The FDM mass function: suppression below the cutoff

Assemble the pieces — a cutoff transfer function, a sharp-k window, Sheth–Tormen collapse — and out comes the fuzzy-dark-matter halo mass function: identical to $\Lambda$CDM at high mass, falling off a cliff below the half-mode mass. This is the direct answer to Task 1.

Putting it together

The FDM mass function is the same $dn/d\ln M=(\bar\rho/M)f(\sigma)|d\ln\sigma/d\ln M|$ as CDM, with one change: $\sigma(M)$ is computed from the FDM power spectrum (CDM transfer $\times$ HBG cutoff, §6.3) using a sharp-k window (§6.4). Because $\sigma(M)$ flattens below the cutoff, the Gaussian tail above $\delta_c$ collapses and the abundance plummets (figure).

Reading the figure

Above the half-mode mass, FDM and $\Lambda$CDM are indistinguishable — the same big halos form. Below $M_{1/2}$, the FDM curve turns over and drops steeply: small halos are dramatically rarer, and the deficit deepens toward high redshift (the mass function is even more suppressed early, when small halos would otherwise dominate).

The FDM halo mass function (dashed) vs $\Lambda$CDM (solid) across $z=3$–$20$: identical at high mass, cutting off sharply below the half-mode mass (crimson). A lighter boson cuts off at higher mass.

The characteristic scales

Worked example — the numbers we report

For our two fiducial masses the pipeline gives:

Boson mass	Half-mode mass $M_{1/2}$ [$M_\odot/h$]	$k_{1/2}$ [$h$/Mpc]
$m=8\times10^{-23}$ eV ($m_{22}=0.8$)	$4.85\times10^{10}$	6.14
$m=1\times10^{-22}$ eV ($m_{22}=1.0$)	$3.61\times10^{10}$	6.77

Consistent with our GAMER runs at $m_{22}=0.8$: a minimum halo mass $M_{\min}\approx3\times10^{8}\,M_\odot$ and delayed first collapse ($z_{\rm ff}\approx15.7$ vs $z\approx50$ for CDM).

A caveat, stated honestly

The result is semi-analytic Press–Schechter (the method named), not counted from our boxes — our zoom volumes hold too few halos for HMF statistics. The sharp-k constant ($c\approx2.5$) and the transfer's node-truncation shift the cutoff mass by tens of percent, not its existence or scaling. And it is validated: axionCAMB (§6.5) confirms it to $\sim$1–15%.

Cumulative counts $n(>M)$ and the 2-D $(M,z)$ suppression map — the FDM deficit deepens toward high redshift.

In our research

This is Task 1 — the direct answer to Sandro's first question. The curves come from our validated pipeline (colossus-checked to 0.1%), and our own GADGET-4 and GAMER simulation points overlay them (Topic 7.2, Fig 7-2b): CDM on the curve, FDM suppressed.

Key references

Kulkarni & Ostriker (2022), What is the halo mass function in FDM? (arXiv:2011.02116).
May & Springel (2023), The halo mass function… with FDM, MNRAS 524, 4256 (arXiv:2209.14886).
Hu, Barkana & Gruzinov (2000), Phys. Rev. Lett. 85, 1158.

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