Jerusalem WS lecture notes: 03. Structure Formation: From Linear to Non-Linear

Given by Frank van den Bosch, slides here.

• cosmological [over]density field: moments of the PDF
• the ergodic hypothesis: ensemble average is equal to spatial average taken over one realisation of the random field (we only have one Universe, but it consists of many random subvolumes). Basically, spatial correlations decay rapidly w/ separation, so the subvolumes are statistically independent.
• random Gaussian field: distribution of arbitrary set of N points is an N-variate Gaussian. covariance matrix is the 2nd moment: 2-point correlation function. We only need this one moment to specify the N-point probability function.
• 2PCF: 0 for Poisson distribution, != 0 for clustered distribution. Gravity makes things cluster, so we get positive CF for small radii, negative for large radii, then turns towards zero.
• Higher-order CFs: irreducible CFs cannot be obtained from lower order reduced CFs. Connected moments of all higher order CFs are 0 for a Gaussian field: one can use this to test for Gaussianity.
• the matter field in Fourier space: V is volume over which the Universe is assumed to be periodic. k -- modes.
• power spectrum (Z-H)
• Linear perturbation growth: Silk damping: e- and photons are no longer coupled, photons move away, structure starts growing. Jeans length for DM is very low: free streaming. When perturbations become of order unity, growth becomes non-linear.
• Non-linear regime: perturbation theory (and Gaussianity assumption) is no longer valid. Modes couple to each other: no analytic solutions, higher order moments are required for complete specification of the density field: higher-order perturbation theory.
• Numerical simulation, the Halo Model
• Over-simplified model: top-hat spherical collapse. No shell crossing, mass conservation. Evolution depends only on the mass inside the shell. parametric form: implies maximum radius before turnaround. Collapse -- shell crossing, after it mass is not conserved.
• By using linear theory, we can identify regions that had already collapsed.
• Beyond shell crossing: collapse is not spherical, hence BHs don't form. Virialization: halo is in equillibrium as a result. We can compute overdensity of the virialised DM halo: 178: haloes finding at 200.
• Zel'dovich approximation:
  • Lagrangian picture: works in mildly non-linear regime, calculated displacement of particles from the initial conditions.
  • no spherical symmetry is required, mass-conservation is valid until shell crossing.
  • deformation tensor in the initial density field, eigenvalues: positive: contracting coordinate, neg: expansion. No over-simplified geometry, works in qusi-linear regime. Collapse happens first in one direction (largest eigenvalue) --> pancake.
  • ellipsoidal overdensity --> sheet (pancake) -- > filament --> halo (longest axis 'halo formation')
• Relaxation and virialisation:
  • 2 body-relaxation time: time required for a particle to change its E_k by about its initial amount due to 2 body interactions. How can galaxies appear relaxed, if galaxies and haloes have 2body relaxation times larger than t_H. 4 mechanisms:
  • Phase mixing: different phases, same energies: mixing in phase space, goes linearly with time. Completely reversible: no information is lost.
  • chaotic mixing --> the book [MBW]
  • violent relaxation: potential is time-dependent, particles gain or lose energy. Overall, energy distribution is broadened by the process. No segregation by mass, quite opposite to collisional (2-body) relaxation. e.g. stars don't sink to the center. Mixing on coarse-grained level. Self-limiting: no all knowledge of initial conditions is erased.
  • Landau damping --> the book