Fit HVF

I plan on fitting the halo $v_{\rm peak}$ function from the simulations to use in the abundance matching step. Here is my set-up for the $512^3$ simulations (I will have to update this with the higher resolution simulations)

Functional Form

I am following the functional form from Comparat et al 2017 (originally from Rodriguez-Puebla et al. 2016):

$$\log_{10} \left[ \dfrac{v_{\rm peak}^3}{H(z)^3} \dfrac{dn}{d \ln v_{\rm peak} }\right] = \log_{10} \left[ A \left( 1+ \dfrac{v_{\rm peak}}{v_{\rm cut}}\right)^{-\beta} \exp \left[ \left(-\dfrac{v_{\rm peak}}{v_{\rm cut}} \right)^{\alpha} \right] \right]$$

There are four parameters to fit $A$, $v_{\rm peak}$, $\alpha$ and $\beta$. I will use velocity units of km/s and have $H(z)$ in units of km/s/Mpc. The number density, $n$ is in ${\rm Mpc}^{-3}$

As in Comparat et al 2017 I will also consider distinct and satellite halos separately. Each parameter can be expressed as a polynomial in redshift $p(z)=p_0 + p_1 z + ...$

Fit at redshift zero

First, here is the fit for the final snapshot, using all the halos with $\log_{10} v_{\rm peak}>2.2$

Looks like it is working pretty well (clearly it will be better with the higher resolution simulations).

Redshift trends

Next, I will look at how the four parameters vary with redshift… I will need to calculate $v_{\rm peak}$ for the halos in each snapshot. Currently I have a script to do this on a single snapshot, by reading in halo catalogs, and using the merger trees to find $v_{\rm max}(z)$. I will work on automating this to run on all the snapshots.

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