D
(q 123 ) = 8π 2 k 1 k 2 k 3 δk 3 . (7.17)
The quantities∆ o F (k i) appearing in the Eq. (7.16) denotes the ‘observed’ Ly-α flux fluctuations
in Fourier space. The observed quantityδ o F (r) is given by the continuous fieldδ F(r)
sampled along skewers corresponding to line of sight to bright quasars. We therefore
have
δ o F (r) = δ F(r)ρ(r), (7.18)
where the sampling window function ρ(r) is defined as
∑
a
ρ(r) = N
w a δ 2(r D ⊥ −r ⊥a )
∑
a w , (7.19)
a
and N is a normalization factor such that ∫ dVρ(r) = 1. The summation extends up to
N Q , the total number of quasar skewers in the field which are assumed to be distributed
with sky locations r ⊥a . The weights w a introduced in ρ(r) are in general related to the
pixel noise and can be chosen with a posteriori criterion of minimizing the variance. The
assumption is, the line of sight direction is continuous and the survey measures the Ly-α
forest in ρ(r) spatial window. In Fourier space, we then have
∆ o F (k) = ˜ρ(k)⊗∆ F(k)+∆ Fnoise (k), (7.20)
where ˜ρ is the Fourier transform of ρ, and∆ Fnoise (k) denotes a possible noise term.
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