Gravitational Lensing by a Massive Object in a Dark Matter Halo. I. Critical Curves and Caustics

The three-dimensional density profile of a spherical dark matter halo can be described by the NFW profile (Navarro et al.1996),

$$\begin{eqnarray}&&\rho (r)={\rho }_{{\rm{s}}}{\left(\displaystyle \frac{r}{{r}_{{\rm{s}}}}\right)}^{-1}{\left(1+\displaystyle \frac{r}{{r}_{{\rm{s}}}}\right)}^{-2},\end{eqnarray} \tag{ 1 }$$

where r is the three-dimensional radial distance from the center, r_s is the scale radius, and ρ_s is the characteristic density such that ρ(r_s) = ρ_s/4. At small radii r ≪ r_s the density diverges as ρ ∼ r⁻¹, at large radii r ≫ r_s the density drops as ρ ∼ r⁻³, and at the scale radius ${d}\,{\rm{ln}}\,\rho \,/{d}\,{\rm{ln}}\,r\,{| }_{r={r}_{{\rm{s}}}}=-2$. The halo mass enclosed in a sphere of radius r is

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}M(r) & = & 4\pi {\displaystyle \int }_{0}^{r}\,r{{\rm{{\prime} }}}^{2}\rho ({r}^{{\rm{{\prime} }}})\,{d}{r}^{{\rm{{\prime} }}}\\ & = & 4\pi {r}_{{\rm{s}}}^{3}{\rho }_{{\rm{s}}}\left[{\rm{ln}}\left(1+\displaystyle \frac{r}{{r}_{{\rm{s}}}}\right)-\displaystyle \frac{r}{r+{r}_{{\rm{s}}}}\right].\end{array}\end{array}\end{eqnarray} \tag{ 2 }$$

Due to the logarithmic divergence of the NFW halo mass for r ≫ r_s, the profile is extended only to a certain distance such as r₂₀₀, at which the mean density within the enclosed sphere is 200 times the critical density at the redshift of the halo, ρ_crit(z). The ratio between the two characteristic radii defines the concentration parameter of the halo, c_s = r₂₀₀/r_s.

Despite its density divergence at the center and its mass divergence at large radii, for a broad range of intermediate radii the NFW profile presents a good fit to cold dark matter halo profiles. For galaxy cluster halos this agreement has been demonstrated by cluster lensing analyses (Okabe et al. 2013; Umetsu & Diemer 2017) and by X-ray emission analyses (Ettori et al. 2013).

In order to compute light deflection by the NFW halo we first integrate Equation (1) along the line of sight to obtain the NFW surface density. We express $r={r}_{{\rm{s}}}\sqrt{{x}^{2}+{l}^{2}}$ in terms of the distances x projected in the plane of the sky and l along the line of sight, both in units of the scale radius r_s. The convergence

$$\begin{eqnarray}&&\kappa (x)=\displaystyle \frac{{r}_{{\rm{s}}}}{{{\rm{\Sigma }}}_{{\rm{cr}}}}{\int }_{-\infty }^{\infty }\,\rho ({r}_{{\rm{s}}}\sqrt{{x}^{2}+{l}^{2}})\,{d}l\end{eqnarray} \tag{ 3 }$$

is defined as the surface density expressed in units of the critical surface density

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{cr}}=\displaystyle \frac{{c}^{2}}{4\pi G}\displaystyle \frac{{D}_{{\rm{s}}}}{{D}_{{\rm{l}}}\,{D}_{\mathrm{ls}}},\end{eqnarray} \tag{ 4 }$$

where c is the speed of light, G is the gravitational constant, and D_l, D_s, and D_ls are the angular diameter distances from the observer to the lens (the halo in our case), from the observer to the source of light (e.g., a background galaxy or quasar), and from the lens to the source, respectively.

For the NFW density from Equation (1) the integral in Equation (3) can be obtained analytically (e.g., Bartelmann 1996; Wright & Brainerd 2000; Keeton 2001; Golse & Kneib 2002), yielding the NFW convergence as a function of the plane-of-the-sky radial position x:

$$\begin{eqnarray}&&\kappa (x)=2\,{\kappa }_{{\rm{s}}}\,\displaystyle \frac{1-{ \mathcal F }(x)}{{x}^{2}-1},\end{eqnarray} \tag{ 5 }$$

where the dimensionless halo convergence parameter κ_s = ρ_s r_s/Σ_cr and the function

$$\begin{eqnarray}{ \mathcal F }(x)=\left\{\begin{array}{cc}\displaystyle \frac{\mathrm{arctanh}\sqrt{1-{x}^{2}}}{\sqrt{1-{x}^{2}}} & \mathrm{for}\,x\lt 1,\\ 1 & \,\mathrm{for}{\text{}}\ x\,=\,1,\\ \displaystyle \frac{\arctan \sqrt{{x}^{2}-1}}{\sqrt{{x}^{2}-1}} & \mathrm{for}\,x\gt 1.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

