Abundance Matching More Thoughts

I have been thinking a bit about the plot below—why does the SMF not quite match at the high mass end?

Thinking about this more, this is because once scatter is added to the $v_{\rm peak}$ values, the HVF of the simulation data changes. This is not what I want.

Therefore, as an update to the method, rather than match the scattered $v_{\rm peak}$ values to the galaxy mass exactly (by solving $\int_x^\infty n(x') dx' = \int_{M_*}^\infty \phi (M_*') dM_*’$), I instead randomly generate $M_{\rm gal}$ values from the SMF, and then rank order the resulting masses—this ensures the SMF is preserved.

This requires setting a minimum mass in the SMF: Therefore, I only consider $v_{\rm peak}$ values above $2.1 \, \rm{dex}$ (this value was chosen, since for the $512^3$ simulation the HVF looks incomplete below this limit). Then, solving $\int_x^\infty n(x') dx' = \int_{M_*}^\infty \phi (M_*') dM_*’$, this sets the minimum galaxy mass to be $9.79$.

Current Abundance Matching Implementation

BASIC ABUNDANCE MATCHING PROCEDURE

1. Get the minimum galaxy mass (corresponding to the minimum $v_{\rm peak}$ value)

2. Randomly generate $N$ $M_{\rm gal}$ values so that they match the SMF (where $N$ is the number of halos with $v_{\rm peak}$ above the minimum value)

3. Rank-order the galaxy masses, and assign them to each halo (so that the halo with the largest $v_{\rm peak}$ has the largest galaxy masses)

GENERATE A MAPPING IN THE SCATTER

1. Randomly generate $10^7$ $v_{\rm peak}$ values from the HVF function (I am using a minimum $v_{\rm peak}$ value of 1.5 dex for this step)

2. Loop through different fixed scatter values in $v_{\rm peak}$: I am using 100 values linearly spaced between 0 and 0.5.

3. For a given scatter value, add guassian errors to the $v_{\rm peak}$ values

4. Perform abundance matching (as outlined above) using these scattered values

5. Bin the resulting masses, and measure the scatter in $M_{\rm gal}$ as a function of $v_{\rm peak}$

6. Repeat steps 3-5 for each scatter value

ABUNANCE MATCHING WITH SCATTER

1. Choose a constant scatter in $M_{\rm gal}$

2. For each vpeak value use the mapping from above to get the corresponding scatter in $v_{\rm peak}$

3. Add scatter to the $v_{\rm peak}$ values, and use these scattered values to perform the abundance matching step

Results

Here is the updated plot:

The SMF looks better, but the stellar mass–$M_{\rm peak}$ relation looks worse… I think this is just a resolution issue though, and I expect it will look better with the 1024 simulations.

---

« Abundance Matching 512 Sims 2

Fit HVF »