Astro 321: Problem Set 3

Due Jan. 30

This problem set will begin a series which builds your standard toolkit for structure formation calculations. The motivation of some pieces may seem mysterious for now, but save the codes you build here because you will need them again later in the course.

1 Problem 1: Power Spectra

• Define the power spectrum today as

Δ^{2}(k) ≡ \frac{k^{3}}{2π^{2}} P (k)
= \frac{4}{25}δ_{ζ}^{2}(\frac{k}{H_{0}} )^{4} (\frac{k}{k_{norm}} )^{n−1} T^{2}(k)  (1)

For reference, δ_{ζ} is the amplitude of the initial curvature fluctuations at a normalization scale of k_{norm}. The current best normalization comes from WMAP at a scale of k_{norm} = 0.05 Mpc^{−1} and a value of δ_{ζ} = 5.07×10^{−5} [WMAP’s normalization parameter A = (1.84δ_{ζ} × 10^{4})^{2} and is chosen to make the observed number of order unity]

Here n is the spectral index of the initial density fluctuations and T (k) is the transfer function which defines the linear response (through gravitational perturbation theory) to the initial perturbations. We will see later where that comes from; for now take it to be defined as

T (k(q)) = \frac{L(q)}{L(q) + C(q)q^{2}}
L(q) = ln(e + 1.84q)
C(q) = 14.4 + \frac{325}{1 + 60.5q^{1.11}}  (2)

q scales k to the horizon size at matter radiation equality and is given for historical reasons by

q = \frac{k}{Ω_{m}h^{2}Mpc^{−1}} (T_{CMB}/2.7K)^{2}  T_{CMB} = 2.728K  (3)

• Calculate η(a_{eq}) and show that kη(a_{eq}) has the above scaling, q ∝ kη(a_{eq}) in its scalings with Ω_{m}, h, T_{CMB}. For what value of q is kη(a_{eq}) = 1? At this value of q, what is the value of the transfer function T?

• Write a code in the language of your choice to generate Δ^{2}(k) with k in units of h Mpc^{−1}. Leave Ω_{m}, n, h as adjustable parameters. Notice that value of q with k in h Mpc^{−1} depends on Ω_{m}h, which is usually called the shape parameter “Γ” in the literature.

• For an Ω_{m} = 1/3, h = 1/\sqrt{2}, n = 1 cosmology, plot your result of Δ^{2}(k). For what k in h Mpc^{−1} is Δ^{2} = 1.