The NFW convergence decreases monotonically and smoothly with the radial position x throughout its range. As shown in Equation (A3), it has a logarithmic divergence for x → 0, where $\kappa (x)\approx -2\,{\kappa }_{{\rm{s}}}\,\mathrm{ln}x$. It drops to κ(1) = 2 κ_s/3 at the scale radius, as shown in Equation (A9), and decreases further to zero as κ(x) ≈ 2 κ_s x⁻² for x ≫ 1. The unit convergence radius x₀ has a special significance from the perspective of lensing. It can be determined for a given value of κ_s by setting κ(x₀) = 1 in Equation (5):

$$\begin{eqnarray}&&\displaystyle \frac{1-{x}_{0}^{2}}{{ \mathcal F }({x}_{0})-1}=2\,{\kappa }_{{\rm{s}}},\end{eqnarray} \tag{ 7 }$$

which can be solved numerically. The solutions are illustrated and discussed further in Section 2.3.

The gravitational field of the halo deflects a light ray passing at point x in the plane of the sky by the deflection angle

$$\begin{eqnarray}&&{\boldsymbol{\alpha }}({\boldsymbol{x}})=\displaystyle \frac{4\,G\,{M}_{\mathrm{cyl}}(x\,{r}_{{\rm{s}}})}{{c}^{2}\,{r}_{{\rm{s}}}}\,\displaystyle \frac{{\boldsymbol{x}}}{{x}^{2}},\end{eqnarray} \tag{ 8 }$$

where M_cyl(x r_s) is the mass within a radius x r_s along the line of sight through the halo center:

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{M}_{{\rm{cyl}}}(x\,{r}_{{\rm{s}}}) & = & 2\,\pi \,{r}_{{\rm{s}}}^{2}\,{{\rm{\Sigma }}}_{{\rm{cr}}}{\displaystyle \int }_{0}^{x}\,\kappa ({x}^{{\rm{{\prime} }}}){x}^{{\rm{{\prime} }}}\,{d}{x}^{{\rm{{\prime} }}}\\ & = & 4\,\pi \,{r}_{{\rm{s}}}^{3}\,{\rho }_{{\rm{s}}}\left[{\rm{ln}}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right],\end{array}\end{array}\end{eqnarray} \tag{ 9 }$$

where we use the NFW convergence from Equation (5). The deflection angle for the NFW halo thus is

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{\boldsymbol{\alpha }}({\boldsymbol{x}}) & = & \displaystyle \frac{16\,\pi \,G\,{r}_{{\rm{s}}}^{2}\,{\rho }_{{\rm{s}}}}{{c}^{2}}\left[{\rm{ln}}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right]\displaystyle \frac{{\boldsymbol{x}}}{{x}^{2}}\\ & = & \displaystyle \frac{4\,{\kappa }_{{\rm{s}}}\,{r}_{{\rm{s}}}\,{D}_{{\rm{s}}}}{{D}_{{\rm{l}}}\,{D}_{{\rm{ls}}}}\left[{\rm{ln}}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right]\,\displaystyle \frac{{\boldsymbol{x}}}{{x}^{2}},\end{array}\end{array}\end{eqnarray} \tag{ 10 }$$

where we use Equation (4) and the definition of κ_s under Equation (5) to get the second expression. The deflection angle can be used in the general lens equation,

$$\begin{eqnarray}&&{\boldsymbol{\beta }}={\boldsymbol{\theta }}-\displaystyle \frac{{D}_{\mathrm{ls}}}{{D}_{{\rm{s}}}}{\boldsymbol{\alpha }},\end{eqnarray} \tag{ 11 }$$

connecting the angular position β of a background source with the angular position θ of its image formed by the lens. In gravitational lensing the angles are often expressed in units of the Einstein radius ¹ of the lens,

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=\sqrt{\displaystyle \frac{4\,G\,{M}_{\mathrm{NFW}}}{{c}^{2}}\,\displaystyle \frac{{D}_{\mathrm{ls}}}{{D}_{{\rm{l}}}\,{D}_{{\rm{s}}}}},\end{eqnarray} \tag{ 12 }$$

where we set M_NFW = M(c_s r_s) as the total mass within radius r₂₀₀ using Equation (2):

$$\begin{eqnarray}&&{M}_{{\rm{NFW}}}=4\,\pi \,{r}_{{\rm{s}}}^{3}\,{\rho }_{{\rm{s}}}\,[\,{\rm{ln}}(1+{c}_{{\rm{s}}})-{c}_{{\rm{s}}}/(1+{c}_{{\rm{s}}})\,].\end{eqnarray} \tag{ 13 }$$

