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| Boltzmann equation (cosmology) | |
|---|---|
| Name | Boltzmann equation (cosmology) |
| Field | Statistical mechanics; Cosmology |
| Introduced | 19th century |
| Applications | Cosmic Microwave Background, Big Bang nucleosynthesis, Dark matter |
Boltzmann equation (cosmology) is the kinetic equation that governs the phase-space evolution of particle distribution functions in an expanding Friedmann–Lemaître–Robertson–Walker spacetime, used to describe processes from Big Bang thermodynamics to the formation of the Cosmic Microwave Background anisotropies and the freeze-out of dark matter species. It connects microscopic scattering rates from Quantum Field Theory and particle physics experiments at facilities like CERN to macroscopic observables measured by missions such as Planck (spacecraft) and WMAP. The cosmological Boltzmann framework interfaces with models proposed by groups at institutions like Institute for Advanced Study, Kavli Institute for Cosmology, and collaborations such as Sloan Digital Sky Survey.
The cosmological Boltzmann equation generalizes the classical Ludwig Boltzmann formulation to an expanding background described by Alexander Friedmann and Georges Lemaître solutions, incorporating redshift, Hubble expansion from Edwin Hubble, and general relativistic corrections as in Albert Einstein's framework. It is foundational for interpreting observations by missions including COBE, Planck (spacecraft), WMAP, and upcoming probes like Euclid (spacecraft) and James Webb Space Telescope. Historical developments tie to contributions by Lev Landau, Enrico Fermi, George Gamow, and later formalizations by researchers at Harvard University, Princeton University, and California Institute of Technology. The equation links scattering amplitudes computed via techniques developed by Richard Feynman, Julian Schwinger, and later effective field theory methods associated with Ken Wilson.
In cosmology the Boltzmann equation is written for the single-particle distribution f(x^i, p_j, t) on a phase space influenced by Albert Einstein's metric; in perturbation theory one expands around the homogeneous Alexander Friedmann background. The Liouville operator includes geodesic motion derived from the Einstein field equations and the distribution couples to metric perturbations often decomposed using the formalism of James Peebles and P. J. E. Peebles or via synchronous and conformal Newtonian gauges favored in numerical codes developed at University of Cambridge and Stanford University. Collision integrals are expressed using scattering matrix elements from Murray Gell-Mann and Francis Low techniques, and incorporate statistics (Bose–Einstein, Fermi–Dirac) whose quantum origins trace to Satyendra Nath Bose and Enrico Fermi.
Collision terms encode interactions such as Compton scattering between photons and electrons (analysis rooted in studies by Arthur Compton), Thomson scattering approximations, electron–positron annihilation tied to experiments at SLAC National Accelerator Laboratory, neutrino decoupling influenced by Wolfgang Pauli and Enrico Fermi weak interaction theory, and annihilation or scattering processes for hypothetical particles proposed in frameworks by Steven Weinberg and Sheldon Glashow. Cross sections derive from perturbative Quantum Chromodynamics and Electroweak interaction calculations, often employing techniques developed by Gerard 't Hooft and Martinus Veltman. Treatment of multi-particle interactions borrows from methods used by Lev Landau and Evgeny Lifshitz in kinetic theory.
Boltzmann solvers evolve photon, baryon, and neutrino distribution multipoles to predict temperature and polarization anisotropies observed by COBE, WMAP, and Planck (spacecraft), with theoretical underpinnings in work by Wayne Hu, N. Sugiyama, and Ruth Durrer. Recombination physics integrates atomic transition rates from quantum calculations influenced by Niels Bohr and laboratory spectroscopy groups at Max Planck Institute for Astrophysics; recombination codes parallel development with radiative transfer formalisms used in Subrahmanyan Chandrasekhar's studies. Line-of-sight integration techniques used by modern analyses build on methods applied in Seljak and Zaldarriaga and collaborations including European Space Agency science teams.
Thermal freeze-out and freeze-in calculations for weakly interacting massive particles (WIMPs) and other candidates use Boltzmann equations to compute relic densities compared against cosmological constraints from Planck (spacecraft) and Large Hadron Collider bounds from ATLAS and CMS. Models invoking axions connect to theoretical work by Peccei–Quinn, Roberto Peccei, and Helen Quinn and experiments at CERN and Fermilab; sterile neutrino scenarios build on concepts by Bruno Pontecorvo and searches by collaborations like IceCube. Analytic approximations such as the Lee–Weinberg calculation and unitarity limits trace to contributions by Benjamin Lee and Steven Weinberg.
Practical cosmological predictions rely on numerical implementations like CMBFAST lineage, CAMB, and CLASS developed by teams at University of Oxford, Institut d'Astrophysique de Paris, and Observatoire de Paris, with codes interfacing to parameter estimation pipelines used by Planck Collaboration, SDSS, and DESI teams. These codes solve hierarchies of multipole equations using techniques from computational physics pioneered at Los Alamos National Laboratory and Lawrence Berkeley National Laboratory, including stiff integrator schemes, implicit solvers, and matrix methods related to algorithms used in Numerical Recipes and by researchers at SIMON FRASER UNIVERSITY and ETH Zurich. Validation employs comparison to Monte Carlo samplers like CosmoMC and optimization by Markov chain methods stemming from work by Metropolis and Hastings.
Useful approximations include tight-coupling expansions developed by George R. Blumenthal and colleagues, fluid approximations linking to Andrei Sakharov-era insights, and moment truncations inspired by Hannes Alfvén and Lev Landau. In nonrelativistic limits the equation reduces to Fokker–Planck forms related to studies by Adrian Fokker and Max Planck, while relativistic linear perturbation solutions connect to Green's function methods used by Claude Shannon-era signal theory practitioners. Analytic solutions exist in simplified scenarios studied by Eddington and extended by modern theorists at Princeton University and Institute for Advanced Study for pedagogical models.