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A dark matter halo is a gravitationally bound concentration of dark matter. Its internal motions are largely supported by a distribution of particle orbits, unlike the cooling gas that can settle into a luminous galactic disk. Halo profiles, abundances and spatial correlations provide useful ingredients for modeling nonlinear clustering. The halo model uses these ingredients as an approximate statistical description of matter.^[ref]

The halo model represents matter as a population of dark matter halos with specified profiles, abundance and clustering. Pairs of mass elements inside the same halo generate the one-halo term; pairs in distinct halos generate the two-halo term. This connects internal object structure with large-scale statistics. The decomposition is useful, but assumptions about halo boundaries and correlations make the simplest implementation approximate.^[ref]

The Press-Schechter formalism estimates dark matter halo abundance by comparing a smoothed linear density contrast with the linear spherical-collapse threshold. A Gaussian upper-tail probability supplies the collapsed fraction, with a factor of two in the original mass accounting. Differentiating that fraction by smoothing mass gives the Press-Schechter halo mass function. It connects initial fluctuations with object counts, while idealized spherical collapse and the treatment of nested regions limit its accuracy.^[ref]

The Press-Schechter halo mass function estimates the comoving number density of dark matter halos per mass interval. Let $\bar{\rho }$ be the mean comoving matter mass density, $\sigma (M,z)$ the root-mean-square linearly evolved density contrast smoothed on mass scale $M$, and ${\delta }_c$ the linear spherical-collapse threshold. Define $\nu ={\delta }_c/\sigma (M,z)$. For a Gaussian distribution of the initial smoothed density contrast, the doubled upper-tail prescription givesAssume $\sigma$ decreases with $M$. Identifying the mass fraction in a narrow interval with $[M/\bar{\rho }](dn/dM)dM$ givesThe minus sign makes the abundance positive for the stated decreasing variance. The exponential suppresses rare high-$\nu$ objects, while the derivative converts the smoothing scale into a mass interval. This is a statistical collapse model: spherical dynamics and its treatment of nested collapsed regions limit the accuracy of the predicted abundance.^[ref]