If we introduce the angular scale radius in units of the Einstein radius ${\theta }_{{\rm{s}}}^{* }={r}_{{\rm{s}}}/({D}_{{\rm{l}}}\,{\theta }_{{\rm{E}}})$, we may write the lens equation for the NFW profile as

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{{\boldsymbol{\beta }}}^{\ast } & = & {{\boldsymbol{\theta }}}^{\ast }-{\left[{\rm{l}}{\rm{n}}(1+{c}_{{\rm{s}}})-{c}_{{\rm{s}}}/(1+{c}_{{\rm{s}}})\,\right]}^{-1}\,\\ & & \,\times \,\left[{\rm{l}}{\rm{n}}\displaystyle \frac{{\theta }^{\ast }}{2\,{\theta }_{{\rm{s}}}^{\ast }}+{ \mathcal F }\,\left(\displaystyle \frac{{\theta }^{\ast }}{{\theta }_{{\rm{s}}}^{\ast }}\right)\right]\displaystyle \frac{{{\boldsymbol{\theta }}}^{\ast }}{{\left({\theta }^{\ast }\right)}^{2}},\end{array}\end{array}\end{eqnarray} \tag{ 14 }$$

where β ^* = β /θ_E = and θ ^* = θ /θ_E. Alternatively, we may express the angles in units of the angular scale length of the halo, r_s/D_l. In these units the source position y = β D_l/r_s and the image position x = θ D_l/r_s, so that the lens equation has the form

$$\begin{eqnarray}&&{\boldsymbol{y}}={\boldsymbol{x}}-4\,{\kappa }_{{\rm{s}}}\left[\mathrm{ln}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right]\,\displaystyle \frac{{\boldsymbol{x}}}{{x}^{2}}.\end{eqnarray} \tag{ 15 }$$

Equation (14), which is expressed in familiar lensing units, explicitly involves two parameters, c_s and ${\theta }_{{\rm{s}}}^{* }$. In the rest of this work we use the more compact Equation (15), which involves a single parameter κ_s that is related to the two parameters by

$$\begin{eqnarray}&&{\kappa }_{{\rm{s}}}={\left(2{\theta }_{{\rm{s}}}^{\ast }\right)}^{-2}\,{\left[{\rm{ln}}(1+{c}_{{\rm{s}}})-{c}_{{\rm{s}}}/(1+{c}_{{\rm{s}}})\,\right]}^{-1}.\end{eqnarray} \tag{ 16 }$$

This expression can be derived using Equations (4), (12), and (13) in the definition of κ_s above Equation (6).

In Figure 1 we illustrate the conversion by κ_s = const. contours in a ${\theta }_{{\rm{s}}}^{* }$ versus c_s plot including sample observational data. The plot range is set to include the parameter combinations of 19 observed galaxy clusters from the CLASH survey (Merten et al. 2015), marked here by crosses. We use the Merten et al. cluster scale radii r_s, masses M_200c for M_NFW, concentrations c_200c for c_s, and cluster redshifts z to compute the angular diameter distance d_A(z) = D_l in an FLRW universe with PLANCK 2015 cosmological parameters (Planck Collaboration et al. 2016). To obtain ${\theta }_{{\rm{s}}}^{* }$ we compute the angular Einstein radius θ_E for asymptotically distant sources, replacing D_ls/D_s → 1. This replacement does not hold exactly for angular diameter distances for which the asymptotic ratio is lens redshift dependent. However, the approximation overestimates the Einstein radii of the clusters in the sample merely by 3%–11%.

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Many important lensing quantities are obtained by computing the Jacobian of the lens equation: the inverse of its absolute value yields the magnification of a point-source image at x , its sign indicates the image parity, and its zero contour defines the critical curve, which in turn yields the caustic when mapped back to the source plane positions y .

