<p>Press-Schechter formalism allows one to universally quantify the number of massive dark matter halos in a given volume -- <strong>The Halo Mass Function</strong>. The density contrast is given by the rms mass variance in Fourier space, which is the dependent on the linear power spectrum <span class="math-container">$P_{\rm lin}(k,z)$</span> in a smooth window <span class="math-container">$\widetilde{W}(R;k)$</span>
<span class="math-container">\begin{align}
\sigma^{2}(R,z) 
= \frac{1}{2\pi^{2}} 
\int_{0}^{\infty} dk\ k^{2} P_{\rm lin}(k,z) \left| \widetilde{W}(k;R)\right|^{2}
\end{align}</span></p>

<p>The halo count can be scaled via observations in a defined window region of <span class="math-container">$R = 8\ h^{-1}\rm Mpc$</span>. We can solve for this analytical/numerically and measure it from CMB fluctuations, given by the parameter <span class="math-container">$\sigma_{8}$</span>. The power spectrum, as well as the halo mass function, is then scaled through the normalization of the analytical variance and the observed variance.</p>

<p>My question is then, <em>why do we choose a smoothing radius of</em> <span class="math-container">$R = 8\ h^{-1}\rm Mpc$</span>? Is it arbitrarily chosen to compute the number variance in a cluster, or is their more too that? What prevents me from defining the window radius of <span class="math-container">$R = 6\ h^{-1}\rm Mpc$</span> (just ignore the fact that the  WMAP and PLANCK results give <span class="math-container">$\sigma_{8}$</span> in the radius of <span class="math-container">$8$</span>)?</p>

CC BY-SA 4.0
2019-02-17T21:14:48.450Z
Mo, Bosch, & White discuss in Cha. 6 various statistical measures that can be used to probe the power spectrum, and write that some are better suited for normalizing the linear power spectrum than $\sigma_8$. But they also write that the amplitude of a power spectrum usually is represented by $\sigma_8$, "largely for historical reasons". My guess is that 6 or 10 would be just as fine, but 0.8 or 80 would be so small/large that the variance would be inconveniently large/small.
pela
CC BY-SA 4.0
2019-02-18T16:14:14.090Z
Thanks for the reply! Right, I was also thought that is is used for some convenience. I also heard before from my undergrad advisor that it is chosen for "historical reasons"... but that doesn't seem like a very fulfilling reasoning in either case.
iron2man
CC BY-SA 4.0
2019-02-19T17:20:19.740Z
Astronomy, and physics in general, is full of conventions that stick for historical reasons 
pela
<cosmology><cosmic-microwave-background>
Motivation for using $\sigma_{8}$?
iron2man
# Motivation for using $\sigma_{8}$?

(Asked by: username_0 on 2019-02-17)

Press-Schechter formalism allows one to universally quantify the number of massive dark matter halos in a given volume -- **The Halo Mass Function**. The density contrast is given by the rms mass variance in Fourier space, which is the dependent on the linear power spectrum $P\_{\rm lin}(k,z)$ in a smooth window $\widetilde{W}(R;k)$
\begin{align}
\sigma^{2}(R,z)
= \frac{1}{2\pi^{2}}
\int\_{0}^{\infty} dk\ k^{2} P\_{\rm lin}(k,z) \left| \widetilde{W}(k;R)\right|^{2}
\end{align}

The halo count can be scaled via observations in a defined window region of $R = 8\ h^{-1}\rm Mpc$. We can solve for this analytical/numerically and measure it from CMB fluctuations, given by the parameter $\sigma\_{8}$. The power spectrum, as well as the halo mass function, is then scaled through the normalization of the analytical variance and the observed variance.

My question is then, *why do we choose a smoothing radius of* $R = 8\ h^{-1}\rm Mpc$? Is it arbitrarily chosen to compute the number variance in a cluster, or is their more too that? What prevents me from defining the window radius of $R = 6\ h^{-1}\rm Mpc$ (just ignore the fact that the WMAP and PLANCK results give $\sigma\_{8}$ in the radius of $8$)?

**Tags:** cosmology, cosmic-microwave-background
> **Comments:**
> **username_1**: Mo, Bosch, & White discuss in Cha. 6 various statistical measures that can be used to probe the power spectrum, and write that some are better suited for normalizing the linear power spectrum than $\sigma_8$. But they also write that the amplitude of a power spectrum usually is represented by $\sigma_8$, "largely for historical reasons". My guess is that 6 or 10 would be just as fine, but 0.8 or 80 would be so small/large that the variance would be inconveniently large/small.  
> **username_0**: Thanks for the reply! Right, I was also thought that is is used for some convenience. I also heard before from my undergrad advisor that it is chosen for "historical reasons"... but that doesn't seem like a very fulfilling reasoning in either case.  
> **username_1**: Astronomy, and physics in general, is full of conventions that stick for historical reasons   

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## 0 answers

