The SP → Vlasov–Poisson correspondence (the classical limit)

Why does fuzzy dark matter mimic cold dark matter on large scales? Because in the limit $\hbar/m\to0$ the Schrödinger–Poisson system reduces to the Vlasov–Poisson equations that govern collisionless CDM. This correspondence tells us exactly where FDM and CDM agree and where they must differ.

Two descriptions of gravitating matter

Cold dark matter is a collisionless gas obeying the Vlasov–Poisson equations for the phase-space distribution $f(\mathbf x,\mathbf p,t)$. Fuzzy dark matter is a wave obeying Schrödinger–Poisson. These look completely different — a distribution function versus a complex field — yet they describe the same gravitating matter in overlapping regimes.

The Husimi bridge

The link is a phase-space smoothing (the Husimi/Wigner transform) that turns the wavefunction into an effective distribution function on scales coarser than the de Broglie wavelength. Mocz et al. (2018) proved that as $\hbar/m\to0$ the SP dynamics converges to Vlasov–Poisson: the gravitational potential of the wave approaches the classical collisionless answer.

Agreement on large scales

This is why FDM and CDM give the same cosmic web, the same big halos, the same large-scale power spectrum (figure). Above the de Broglie scale the wave's granular texture averages out and it behaves as a classical collisionless fluid. The correspondence guarantees the agreement — it is not a coincidence.

Same initial conditions, two dark matters: on large scales FDM (right) matches $\Lambda$CDM (left) — the SP$\to$Vlasov correspondence — and differs only on small scales, where quantum pressure smooths structure.

Where they must differ

The correspondence also pinpoints the differences: they live precisely where $\hbar/m$ cannot be neglected — on scales near and below the de Broglie wavelength. There the wave retains order-unity interference fringes and quantum pressure at all resolutions, producing the soliton core and the small-scale cutoff. Smoothed quantities (potentials, large-scale density) converge cheaply; granular quantities (core masses, granule amplitudes) demand that the de Broglie scale be genuinely resolved.

Worked example — a quantum algorithm connection

The classical Vlasov problem is so costly ($O(n_{\rm gr}^6)$ memory for a 6-D phase space) that Miyamoto et al. (2024) proposed a quantum algorithm to solve it. Because FDM's classical limit is this Vlasov dynamics, quantum-computational methods for phase-space cosmology are a long-horizon route to the same physics — the adjacent frontier catalogued in our literature survey (§24.6).

The FDM/$\Lambda$CDM power ratio: unity on large scales (the correspondence), plunging on small scales (quantum pressure).

In our research

This correspondence is the logical backbone of the whole comparison: it is why GADGET-4 (a Vlasov/N-body CDM code) is a valid control for our wave codes, and why our figures show FDM and CDM identical at high mass and divergent below the cutoff. It also frames the practical challenge (Topic 9.5): smoothed quantities converge easily, granular ones do not.

Key references

Mocz et al. (2018), SP–VP correspondence, Phys. Rev. D 97, 083519 (arXiv:1801.03507).
Widrow & Kaiser (1993), Using the Schrödinger equation to simulate CDM, ApJ 416, L71.
Miyamoto et al. (2024), Quantum algorithm for the Vlasov simulation, Phys. Rev. Research 6, 013200 (arXiv:2310.01832).

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