For the NFW profile, the Jacobian of Equation (15) is

$$\begin{eqnarray}&&\begin{array}{l}\begin{array}{l}det\,J({\boldsymbol{x}})\,=\,\left|\frac{{\rm{\partial }}\,{\boldsymbol{y}}}{{\rm{\partial }}\,{\boldsymbol{x}}}\right|=\left\{1-\frac{4\,{\kappa }_{{\rm{s}}}}{{x}^{2}}\left[{\rm{l}}{\rm{n}}\frac{x}{2}+{ \mathcal F }(x)\right]\right\}\\ \times \left\{1+\frac{4\,{\kappa }_{{\rm{s}}}}{{x}^{2}}\left[{\rm{l}}{\rm{n}}\frac{x}{2}+{ \mathcal F }(x)\right]-4\,{\kappa }_{{\rm{s}}}\frac{{ \mathcal F }(x)-1}{1-{x}^{2}}\right\},\end{array}\end{array}\end{eqnarray} \tag{ 17 }$$

which, due to symmetry, is a purely radial function of image position x. For x → 0 at the halo center $\det \,J({\boldsymbol{x}})\,\approx 4\,{\kappa }_{{\rm{s}}}^{2}\,{\mathrm{ln}}^{2}x\,\to \infty$, as shown in Equation (A4). For x → ∞ the Jacobian asymptotically reaches unity, $\det \,J({\boldsymbol{x}})\to 1$.

The factorized form of the Jacobian indicates that the critical curve $\det \,J({\boldsymbol{x}})=0$ consists of solutions of two simpler equations. The factor in the first braces yields the tangential critical curve, which is a circle ∣ x ∣ = x_T with radius obtained by numerically solving

$$\begin{eqnarray}&&1-\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}_{{\rm{T}}}^{2}}\left[\,\mathrm{ln}\displaystyle \frac{{x}_{{\rm{T}}}}{2}+{ \mathcal F }({x}_{{\rm{T}}})\right]=0,\end{eqnarray} \tag{ 18 }$$

an equation with a single solution for any value of κ_s. If we introduce the mean convergence within a circle of radius x,

$$\begin{eqnarray}&&\bar{\kappa }(x)=\displaystyle \frac{1}{\pi \,{x}^{2}}\,{\int }_{0}^{x}2\pi \,{x}^{{\rm{{\prime} }}}\kappa ({x}^{{\rm{{\prime} }}})\,{d}{x}^{{\rm{{\prime} }}}=\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}^{2}}\left[\,{\rm{ln}}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right],\end{eqnarray} \tag{ 19 }$$

where we take into account Equation (9), we see that Equation (18) implies $\bar{\kappa }({x}_{{\rm{T}}})=1$. The tangential critical curve thus encloses a circle with unit mean convergence. Recalling that the NFW convergence is a monotonically decreasing function, this means that $\kappa ({x}_{{\rm{T}}})\lt \bar{\kappa }({x}_{{\rm{T}}})=1$ and thus, x_T > x₀, where the radius of unit convergence x₀ is given by Equation (7). Substituting the critical curve ${\boldsymbol{x}}={x}_{{\rm{T}}}(\cos \varphi ,\sin \varphi )$ with φ ∈ [0, 2π] in Equation (15) and using Equation (18), we obtain the corresponding part of the caustic:

$$\begin{eqnarray}&&{\boldsymbol{y}}={\boldsymbol{x}}\left\{1-\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}_{{\rm{T}}}^{2}}\left[\,\mathrm{ln}\displaystyle \frac{{x}_{{\rm{T}}}}{2}+{ \mathcal F }({x}_{{\rm{T}}})\right]\right\}=(0,0).\end{eqnarray} \tag{ 20 }$$

The tangential part of the caustic consists of a single point at the origin.

Setting the second braces in Equation (17) equal to zero yields the radial critical curve. This is another circle with radius x_R obtained by numerically solving

$$\begin{eqnarray}&&1+\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}_{{\rm{R}}}^{2}}\left[\,\mathrm{ln}\displaystyle \frac{{x}_{{\rm{R}}}}{2}+{ \mathcal F }({x}_{{\rm{R}}})\right]-4\,{\kappa }_{{\rm{s}}}\,\displaystyle \frac{{ \mathcal F }({x}_{{\rm{R}}})-1}{1-{x}_{{\rm{R}}}^{2}}=0,\end{eqnarray} \tag{ 21 }$$

which also has a single solution for any κ_s. The left-hand side can be written as a simple combination of Equations (19) and (5), yielding

$$\begin{eqnarray}&&1+\bar{\kappa }({x}_{{\rm{R}}})-2\,\kappa ({x}_{{\rm{R}}})=0.\end{eqnarray} \tag{ 22 }$$

Therefore, the convergence at the radius of the radial critical curve

$$\begin{eqnarray}&&\kappa ({x}_{{\rm{R}}})=\displaystyle \frac{1}{2}\left[1+\bar{\kappa }({x}_{{\rm{R}}})\right]\end{eqnarray} \tag{ 23 }$$

is the average of the mean convergence at the radius and 1. For the monotonically decreasing NFW convergence $\bar{\kappa }({x}_{{\rm{R}}})\,\gt \kappa ({x}_{{\rm{R}}})\gt 1$ and thus, x_R < x₀. Substituting the radial critical curve ${\boldsymbol{x}}={x}_{{\rm{R}}}(\cos \varphi ,\sin \varphi )$ with φ ∈ [0, 2π] in Equation (15) and using Equation (21), we obtain the corresponding part of the caustic,

