Halo Density Profiles

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Halo Density Profiles

This page describes the spherical matter density profiles used to model dark matter halos, implemented by the abstract base class NcHaloDensityProfile and its concrete subclasses (NFW, Einasto, Hernquist). It covers the common parametrization shared by all profiles, the projected (two-dimensional) quantities derived from them, and the closed-form expressions provided by each family. The projected quantities feed directly into the weak-lensing observables of Weak-Lensing Surface Mass Density.

Dimensionless Profile

Every profile is written in terms of a dimensionless three-dimensional density \(\hat\rho(x)\), \[ \hat\rho(x) \equiv \frac{\rho(x\,r_s)}{\rho_s}, \qquad \rho(r) = \rho_s\,\hat\rho\!\left(\frac{r}{r_s}\right), \] where \(\rho(r)\) is the physical density, \(\rho_s\) is the density scale, and \(r_s\) is the scale radius. A subclass only needs to supply \(\hat\rho(x)\); everything else can be derived from it, numerically if necessary.

Parametrization

The pair \((\rho_s,\,r_s)\) is fixed by a more physical parametrization in terms of a halo mass \(M_\Delta\) and a concentration \(c_\Delta\), defined relative to an overdensity \(\Delta\) and a background density \(\rho_\mathrm{bg}\). These are supplied by a halo mass summary object NcHaloMassSummary attached to the profile at construction time.

Scale Radius

The mass–radius relation defines \(r_\Delta\) through \[ M_\Delta = \frac{4\pi}{3}\,r_\Delta^3\,\Delta\,\rho_\mathrm{bg}, \] and the concentration is \(c_\Delta \equiv r_\Delta/r_s\). The scale radius then follows directly from \((M_\Delta,\,c_\Delta)\), \[ r_s = \frac{1}{c_\Delta}\left(\frac{3M_\Delta}{4\pi\,\Delta\,\rho_\mathrm{bg}}\right)^{1/3} = \frac{r_{s0}}{(\Delta\,\rho_\mathrm{bg})^{1/3}}, \qquad r_{s0} \equiv \frac{1}{c_\Delta}\left(\frac{3M_\Delta}{4\pi}\right)^{1/3}. \] The constant part \(r_{s0}\) is independent of redshift; the only time dependence enters through \(\Delta\,\rho_\mathrm{bg}(z)\).

Density Scale

Integrating the profile out to \(r_\Delta\) and matching the mass definition gives \[ M_\Delta = \int_0^{r_\Delta} 4\pi r^2 \rho(r)\,\mathrm{d}r = 4\pi r_s^3 \rho_s\, I_{x^2\hat\rho}(c_\Delta), \qquad I_{x^2\hat\rho}(c_\Delta) \equiv \int_0^{c_\Delta} x^2\,\hat\rho(x)\,\mathrm{d}x. \] Equating this to the mass–radius relation fixes the density scale, \[ \rho_s = \frac{c_\Delta^3\,\Delta\,\rho_\mathrm{bg}}{3\,I_{x^2\hat\rho}(c_\Delta)}, \qquad \rho_{s0} \equiv \frac{\rho_s}{\Delta\,\rho_\mathrm{bg}} = \frac{c_\Delta^3}{3\,I_{x^2\hat\rho}(c_\Delta)}, \] where \(\rho_{s0}\), like \(r_{s0}\), is the redshift-independent part. A subclass may provide \(I_{x^2\hat\rho}\) in closed form; otherwise it is integrated numerically from \(\hat\rho\).

Projected Quantities

The line-of-sight projection of the profile yields the surface mass density \[ \Sigma(R) = \int_{-\infty}^{\infty} \rho\!\left(\sqrt{R^2 + z^2}\right)\mathrm{d}z = r_s\,\rho_s\,\hat\Sigma(R/r_s), \qquad \hat\Sigma(X) \equiv 2\int_0^\infty \hat\rho\!\left(\sqrt{X^2 + u^2}\right)\mathrm{d}u, \] with \(X = R/r_s\). The total mass inside an infinite cylinder of projected radius \(R\) is \[ \overline{M}(R) = \int_0^R \Sigma(R')\,2\pi R'\,\mathrm{d}R' = 2\pi r_s^3 \rho_s\,\hat{\overline{M}}(<X), \qquad \hat{\overline{M}}(X) \equiv \int_0^X \hat\Sigma(X')\,X'\,\mathrm{d}X'. \] As with the spherical mass, a subclass may implement \(\hat\Sigma\) and \(\hat{\overline{M}}\) directly, or rely on numerical integration of \(\hat\rho\).

Numerical Computation

When a subclass does not provide closed forms for the spherical mass, surface density, or cylinder mass, they are computed by integrating \(\hat\rho\) over the interval \((X_i,\,X_f)\) set by the lnXi and lnXf properties, to the relative tolerance reltol. These are controlled with nc_halo_density_profile_set_lnXi, nc_halo_density_profile_set_lnXf, and nc_halo_density_profile_set_reltol.

