Spherical Collapse Model

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 ${\displaystyle \rho }$, mass ${\displaystyle M}$, and radius ${\displaystyle r}$ (given in physical coordinates) which are related by ${\displaystyle \rho =M/(4\pi r^{3}/3)}$.

To model the collapse of the spherical region, we can either use Newton's law or the second Friedmann equation

${\displaystyle {\ddot {r}}^{2}=-{\frac {GM}{r^{2}}}~.}$

The effect of the accelerated expansion of the Universe can be included in the Friedmann equation if desired, but it is a subdominant effect. The above equation admits a parameteric solution

${\displaystyle r(\theta )=A(1-\cos \theta )~,\qquad t(\theta )=B(\theta -\sin \theta )~,}$

in terms of a parameter ${\displaystyle \theta }$ where time ${\displaystyle t=0}$ corresponds to ${\displaystyle \theta =0}$ and an increasing time corresponds to an increasing ${\displaystyle \theta }$. The coefficients ${\displaystyle A,B}$ 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 (${\displaystyle \theta =0}$), but then it slows down, turns around (${\displaystyle \theta =\pi }$), and ultimately collapses (${\displaystyle \theta =2\pi }$).

If we split the density into a background ${\displaystyle {\bar {\rho }}}$ and perturbation ${\displaystyle \delta }$ by ${\displaystyle \rho (\theta )={\bar {\rho }}\left[1+\delta (\theta )\right]}$, we can solve for the fully nonlinear perturbation

${\displaystyle 1+\delta ={\frac {9}{2}}{\frac {(\theta -\sin \theta )^{2}}{(1-\cos \theta )^{3}}}~.}$

Initially ${\displaystyle \delta =0}$, at the turn-around point ${\displaystyle \delta \approx 4.55}$, and at collapse ${\displaystyle \delta =\infty }$.

Alternatively, if one considers linear perturbations, or equivalently small times ${\displaystyle \theta \ll 1}$, the above equation gives us an expression for linear perturbations

${\displaystyle \delta _{L}={\frac {3}{20}}\theta ^{2}~.}$

We can then extrapolate the linear perturbation into nonlinear regimes (more on the usefulness of this below). At turn-around ${\displaystyle \delta _{L}=1.06}$ and at collapse we get the spherical collapse threshold

${\displaystyle \delta _{L}\approx 1.69~.}$

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 ${\displaystyle \delta _{L}\geq 1.69}$ can be thought of as a collapsing region that will form a halo.

• Galaxy formation and evolution – From a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time
• Press–Schechter formalism – A mathematical model used to predict the number of dark matter halos of a certain mass.