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{y}} & = & {\boldsymbol{x}}\left\{1-\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}_{{\rm{R}}}^{2}}\left[\mathrm{ln}\displaystyle \frac{{x}_{{\rm{R}}}}{2}+{ \mathcal F }({x}_{{\rm{R}}})\right]\right\}\\ & = & -2\,\left[2\,{\kappa }_{{\rm{s}}}\,\displaystyle \frac{{ \mathcal F }({x}_{{\rm{R}}})-1}{1-{x}_{{\rm{R}}}^{2}}-1\right]\,{x}_{{\rm{R}}}(\cos \varphi ,\sin \varphi ),\end{array}\end{eqnarray} \tag{ 24 }$$

where the expression in the last square brackets is equal to κ(x_R) − 1, which is positive. The radial part of the caustic is thus a circle with radius

$$\begin{eqnarray}&&{y}_{{\rm{R}}}=2\,{x}_{{\rm{R}}}[\kappa ({x}_{{\rm{R}}})-1].\end{eqnarray} \tag{ 25 }$$

Figure 2 shows the critical curve in the image plane (black, left panel) and the caustic in the source plane (red, right panel) for an NFW density profile with κ_s ≈ 0.239035. This value is obtained from Equation (16) for the combination

$$\begin{eqnarray}&&\{{c}_{{\rm{s}}},{\theta }_{{\rm{s}}}^{* }\}=\{3.5,1.2\}\end{eqnarray} \tag{ 26 }$$

chosen to represent the Merten et al. (2015) cluster data and marked by a diamond in Figure 1. The outer tangential and inner radial critical curves are plotted over a color map of the Jacobian $\det \,J({\boldsymbol{x}})$, marked red where positive and blue where negative. Lighter areas indicate regions with higher image magnification (brighter images), darker areas regions with lower image magnification (dimmer images). Inside the radial critical curve the Jacobian is positive, increasing divergently toward the origin. Any images appearing close to the origin are thus strongly demagnified. In the annulus between the radial and tangential critical curves the Jacobian is negative; thus any images appearing here have negative parity. Outside the tangential critical curve the Jacobian is positive, increasing monotonically outward, and approaching 1 asymptotically.

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The caustic is plotted over a gray-scale magnification map, showing the total point-source magnification

$$\begin{eqnarray}&&A({\boldsymbol{y}})=\sum _{i}| \det \,J({{\boldsymbol{x}}}_{i}){| }^{-1}\end{eqnarray} \tag{ 27 }$$

of all images x _i formed by the lens for a point source at y . The magnification map is computed by inverse ray shooting (e.g., Kayser et al. 1986), with the color ranging from black for the lowest magnification A = 1 to white for the highest magnification A ≥ 1000. The magnification diverges at the caustic, here more prominently at the central point (tangential caustic). At the circular radial caustic the magnification remains finite from the outer side but diverges from the inner side in an extremely narrow high-magnification region (see also Martel & Shapiro 2003). Clearly, sources positioned outside the radial caustic are magnified substantially less than those positioned inside.

Although Figure 2 is plotted for a single value of the NFW convergence parameter κ_s, the general character of the critical curve, caustic, Jacobian, and magnification maps does not change for other values. What changes is the radii of the tangential and radial critical curves, x_T and x_R, respectively, and the radial caustic radius y_R (Bartelmann 1996; Martel & Shapiro 2003). Figure 3 shows the dependence of these radii and the unit convergence radius x₀ on κ_s. All the plotted radii are simple monotonically increasing functions of κ_s. The vertical dotted–dashed line indicates the value κ_s ≈ 0.239035 chosen for illustration in Figure 2 as well as in the rest of this work.

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While the values of the radii for general κ_s plotted in Figure 3 have to be computed numerically, for κ ≲ 0.2 they may be approximated by analytic expressions using Equations (A1) and (A2) from Appendix A.1 in Equations (18), (7), (21), and (24). For the critical-curve radii (Dúmet-Montoya et al. 2013) and the unit convergence radius we find

$$\begin{eqnarray}&&\left\{{x}_{{\rm{T}}},{x}_{0},{x}_{{\rm{R}}}\right\}\approx 2\,{{e}}^{-(1+3{\kappa }_{{\rm{s}}})/(2{\kappa }_{{\rm{s}}})}\left\{{e},\,\sqrt{{e}},\,1\right\}\end{eqnarray} \tag{ 28 }$$

and for the (radial) caustic radius we find

$$\begin{eqnarray}&&{y}_{{\rm{R}}}\approx 4\,{\kappa }_{{\rm{s}}}\,{{e}}^{-(1+3{\kappa }_{{\rm{s}}})/(2{\kappa }_{{\rm{s}}})}.\end{eqnarray} \tag{ 29 }$$

These approximations are marked by dotted curves in Figure 3. All four radii shrink exponentially fast for low values of κ_s. Nevertheless, the ratios between the radii x_T, x₀, and x_R are constant in this regime, given by the factors in braces on the right-hand side of Equation (28).

