kgrad (Kappa Gradient)

:::info Migrated from Old Wiki This content was migrated from the old wiki using GitHub Copilot and should be double-checked for accuracy. :::

Overview

The kgrad profile implements a constant external convergence (κ) gradient with an optional constant sheet. This is useful for modeling the effects of large-scale structure or nearby mass concentrations that produce a linear gradient in the convergence across the lens system.

This profile is similar to the kappa profile but parametrized by gradient magnitude and direction instead of separate x and y slopes.

Mathematical Definition

Convergence

$$\kappa(x,y) = k_0 + a(x - x_0) + b(y - y_0)$$

where:

• $k_0$ = constant convergence sheet
• $x_0, y_0$ = centroid of the gradient
• $a, b$ = gradients in x and y directions

Parametrization

The profile is parametrized by:

• $k_0$ = constant sheet
• $\nabla\kappa$ = gradient magnitude $|\nabla\kappa| = \sqrt{a^2 + b^2}$
• $\theta$ = angle/direction of the gradient

Angle Convention

Following the external shear angle convention:

• $\theta = 0°$ → gradient along +y axis
• $\theta = 90°$ → gradient along +x axis

For a gradient purely in the y-direction:

$$\kappa(x,y) = k_0 + b(y - y_0)$$

where $b = \nabla\kappa$ (the gradient magnitude parameter).

Lens Potential

The lens potential corresponding to $\kappa(x,y) = k_0 + b(y - y_0)$ is:

$$\psi(x,y) = \frac{1}{2}k_0(x'^2 + y'^2) + \frac{1}{4}b y'^3 + \frac{1}{4}b x'^2 y'$$

where $x' = x - x_0$ and $y' = y - y_0$.

Deflection Angles

$$\alpha_x(x,y) = k_0 x' + \frac{1}{2}b x' y'$$ $$\alpha_y(x,y) = k_0 y' + \frac{1}{4}b(3y'^2 + x'^2)$$

:::note Potential Ambiguity The above potential is one valid solution for $\kappa = k_0 + b(y-y_0)$. The general solution is not unique due to freedom in third-order terms:

$$\psi = v y^3 + w x^2 y \quad \Rightarrow \quad \kappa = 3v y + w y$$

Any combination satisfying $3v + w = b$ is valid. The symmetric choice ($v = w = \frac{b}{4}$) is used in GLEE.

Terms like $c x' + d y' + e$ (constant deflections/potential offsets) have no effect on time delays and are omitted. :::

GLEE Configuration

Basic Usage

kgrad
  4.400000  #x-coord   exact:
  4.000000  #y-coord   exact:
  0.000000  #k0        exact:
  2.2009e-05  #kgrad   flat:0,1  step:0.03
  1.592562  #theta     noprior:  step:0.05

Parameter Summary

#	Parameter	Description	Units	Notes
1	x	x-coordinate of centroid	arcsec	Gradient reference point
2	y	y-coordinate of centroid	arcsec	Gradient reference point
3	k0	Constant convergence sheet	dimensionless	Can be 0
4	kgrad	Convergence gradient magnitude $\|\nabla\kappa\|$	$\text{arcsec}^{-1}$	$\geq 0$
5	theta	Direction of gradient (radians, CCW from +y axis)	radians	$[0, 2\pi)$

Physical Interpretation

Constant Sheet ($k_0$)

• Uniform convergence across the field
• Does not produce deflection (only magnification)
• Often degenerate with other parameters (mass-sheet degeneracy)

Gradient ($\nabla\kappa$)

• Linear change in convergence across the lens
• Produces position-dependent deflection
• Sources:
  • Nearby large-scale structure (filaments, groups)
  • Asymmetric matter distribution around lens
  • Second-order tidal field expansion

Gradient Direction ($\theta$)

• $\theta = 0°$: convergence increases in +y direction
• $\theta = 90°$: convergence increases in +x direction
• Can be constrained by environmental studies (weak lensing, galaxy counts)

Use Cases

Environmental Effects:

• Model the influence of a nearby cluster or group
• Account for line-of-sight structure
• Combine with weak lensing analysis of surrounding field

Perturbation Analysis:

• Test sensitivity of time delays to external κ gradients
• Quantify systematic uncertainties from line-of-sight structure

Mass-Sheet Degeneracy Breaking:

• Constant $k_0$ combined with gradient can partially break degeneracies
• Use with independent constraints (e.g., velocity dispersion)

Typical Values

For galaxy-scale strong lenses:

• $k_0$: typically $-0.1$ to $+0.1$
• $\nabla\kappa$: typically $10^{-5}$ to $10^{-3}$ arcsec$^{-1}$ (small!)
• Gradient is usually a small perturbation to the main lens

Comparison with Related Profiles

Profile	Parametrization	Use Case
kgrad	Magnitude + angle	Clean parametrization for gradients
kappa	Separate $\partial\kappa/\partial x$, $\partial\kappa/\partial y$ slopes	Direct mapping from theory
shear	External shear γ + PA	Quadrupole distortion (no monopole)

:::tip Relationship to Shear External shear produces a quadrupole pattern (no net convergence gradient). kgrad produces a dipole pattern (linear gradient). They are complementary:

• Shear: models tidal field at quadrupole order
• kgrad: models tidal field at dipole order :::

Related Profiles

• External Shear - External shear profile (quadrupole component)

References

• Keeton, C. R. (2003). Computational Methods for Gravitational Lensing. arXiv:astro-ph/0102340