The CDM transfer function and power spectrum
The present-day power spectrum of CDM perturbations can be written as
2
P (k) = 2πδH
k ns
T 2 (k),
H0ns+3
where δH is a normalization constant, ns is the spectral index of primordial
perturbations and T (k) is the transfer function. In the days before fast computer programs for calculating the transfer function a popular approximation
to T (k) was the BBKS (Bardeen, Bond, Kaiser and Szalay) formula
T (x) =
i−1/4
ln(1 + 0.171x) h
1 + 0.284x + (1.18x)2 + (0.399x)3 + (0.490x)4
,
0.171x
where x = k/keq and keq = (0.073Ωm0h) h Mpc−1 .
a) Plot T (x) for Ωm0 = 1.0, h = 0.5, and for Ωm0 = 0.3, h = 0.7.
b) Make a log-log plot of the power spectrum for the same two models.
Use ns = 1, and δH = 1.9 × 10−5 for Ωm0 = 1, δH = 4.6 × 10−5 for
Ωm0 = 0.3.
c) How well does T (x) agree with the simple-minded results we derived
in the lectures in the limits x ≪ 1 and x ≫ 1?
A popular measure of the amplitude of the density perturbations is the
RMS overdensity in a sphere of radius R, defined as
σR2 = hδR2 (x)i,
with
δR (x) =
Z
d3 x′ δ(x′ )WR (x − x′ ).
where WR (x) is equal to 1 for x < R and vanishes otherwise. One can show
(you don’t have to!) that
σR2
1
= 2
2π
Z
where
W (x) =
∞
dkk 2 P (k)W 2 (kR),
0
3
(sin x − x cos x).
x3
1
d) Write a program that calculates σ8 , that is σR with R = 8 h−1 Mpc for
the two models you looked at in a) and b).
e) Go to the LAMBDA archive on the internet and look at the tables of
derived cosmological parameters from the WMAP satellite. Which of
the two models agrees best with the values for σ8 you find there?
2