The lensing effect of an additional compact object with mass distributed in a region much smaller than the halo scale radius r_s can be modeled by adding a point-mass deflection term

$$\begin{eqnarray}&&{{\boldsymbol{\alpha }}}_{{\rm{P}}}({\boldsymbol{\theta }})=\displaystyle \frac{4\,G\,{M}_{{\rm{P}}}}{{c}^{2}\,{D}_{{\rm{l}}}}\,\displaystyle \frac{{\boldsymbol{\theta }}-{{\boldsymbol{\theta }}}_{{\rm{P}}}}{| {\boldsymbol{\theta }}-{{\boldsymbol{\theta }}}_{{\rm{P}}}{| }^{2}}\end{eqnarray} \tag{ 30 }$$

to the deflection angle in Equation (11). Here M_P is the mass of the object and θ _P is its angular position from the halo center. If the object acted as an isolated lens, its region of influence could be measured by its angular Einstein radius

$$\begin{eqnarray}&&{\theta }_{\mathrm{EP}}=\sqrt{\displaystyle \frac{4\,G\,{M}_{{\rm{P}}}}{{c}^{2}}\,\displaystyle \frac{{D}_{\mathrm{ls}}}{{D}_{{\rm{l}}}\,{D}_{{\rm{s}}}}}.\end{eqnarray} \tag{ 31 }$$

In our case, with the object embedded in the NFW halo, we will use θ_EP for comparison and illustration purposes.

Adding the point-mass term in units of the NFW halo Einstein radius to Equation (14), we obtain the full lens equation:

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}{{\boldsymbol{\beta }}}^{\ast }={{\boldsymbol{\theta }}}^{\ast }-{\left[{\rm{l}}{\rm{n}}(1+{c}_{{\rm{s}}})-{c}_{{\rm{s}}}/(1+{c}_{{\rm{s}}})\,\right]}^{-1}\,\\ \times \,\left[{\rm{l}}{\rm{n}}\displaystyle \frac{{\theta }^{\ast }}{2\,{\theta }_{{\rm{s}}}^{\ast }}+{ \mathcal F }\,\left(\displaystyle \frac{{\theta }^{\ast }}{{\theta }_{{\rm{s}}}^{\ast }}\right)\right]\,\displaystyle \frac{{{\boldsymbol{\theta }}}^{\ast }}{{\left({\theta }^{\ast }\right)}^{2}}-\displaystyle \frac{{M}_{{\rm{P}}}}{{M}_{{\rm{N}}{\rm{F}}{\rm{W}}}}\,\displaystyle \frac{{{\boldsymbol{\theta }}}^{\ast }-{{\boldsymbol{\theta }}}_{{\rm{P}}}^{\ast }}{| {{\boldsymbol{\theta }}}^{\ast }-{{\boldsymbol{\theta }}}_{{\rm{P}}}^{\ast }{| }^{2}},\end{array}\end{array}\end{eqnarray} \tag{ 32 }$$

where ${{\boldsymbol{\theta }}}_{{\rm{P}}}^{* }={{\boldsymbol{\theta }}}_{{\rm{P}}}/{\theta }_{{\rm{E}}}$. The influence of the added object is proportional to its relative mass and drops inversely proportionally to the angular separation from its position. Similarly, we may add the term in units of the angular scale length of the halo to Equation (15) and obtain the lens equation in the form

$$\begin{eqnarray}&&{\boldsymbol{y}}={\boldsymbol{x}}-4\,{\kappa }_{{\rm{s}}}\left[\mathrm{ln}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right]\,\displaystyle \frac{{\boldsymbol{x}}}{{x}^{2}}-{\kappa }_{{\rm{P}}}\,\displaystyle \frac{{\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\rm{P}}}}{| {\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\rm{P}}}{| }^{2}},\end{eqnarray} \tag{ 33 }$$

where the point-mass position x _P = θ _P D_l/r_s. The newly introduced dimensionless mass parameter κ_P can be expressed in several equivalent ways,

