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

properties of the matter distribution, although the values of the hierarchical
amplitudes may change arbitrarily. In particular [235],
σ 2 g =b2 1 σ2
Sg,3 =b −1
1 (S3 + 3c2)
Sg,4 =b −2
1 S4 + 12c2S3 + 4c3 + 12c 2
2
Sg,5 =b −3
1 S5 + 20c2S4 + 15c2S 2 3
+60c 3
2
+
30c3 + 120c 2 2
S3 + 5c4 + 60c2c3 +
, (526)
where ck ≡ bk/b1. As pointed out in [235], this framework encompasses the
model of bias as a sharp threshold clipping [360,523,21,615], where δg = 1 for
δ > νσ and δg = 0 otherwise. Although it does not have a series representation
around δ = 0, such a clipping applied to a Gaussian background produces
a hierarchical result with Sg,p = p p−2 in the limit ν ≫ 1, σ ≪ 1. This is
the same result as we obtain from Eq. (526) for an exponential biasing of
a Gaussian matter distribution, δg = exp(αδ/σ), which is equivalent to the
sharp threshold when the threshold is large and fluctuations are weak [21,615].
The exponential bias function has an expansion F =
k(αδ/σ) k /k! and thus
bk = b k 1, independently of α and σ. With Sp = 0, the terms induced in Eq. (526)
by bk alone also give Sg,p = p p−2 .
As a result of Eq. (526), it is clear that for high order correlations, p > 2,
a linear bias assumption cannot be a consistent approximation even at very
large scales, since non-linear biasing can generate higher-order correlations. To
draw any conclusions from the galaxy distribution about matter correlations
of order p, properties of biasing must be included to order p − 1.
Let us make at this stage a general remark. From Eq. (526) it follows that
in the simplest case, when the bias is linear, a value b1 > 1 reduces the
Sp parameters and it may suggest that this changes how the distribution
deviates from a Gaussian (e.g. the galaxy field would be “more Gaussian”
than the underlying density field, given that S3 is smaller). However, this is
obviously an incorrect conclusion, a linear scaling of the density field cannot
alter the degree of non-Gaussianity. The reason is that the actual measure of
non-Gaussianity is encoded not by the hierarchical amplitudes Sp but rather
by the dimensionless skewness B3 = S3σ, kurtosis B4 = S4σ 2 , and so on,
which remain invariant under linear biasing. These dimensionless quantities
are indeed what characterize the probability distribution function, as it clearly
appears in an Edgeworth expansion, Eq. (144).
Since Fourier transforms are effectively a smoothing operation, similar results
to those above hold for Fourier-space statistics at low wavenumbers. In this
regime, the galaxy density power spectrum Pg(k) is given by
Pg(k) = b 2 1 P(k), (527)
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