GNFW (Generalized NFW)

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Overview

The GNFW (Generalized Navarro-Frenk-White) profile extends the standard NFW profile by introducing a variable inner slope parameter $\gamma$. This allows for more flexibility in modeling dark matter halos with different central density profiles.

When $\gamma = 1$, the GNFW profile reduces to the standard NFW profile (without the scale: option).

Mathematical Definition

The 3D density profile is:

$$\rho(r) = \frac{\rho_s}{(r/r_s)^\gamma \cdot (1 + r/r_s)^{3-\gamma}}$$

where:

• $\rho_s$ = characteristic density
• $r_s$ = scale radius
• $\gamma$ = inner slope parameter

Spherical Projection Formulas

For spherical models with $r, r_s$ in angular coordinates:

Convergence

$$\kappa(r) = 0.5 \cdot \theta_e \cdot \chi^{1-\gamma} \left[ (1+\chi)^{\gamma-3} + (3-\gamma) \int_0^1 (y+\chi)^{\gamma-4}(1-\sqrt{1-y^2}) \, dy \right]$$

Deflection Angle

$$\alpha(r) = \theta_e \cdot r_s \cdot \chi^{2-\gamma} \left( \frac{1}{3-\gamma} \, _2F_1[3-\gamma, 3-\gamma, 4-\gamma, -\chi] + \int_0^1 \frac{(y+\chi)^{\gamma-3}(1-\sqrt{1-y^2})}{y} \, dy \right)$$

Lens Potential

$$\phi(r) = \int_0^r \alpha(r') \, dr'$$

where:

• $\chi = r/r_s$ (dimensionless radius)
• $_2F_1$ is the hypergeometric function

Normalization

The normalization parameter $\theta_e$ is defined as:

$$\theta_e = 4\kappa_s = 4\rho_s r_s \frac{D_d}{\Sigma_{\text{crit}}}$$

where:

• $\kappa_s = \rho_s r_s D_d / \Sigma_{\text{crit}}$
• $D_d$ = angular diameter distance to the lens
• $\Sigma_{\text{crit}}$ = critical surface density

:::note No Scale Option Since GNFW requires numerical integration (computationally intensive), the scale: option is not implemented for the strength parameter. The strength remains a direct normalization rather than an equivalent Einstein radius. :::

Ellipticity Implementation

:::warning Important Ellipticity is implemented in the lens potential, not in $\kappa$ directly:

$$\phi(r) \rightarrow \phi(\sqrt{x^2 q + y^2/q})$$

This differs from some other profiles where ellipticity is applied directly to the convergence. :::

GLEE Configuration

Basic Usage

gnfw
  4.0000     #x-coord  
  3.0000     #y-coord
  0.7        #b/a      (q, axis ratio)
  3.6        #theta    (position angle, ccw from +x in radians; 0 = elongated along x)
  0.2        #theta_e  (normalization, see definition above)
  20.0       #r_scale  (rs, scale radius in arcsec)
  1.0        #gamma    (inner slope)

Parameter Summary

#	Parameter	Description	Notes
1	x	x-coordinate of centroid (arcsec)
2	y	y-coordinate of centroid (arcsec)
3	q	Axis ratio (b/a, minor/major)	$0 < q \leq 1$
4	theta	Position angle (radians, CCW from +x axis)
5	theta_e	Normalization ($4\kappa_s$)	No scale: option
6	r_scale	Scale radius $r_s$ (arcsec)
7	gamma	Inner slope	$\gamma = 1$ recovers NFW

Numerical Integration

The deflection angle and lens potential require numerical integration using GSL (GNU Scientific Library) integrators.

Available Integrators

GLEE supports three GSL integration methods:

• QAGS (default) - Adaptive integration with singularities
• QAGP - Adaptive integration with specified singular points
• CQUAD - Doubly-adaptive integration

See GSL Documentation for implementation details.

Selecting an Integrator

Set the environment variable GNFW_INTEGRATOR before running GLEE:

tcsh shell:

setenv GNFW_INTEGRATOR CQUAD

bash shell:

export GNFW_INTEGRATOR=CQUAD

Valid values: QAGS, QAGP, CQUAD. If not set, QAGS is used. Invalid values trigger an error.

Integration Failure Handling

:::caution Automatic Handling When GSL fails to numerically integrate, GLEE does not halt with an error. Instead:

• Deflection angles and lens potentials are set to INFINITY
• MCMC/emcee chains continue running (avoiding periodic halts)

Warning: If integration fails frequently, the chains will be inaccurate. Monitor chain behavior and adjust parameters if needed. :::

Special Cases

Recovery of Standard NFW

When $\gamma = 1$:

$$\rho(r) = \frac{\rho_s}{(r/r_s)(1 + r/r_s)^2}$$

This matches the standard NFW profile (without scale: option).

Related Profiles

• NFW - Standard Navarro-Frenk-White profile
• TNFW - Triaxial NFW profile
• PNFW - Prolate NFW profile
• eNFW - Elliptical NFW with ellipticity in κ

References

• Navarro, J. F., Frenk, C. S., & White, S. D. M. (1996). The Structure of Cold Dark Matter Halos. ApJ, 462, 563
• Zhao, H. (1996). Analytical models for galactic nuclei. MNRAS, 278, 488