$$\begin{eqnarray}&&{\kappa }_{{\rm{P}}}=\displaystyle \frac{{M}_{{\rm{P}}}}{{M}_{\mathrm{NFW}}}{\left({\theta }_{{\rm{s}}}^{* }\right)}^{-2}=\displaystyle \frac{{\theta }_{\mathrm{EP}}^{2}\,{D}_{{\rm{l}}}^{2}}{{r}_{{\rm{s}}}^{2}}=\displaystyle \frac{{M}_{{\rm{P}}}}{\pi \,{r}_{{\rm{s}}}^{2}\,{{\rm{\Sigma }}}_{\mathrm{cr}}},\end{eqnarray} \tag{ 34 }$$

where we use the definition of ${\theta }_{{\rm{s}}}^{* }$ above Equations (14), (31), and (4). The first expression defines the transformation from the parameters appearing in Equation (32). The second expression identifies κ_P as the ratio between the areas (or solid angles) of the point-mass Einstein circle and the halo scale radius circle. The third expression shows that we may interpret κ_P as the convergence corresponding to the surface density of the mass M_P spread out over the area of the halo scale radius circle.

In the rest of this work we will use the lens equation in the more compact form provided by Equation (33). The equation involves four parameters: the convergences κ_s and κ_P, and the two components of the point-mass position x _P. For exploring the lens properties of the model we may always rotate our coordinate axes to position the point mass along the positive horizontal axis, so that x _P = (x_P, 0) with x_P ≥ 0. With this choice of orientation only three free parameters remain: one describing the NFW halo (κ_s) and two describing the point mass (κ_P and x_P).

Computing the determinant of the Jacobian matrix consisting of the partial derivatives ∂y_i /∂x_j of Equation (33) gives us the Jacobian:

$$\begin{eqnarray}&&\begin{array}{l}\det \,J({\boldsymbol{x}})=\left\{1-\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}^{2}}\left[\mathrm{ln}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right]-\displaystyle \frac{{\kappa }_{{\rm{P}}}}{| {\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\rm{P}}}{| }^{2}}\right\}\\ \times \left\{1+\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}^{2}}\left[\mathrm{ln}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right]+\displaystyle \frac{{\kappa }_{{\rm{P}}}}{| {\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\rm{P}}}{| }^{2}}-4\,{\kappa }_{{\rm{s}}}\,\displaystyle \frac{{ \mathcal F }(x)-1}{1-{x}^{2}}\right\}\\ +16\,{\kappa }_{{\rm{s}}}\,{\kappa }_{{\rm{P}}}\,\displaystyle \frac{| \,{\boldsymbol{x}}\times {{\boldsymbol{x}}}_{{\rm{P}}}\,{| }^{2}}{{x}^{4}| {\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\rm{P}}}{| }^{4}}\left\{\mathrm{ln}\displaystyle \frac{x}{2}+{ \mathcal F }(x)-\displaystyle \frac{{x}^{2}}{2}\,\displaystyle \frac{{ \mathcal F }(x)-1}{1-{x}^{2}}\right\},\end{array}\end{eqnarray} \tag{ 35 }$$

written here in a form independent of coordinate-frame orientation. In the orientation with the point mass on the positive horizontal axis, the norm of the cross-product ∣ x × x _P ∣ = x_P ∣x₂∣.

Far from the center of the halo (x = 0) and far from the point mass ( x = x _P) the Jacobian $\det \,J\to 1$. In the general case, with x_P ≠ 0, the Jacobian has two divergences. As shown in Section 2.3, at the center of the halo $\det \,J({\boldsymbol{x}})\,\approx 4\,{\kappa }_{{\rm{s}}}^{2}\,{\mathrm{ln}}^{2}x\to \infty$, while at the position of the point mass $\det \,J({\boldsymbol{x}})\,\approx -{\kappa }_{{\rm{P}}}^{2}/| {\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\rm{P}}}{| }^{4}\to -\infty$.

In the special case where the point mass is positioned at the center of the halo (x_P = 0) the Jacobian loses the entire final term in Equation (35), leaving the factorized part:

$$\begin{eqnarray}&&\begin{array}{l}\det \,J({\boldsymbol{x}})=\left\{1-\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}^{2}}\left[\mathrm{ln}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right]-\displaystyle \frac{{\kappa }_{{\rm{P}}}}{{x}^{2}}\right\}\\ \times \ \left\{1+\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}^{2}}\left[\mathrm{ln}\displaystyle \frac{x}{2}+{ \mathcal F }(x)\right]+\displaystyle \frac{{\kappa }_{{\rm{P}}}}{{x}^{2}}-4\,{\kappa }_{{\rm{s}}}\,\displaystyle \frac{{ \mathcal F }(x)-1}{1-{x}^{2}}\right\}.\end{array}\end{eqnarray} \tag{ 36 }$$

Here the situation is peculiar, since the two opposite divergences coincide at the origin. The stronger one due to the point mass prevails, so that $\det \,J({\boldsymbol{x}})\to -\infty$ as x → 0.

