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Deep-dive physics

Study — the physics behind the campaign

A ground-up tutorial series covering everything needed to understand our fuzzy-dark-matter research — from the expanding universe to solitons, halo mass functions, wave simulations, and the observational tension. All 55 articles published, each with typeset math, figures, worked examples, and a tie-in to what the campaign measured.

1. The expanding universe & ΛCDM

1.1 The Friedmann universe: scale factor, redshift & the cosmic budget
1.2 The ΛCDM concordance model and its parameters
1.3 Linear perturbations and the growth of structure
1.4 Two-point statistics: correlation function & power spectrum
1.5 Cosmic epochs relevant to our work (z=127→0)

2. Dark matter: evidence, problems & candidates

2.1 The evidence for dark matter
2.2 Cold dark matter and its small-scale problems
2.3 The candidate map: WIMPs, warm, axion, fuzzy
2.4 Ultralight bosons: the axiverse & misalignment
2.5 Why ‘fuzzy’: the de Broglie wavelength as a galactic scale

3. The quantum mechanics you need

3.1 Wavefunction, Schrödinger equation & probability current
3.2 Free particles, wave packets & dispersion
3.3 The uncertainty principle and quantum pressure
3.4 Bose–Einstein condensates & the Gross–Pitaevskii equation
3.5 The Madelung (hydrodynamic) formulation

4. The Schrödinger–Poisson system

4.1 From a classical scalar field to Schrödinger–Poisson
4.2 The coupled equations: wave + self-gravity
4.3 Scaling symmetry and dimensionless units
4.4 The quantum Jeans scale and instability
4.5 The SP → Vlasov–Poisson correspondence

5. Solitons: the cored ground state

5.1 The self-gravitating ground state
5.2 The Schive soliton profile and its derivation
5.3 The M⁴ scaling and the soliton family
5.4 A soliton inside a halo: core + envelope
5.5 Computing a soliton: imaginary-time relaxation

6. Linear growth, P(k) & the FDM cutoff

6.1 Growth of perturbations and the growth factor
6.2 The transfer function
6.3 The FDM cutoff: Hu–Barkana–Gruzinov & the half-mode
6.4 σ(M), σ₈ and window functions
6.5 Boltzmann codes: CAMB and axionCAMB

7. The halo mass function

7.1 Spherical collapse and the critical overdensity
7.2 Press–Schechter: the barrier-crossing argument
7.3 Excursion set and Sheth–Tormen
7.4 The FDM mass function: suppression below the cutoff
7.5 Measuring the HMF: halo finders (FOF vs SO)

8. Halo structure: NFW, cusps, cores & core–halo

8.1 The NFW profile
8.2 The concentration–mass relation
8.3 Cusp vs core: the observational problem
8.4 The FDM halo: soliton core + NFW envelope
8.5 The core–halo relation (β≈⅓) and its debate

9. Simulating dark matter

9.1 N-body simulation and GADGET-4
9.2 Initial conditions: from P(k) to particles
9.3 Wave solvers I: pseudo-spectral (JAXiON)
9.4 Wave solvers II: adaptive mesh (GAMER/ELBDM)
9.5 The de Broglie wall and the fluid shortcut
9.6 Differentiable simulation & gradient-based inference

10. Observations & the boson-mass tension

10.1 Dwarf-galaxy cores: a light-boson signal
10.2 The Lyman-α forest: a heavy-boson bound
10.3 Substructure, satellites & stellar streams
10.4 The mass tension: combining the probes
10.5 Statistical inference: likelihoods, SBI & calibration

11. Frontier methods

11.1 Quantum algorithms for cosmological simulation
11.2 Analog gravity: cold-atom simulation
11.3 Hydrodynamics + FDM; mixed dark matter
11.4 Open problems & our contribution