$\sigma_8$ in power-law cosmologies¶

Author: Lehman Garrison (https://lgarrison.github.io)

Update 10/25/2019:¶

Thanks to Daniel Eisenstein for deriving an even simpler expression! The notebook has been updated with this version.

Power-law cosmologies ($P(k) = A k^n$) have an analytic form for $\sigma^2(R)$ with a spherical tophat window $W_T$:

$$\begin{align} \sigma^2(R) &= \int \frac{d^3k}{(2\pi)^3}\tilde W_T^2(kR)P(k) \\ &= A \int_0^\infty \frac{k^2dk}{2\pi^2} \left[\frac{3(\sin(kR) - kR\cos(kR))}{(kR)^3}\right]^2 k^n\\ &= \frac{9 A (n+1) 2^{-n-1} R^{-n-3} \sin \left(\frac{\pi n}{2}\right) \Gamma (n-1)}{\pi^2 (n-3)} \end{align}$$

where we used the Fourier transform $\tilde W_T(kR)$ of the spherical tophat window. The result in the last line (obtained via Mathematica) is only valid for $-3 < n < 1$; otherwise, the variance is undefined.

However, the result is numerically unstable if $n$ is an integer in this range, since the function $\Gamma(n-1)$ in the numerator goes to infinity for 0 and the negative integers.

The $\sigma^2(R)$ function itself is perfectly smooth and finite in this range, as seen below:

For plotting purposes, we avoided exact integers in the $n$ range and used $A=1$.

So, to find a numerically stable variant of the above expression, we need to re-write it to avoid evaluating $\Gamma(n-1)$ at negative integers.

Using some properties of the $\Gamma$ function (https://dlmf.nist.gov/5.5), we can arrive at the following expression:

$$\sigma^2(R) = \frac{9 A R^{-(3+n)}}{2 \pi^{3/2}} \frac{\Gamma[(3+n)/2]}{\Gamma[(2-n)/2] (n-3)(n-1)}$$

It's worth noting that this expression is exact; we have not used any approximations.

Note: a previous version of this notebook had two expressions, one for even poles and one for odd, which are now combined into one expression.