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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 $8$)?

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    $\begingroup$ 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. $\endgroup$ – pela Feb 17 '19 at 21:14
  • $\begingroup$ 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. $\endgroup$ – iron2man Feb 18 '19 at 16:14
  • $\begingroup$ Astronomy, and physics in general, is full of conventions that stick for historical reasons 🙄 $\endgroup$ – pela Feb 19 '19 at 17:20
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The specific radius of $8 h^{−1} \text{Mpc}$ is used because the value of $\sigma_8$ turned out to be close to unity. To quote e.g. from Amendola & Tsujikawa (2010), p. 39:

If the cells have a radius of $8 h^{−1} \text{Mpc}$, it turns out that $\sigma_R$ is close to unity. Conventionally the normalization of the power spectrum is therefore given by quoting $\sigma_8$.

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  • $\begingroup$ That seems plausible! +1 $\endgroup$ – pela Mar 19 '19 at 15:15

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