Spherical Collapse Model
The spherical collapse model describes the evolution of nearly homogeneous matter in the early Universe into collapsed virialized structures - dark matter halos. This model assumes that halos are spherical and dominated by gravity which leads to an analytical solution for several of the halos' properties such as density and radius over time.[1][2][3]
The framework for spherical collapse was first developed to describe the infall of matter into clusters of galaxies.[4] At this time, in the early 1970s, astronomical evidence for dark matter was still being collected, and it was believed that the Universe was dominated by ordinary, visible matter. However, it is now thought that dark matter is the dominating species of matter.
Derivation and key equations
The simplest halo formation scenario involves taking a sufficiently overdense spherical patch, which we call a proto-halo (e.g., Descjacques et al. 2018),[5] of the early Universe and tracking its evolution under the effect of its self-gravity. Once the proto-halo has collapsed and virialized, it becomes a halo.
Since the matter outside this sphere is spherically symmetric, we can apply Newton's shell theorem or Birkhoff's theorem (for a more general description), so that external forces average to zero and we can treat the proto-halo as isolated from the rest of the Universe. The proto-halo has a density , mass , and radius (given in physical coordinates) which are related by .
To model the collapse of the spherical region, we can either use Newton's law or the second Friedmann equation, giving
The effect of the accelerated expansion of the Universe can be included if desired, but it is a subdominant effect.
The above equation admits the explicit solution [6]
where is the maximum radius, assumed to occur at time , and is the quantile function of the Beta distribution, also known as the inverse function of the regularized incomplete beta function . The time
is the free-fall time, where .
Long before the derivation of this explicit solution, the spherical collapse equation has been known to admit a parametric solution [7]
in terms of a parameter . The origin of time, , now occurs at a vanishing radius, and the time increases with increasing . The coefficients are given by the energy contents of the sphere (cf. equation 5.89 in Dodelson et al.).[2] Initially the sphere expands at the rate of the Universe (), but then it slows down, turns around (), and ultimately collapses ().
If we split the density into a background and perturbation by , we can solve for the fully nonlinear perturbation
Initially , at the turn-around point , and at collapse .
Alternatively, if one considers linear perturbations, or equivalently small times , the above equation gives us an expression for linear perturbations
We can then extrapolate the linear perturbation into nonlinear regimes (more on the usefulness of this below). At turn-around and at collapse we get the spherical collapse threshold
Although the halo does not physically have an overdensity of 1.69 at collapse, the above collapse threshold is nevertheless useful. It tells us that if we model the initial (linear) density field and extrapolate into the future, wherever can be thought of as a collapsing region that will form a halo.
See also
- Galaxy formation and evolution
- Press–Schechter formalism – A mathematical model used to predict the number of dark matter halos of a certain mass.
References
- ^ Barkana, Rennan (2018). The Encyclopedia of Cosmology, Volume 1: Galaxy Formation and Evolution. Vol. 2. World Scientific. doi:10.1142/9496. ISBN 9789814656221.
- ^ a b Dodelson, Scott; Schmidt, Fabian (2021). Modern Cosmology (2 ed.). Academic Press. ISBN 978-0-12-815948-4.
- ^ Baumann, Daniel (2022). Cosmology. Cambridge University Press. Bibcode:2022cosm.book.....B. doi:10.1017/9781108937092. ISBN 9781108838078.
- ^ Gunn, James E.; Gott III, J. Richard (1972). "On the infall of matter into clusters of galaxies and some effects on their evolution". The Astrophysical Journal. 176: 1–19. Bibcode:1972ApJ...176....1G. doi:10.1086/151605.
- ^ Desjacques, Vincent; Jeong, Donghui; Schmidt, Fabian (2018). "Large-scale galaxy bias". Physics Reports. 733: 1–193. arXiv:1611.09787. Bibcode:2018PhR...733....1D. doi:10.1016/j.physrep.2017.12.002.
- ^ Obreschkow, Danail (7 June 2024). "From Cavitation to Astrophysics: Explicit Solution of the Spherical Collapse Equation". Phys. Rev. E. 109 (6): 065102. arXiv:2401.05445. doi:10.1103/PhysRevE.109.065102.
- ^ Lin, C.C.; Mestel, L.; Shu, F.H. (Nov 1965). "The Gravitational Collapse of a Uniform Spheroid". Astrophysical Journal. 142: 1431. doi:10.1086/148428.
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