TNFW (Triaxial NFW)

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Overview

The TNFW (Triaxial Navarro-Frenk-White) profile implements a fully triaxial dark matter halo following Oguri et al. (2003). Unlike the standard NFW which is spherical, TNFW allows for three independent axis ratios, providing a more realistic representation of dark matter halos in simulations and observations.

Mathematical Definition

3D Density Profile

In the halo's principal axes coordinates:

$$\rho(R) \propto \frac{1}{(R/R_0)(1 + R/R_0)^2}$$

where the triaxial radius is:

$$R^2 = c^2 \left( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \right), \quad a \leq b \leq c$$

Coordinate Transformation

Observer coordinates $(x', y', z')$ relate to halo coordinates $(x, y, z)$ via rotation angles $\theta$ and $\phi$:

$$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -\sin\phi & -\cos\phi\cos\theta & \cos\phi\sin\theta \\ \cos\phi & -\sin\phi\cos\theta & \sin\phi\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x' \\ y' \\ z' \end{pmatrix}$$

where $z'$ is the line of sight. See Oguri et al. Fig. 1 for angle definitions.

Projected Convergence

Expressing $(x', y', z')$ in units of $R_0$:

$$\kappa(x', y') = \frac{b_{\text{TNFW}}}{2} \cdot f_{\text{GNFW}}(\zeta)$$

where:

$$\zeta^2 = \frac{1}{f} (A x'^2 + B x' y' + C y'^2)$$

with $f, A, B, C$ given by Oguri et al. equations (17–19) and (28–30).

Function $f_{\text{GNFW}}$

$$f_{\text{GNFW}}(r) = \begin{cases} \dfrac{1}{1-r^2} \left( -1 + \dfrac{2}{\sqrt{1-r^2}} \text{arctanh}\left(\sqrt{\dfrac{1-r}{1+r}}\right) \right) & r < 1 \\[1em] \dfrac{1}{r^2-1} \left( 1 - \dfrac{2}{\sqrt{r^2-1}} \arctan\left(\sqrt{\dfrac{r-1}{r+1}}\right) \right) & r > 1 \\[1em] 0.25 & r = 1 \end{cases}$$

Einstein Radius Normalization

The strength is expressed via Einstein radius $r_E$ (for the equivalent spherical case $a=b=c$):

Case: $r_E$ less than 1 (in units of $R_0$)

$$b_{\text{TNFW}} = \frac{1}{\sqrt{f}} \frac{r_E^2}{\ln(r_E/2) + \dfrac{2}{\sqrt{1-r_E^2}} \text{arctanh}\left(\sqrt{\dfrac{1-r_E}{1+r_E}}\right)}$$

Case: $r_E$ equals 1

$$b_{\text{TNFW}} = \frac{1}{\ln(1/2) + 1}$$

Case: $r_E$ greater than 1

$$b_{\text{TNFW}} = \frac{1}{\sqrt{f}} \frac{r_E^2}{\ln(r_E/2) + \dfrac{2}{\sqrt{r_E^2-1}} \arctan\left(\sqrt{\dfrac{r_E-1}{r_E+1}}\right)}$$

Projected Ellipticity

The projected 2D convergence $\kappa$ is elliptical with:

• Axis ratio $q$: given by Oguri et al. equation (35)
• Position angle PA: derived from eigenvector of matrix $\begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}$ (see Sherry's notes for PA computation to ensure $\text{PA} \in (0, \pi)$)

Softening Radius

A softening radius $r_{\text{soft}}$ is used to avoid singularities:

$$R^2 \rightarrow R^2 + r_{\text{soft}}^2$$

Default: $r_{\text{soft}} = 10^{-4}$ (can be changed via Rsoft keyword in config)

GLEE Configuration

Basic Usage

tnfw
  30.000000  #x-coord   exact:
  30.000000  #y-coord   exact:
   0.750000  #a/c       exact:
   0.800000  #b/c       exact:
   0.750000  #theta     exact:
   0.500000  #phi       exact:
  10.000000  #theta_e   exact:
  25.000000  #r_scale   exact:
   0.891120  #q         exact:
   1.721835  #pa        exact:

Parameter Summary

#	Parameter	Description	Input/Output	Range
1	x	x-coordinate of centroid (arcsec)	Input
2	y	y-coordinate of centroid (arcsec)	Input
3	a/c	Short-to-long axis ratio	Input	$0 < a/c \leq 1$
4	b/c	Intermediate-to-long axis ratio	Input	$a/c \leq b/c \leq 1$
5	theta	Viewing angle (radians)	Input
6	phi	Viewing angle (radians)	Input
7	theta_e	Einstein radius $r_E$ (arcsec)	Input
8	r_scale	Scale radius $R_0$ (arcsec)	Input
9	q	Output axis ratio of projected κ	Output (can impose prior)
10	pa	Output position angle of projected κ (radians, CCW from $x'$)	Output (can impose prior)

:::tip Output Parameters Parameters 9 ($q$) and 10 (PA) are outputs computed from the 3D→2D projection. They are included in the parameter list to allow imposing priors based on observed 2D light distributions. :::

Use Cases

Observationally Motivated Constraints:

• Measure axis ratio and PA from 2D light distribution
• Impose priors on output $q$ and PA parameters
• GLEE will explore 3D halo configurations consistent with observed 2D projection

Dark Matter Halo Shape Studies:

• Constrain full 3D shape $(a/c, b/c, \theta, \phi)$ from lensing data
• Test predictions from N-body simulations about halo triaxiality

Related Profiles

• NFW - Standard spherical NFW profile
• PNFW - Prolate NFW (special case: $a=b \leq c$)
• GNFW - Generalized NFW with variable inner slope
• eNFW - Elliptical NFW with ellipticity in κ

References

• Oguri, M., Lee, J., & Suto, Y. (2003). Detailed Cluster Lensing Profiles at Large Radii and the Impact on Cluster Weak Lensing Studies. ApJ, 599, 7. ADS Link