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

Fig. 12. One-loop corrections to the power spectrum of the density field as a function
of spectral index [see Eq. (169)]. Also shown is the one-loop corrections to the velocity
divergence power spectrum, αθ(n). Note that non-linear effects can slow down
the growth of the velocity power spectrum for a broader class of initial conditions
than in case of the density field.
light on aspects of gravitational clustering in the transition to the non-linear
regime.
To characterize the degree of non-linear evolution when including one-loop
corrections, it is convenient to define a physical scale from the linear power
spectrum, the non-linear scale R0, as the scale where the smoothed linear
variance is unity,
σ 2 ℓ (R0)
= d 3 k PL(k, τ) W 2 (kR0) ≡ 1. (167)
For scale-free initial conditions and a Gaussian filter, W(x) = exp(−x2 /2),
Eq. (167) gives R n+3
0 = 2πAa2Γ[(n + 3)/2]. This is related to the non-linear
scale defined from the power spectrum, ∆(knl) = 4πk3 nlP(knl) = 1 by
knlR0 = Γ[(n + 5)/2]. (168)
Figure 12 displays the one-loop correction to the power spectrum in terms of
the function αδ(n) defined by
∆(k) ≡
n+3 2(kR0)
1 + αδ(n) (kR0)
Γ[(n + 3)/2]
n+3
, (169)
which measures the strength of one-loop corrections (and similarly for the
velocity divergence spectrum replacing αδ by αθ). This function has been cal-
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