Units

Distances are in \(\mathrm{Mpc}\), masses in \(\mathrm{M}_\odot\), densities in \(\mathrm{M}_\odot\,\mathrm{Mpc}^{-3}\), and surface mass densities in \(\mathrm{M}_\odot\,\mathrm{Mpc}^{-2}\).

Profile Families

Navarro–Frenk–White

The NFW profile (Navarro et al. 1996) is \[ \hat\rho(x) = \frac{1}{x(1 + x)^2}, \qquad I_{x^2\hat\rho}(x) = \ln(1 + x) - \frac{x}{1 + x}. \] Its projected quantities have closed forms (Wright and Brainerd 2000), \[ \hat\Sigma(X) = \frac{2}{X^2 - 1}\left[1 - \frac{\arctan\sqrt{X^2 - 1}}{\sqrt{X^2 - 1}}\right], \qquad \hat{\overline{M}}(<X) = 2\left[\frac{\arctan\sqrt{X^2 - 1}}{\sqrt{X^2 - 1}} + \ln\frac{X}{2}\right]. \] For \(X < 1\) these are rewritten with \(\mathrm{arctanh}\sqrt{1 - X^2}\), and near \(X = 1\) or \(X = 0\) a Taylor expansion is used to retain double precision. Implemented by NcHaloDensityProfileNFW.

Navarro, J. F., C. S. Frenk, and S. D. M. White. 1996. “The Structure of Cold Dark Matter Halos.” The Astrophys. J. 462 (May): 563. https://doi.org/10.1086/177173.
Wright, C. O., and T. G. Brainerd. 2000. “Gravitational Lensing by NFW Halos.” ApJ 534 (May): 34–40. https://doi.org/10.1086/308744.

Einasto

The Einasto profile (Einasto 1965) introduces a shape parameter \(\alpha\) controlling how steeply the slope changes with radius, \[ \hat\rho(x) = \exp\left[-\frac{2}{\alpha}\left(x^\alpha - 1\right)\right], \] with spherical-mass integral \[ I_{x^2\hat\rho}(x) = \left(\frac{\alpha}{2}\right)^{3/\alpha}\frac{e^{2/\alpha}}{\alpha}\, \Gamma\!\left(\frac{3}{\alpha}\right) \left[1 - \frac{\Gamma\!\left(3/\alpha,\;2x^\alpha/\alpha\right)}{\Gamma\!\left(3/\alpha\right)}\right], \] where \(\Gamma(s)\) and \(\Gamma(s,\,t)\) are the complete and upper incomplete gamma functions. Typical values \(\alpha \approx 0.12\)–\(0.35\) are motivated by \(N\)-body simulations (Gao et al. 2008; Dutton and Macciò 2014). The projected quantities are computed numerically. Implemented by NcHaloDensityProfileEinasto.

Einasto, J. 1965. “On the Construction of a Composite Model for the Galaxy and on the Determination of the System of Galactic Parameters.” Trudy Astrofizicheskogo Instituta Alma-Ata 5 (January): 87–100.
Gao, Liang, Julio F. Navarro, Shaun Cole, et al. 2008. “The redshift dependence of the structure of massive \(\Lambda\) cold dark matter haloes.” Mon. Not. R. Astron. Soc. 387 (2): 536–44. https://doi.org/10.1111/j.1365-2966.2008.13277.x.
Dutton, Aaron A., and Andrea V. Macciò. 2014. “Cold dark matter haloes in the Planck era: evolution of structural parameters for Einasto and NFW profiles.” Mon. Not. R. Astron. Soc. 441 (4): 3359–74. https://doi.org/10.1093/mnras/stu742.

Hernquist

The Hernquist profile is \[ \hat\rho(x) = \frac{1}{x(1 + x)^3}, \qquad I_{x^2\hat\rho}(x) = \frac{x^2}{2(1 + x)^2}, \] with projected surface density and cylinder mass \[ \hat\Sigma(X) = \frac{1}{(X^2 - 1)^2}\left[\frac{(2 + X^2)\,\arctan\sqrt{X^2 - 1}}{\sqrt{X^2 - 1}} - 3\right], \qquad \hat{\overline{M}}(<X) = \frac{X^2}{X^2 - 1}\left[1 - \frac{\arctan\sqrt{X^2 - 1}}{\sqrt{X^2 - 1}}\right], \] using the same \(\mathrm{arctanh}\) rewriting and Taylor expansions near the singular points as the NFW case. Implemented by NcHaloDensityProfileHernquist.

API Reference

See NcHaloDensityProfile for the full base-class reference, and the subclasses NcHaloDensityProfileNFW, NcHaloDensityProfileEinasto, and NcHaloDensityProfileHernquist. The most relevant methods are: