The transfer function

The transfer function is the fingerprint each type of matter leaves on the primordial spectrum. It is the single object that carries all the dark-matter physics into $P(k)$ — and the one place fuzzy dark matter announces itself.

What the transfer function does

Inflation lays down a nearly scale-invariant primordial spectrum $P_{\rm prim}(k)\propto k^{n_s}$. Between then and now, physics reshapes it scale by scale — radiation pressure, baryon oscillations, free-streaming, quantum pressure. The transfer function $T(k)$ encodes that reshaping:

$$P(k)=A\,k^{n_s}\,T^2(k)\,D^2(a),$$

normalized so $T\to1$ on large scales. Everything about the dark-matter model that matters for structure is in $T(k)$.

The CDM transfer function

For cold dark matter we use the Eisenstein & Hu (1998) closed-form fit, which captures the turnover at matter–radiation equality and the baryon-acoustic wiggles to sub-percent accuracy. On small scales $T_{\rm CDM}\to1$: CDM preserves power everywhere (figure, solid).

The FDM transfer function

Fuzzy dark matter multiplies the CDM transfer by a suppression factor that plunges to zero beyond the quantum-Jeans scale (figure, dashed). This is the only place FDM differs from CDM in the linear mass-function calculation — it carves the small-scale cutoff directly into $P(k)$, and hence into every downstream halo count.

Worked example — reading the suppression

At $k=10\ h/$Mpc the CDM transfer is $\sim1$, but for $m_{22}=0.8$ the FDM transfer squared has fallen below $\sim0.01$ — power suppressed by more than 100$\times$. That two-orders-of-magnitude cut is what removes the small halos: with almost no power at high $k$, $\sigma(M)$ flattens and the mass function collapses below the half-mode mass.

Transfer functions: $\Lambda$CDM (Eisenstein–Hu, solid) stays near 1 at small scales, while the FDM versions (dashed) plunge beyond the boson's Jeans scale — a heavier boson cuts off at larger $k$.

Fitting formula vs first principles

These transfer functions are analytic approximations, but excellent ones. The next article (Topic 6.3) derives the FDM form (Hu–Barkana–Gruzinov), and Topic 6.5 shows we cross-checked it against a full Boltzmann solve (axionCAMB), confirming it to $\sim$few percent.

The transfer function applied: $P(k)=P_{\rm prim}T^2$ — $\Lambda$CDM versus the FDM cutoff.

In our research

The product Eisenstein–Hu $\times$ Hu–Barkana–Gruzinov is our linear input for Task 1 (our in-house cosmology_pk.py, validated to $\le0.1\%$ against Hu+2000). The FDM factor is what makes the dashed curves in our mass-function figures peel away from $\Lambda$CDM.

Key references

Eisenstein & Hu (1998), Baryonic features in the matter transfer function, ApJ 496, 605 (arXiv:astro-ph/9709112).
Hu, Barkana & Gruzinov (2000), Phys. Rev. Lett. 85, 1158 (arXiv:astro-ph/0003365).

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