The factor in the first braces in Equation (36) yields the tangential critical curve, which is a circle ∣ x ∣ = x_PT with radius obtained by numerically solving

$$\begin{eqnarray}&&1-\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}_{\mathrm{PT}}^{2}}\left[\,\mathrm{ln}\displaystyle \frac{{x}_{\mathrm{PT}}}{2}+{ \mathcal F }({x}_{\mathrm{PT}})\right]-\displaystyle \frac{{\kappa }_{{\rm{P}}}}{{x}_{\mathrm{PT}}^{2}}=0.\end{eqnarray} \tag{ 37 }$$

The equation has a single solution for any combination of κ_s and κ_P. Substituting the critical curve ${\boldsymbol{x}}={x}_{\mathrm{PT}}(\cos \varphi ,\sin \varphi )$ with φ ∈ [0, 2π] in Equation (33) and using Equation (37), we obtain the corresponding part of the caustic:

$$\begin{eqnarray}&&{\boldsymbol{y}}={\boldsymbol{x}}\left\{1-\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}_{\mathrm{PT}}^{2}}\left[\,\mathrm{ln}\displaystyle \frac{{x}_{\mathrm{PT}}}{2}+{ \mathcal F }({x}_{\mathrm{PT}})\right]-\displaystyle \frac{{\kappa }_{{\rm{P}}}}{{x}_{\mathrm{PT}}^{2}}\right\}=(0,0).\end{eqnarray} \tag{ 38 }$$

The tangential part of the caustic remains unchanged, consisting of the single point at the origin.

Setting the second braces in Equation (36) equal to zero yields the radial critical curve equation. The solutions are circles with radius x_PR obtained by numerically solving

$$\begin{eqnarray}&&1+\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}_{{\rm{PR}}}^{2}}\left[\,{\rm{ln}}\displaystyle \frac{{x}_{{\rm{PR}}}}{2}+{ \mathcal F }({x}_{{\rm{PR}}})\right]+\displaystyle \frac{{\kappa }_{{\rm{P}}}}{{x}_{{\rm{PR}}}^{2}}-4\,{\kappa }_{{\rm{s}}}\,\displaystyle \frac{{ \mathcal F }({x}_{{\rm{PR}}})-1}{1-{x}_{{\rm{PR}}}^{2}}=0.\end{eqnarray} \tag{ 39 }$$

However, here the number of solutions for a given value of κ_s depends on the value of κ_P. Substituting the radial critical curve ${\boldsymbol{x}}={x}_{\mathrm{PR}}(\cos \varphi ,\sin \varphi )$ with φ ∈ [0, 2π] in Equation (33) and using Equation (39), we obtain the corresponding part of the caustic:

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{y}} & = & {\boldsymbol{x}}\left\{1-\displaystyle \frac{4\,{\kappa }_{{\rm{s}}}}{{x}_{\mathrm{PR}}^{2}}\left[\mathrm{ln}\displaystyle \frac{{x}_{\mathrm{PR}}}{2}+{ \mathcal F }({x}_{\mathrm{PR}})\right]-\displaystyle \frac{{\kappa }_{{\rm{P}}}}{{x}_{\mathrm{PR}}^{2}}\right\}\\ & = & -2\left[2\,{\kappa }_{{\rm{s}}}\,\displaystyle \frac{{ \mathcal F }({x}_{\mathrm{PR}})-1}{1-{x}_{\mathrm{PR}}^{2}}-1\right]{x}_{\mathrm{PR}}\left(\cos \varphi ,\sin \varphi \right),\end{array}\end{eqnarray} \tag{ 40 }$$

where the expression in square brackets is equal to κ(x_PR) − 1, which is positive. For each radial critical curve, the corresponding part of the caustic is thus a circle with radius

$$\begin{eqnarray}&&{y}_{\mathrm{PR}}=2\,{x}_{\mathrm{PR}}\,[\kappa ({x}_{\mathrm{PR}})-1].\end{eqnarray} \tag{ 41 }$$

We present the structure of the critical curves and caustics in Figure 4 as a function of κ_P for the cluster with κ_s ≈ 0.239035 chosen for illustration in Figure 2. The radius of the tangential critical curve x_PT is a simple monotonically increasing function of κ_P, which starts on the vertical axis at x_T, the radius of the NFW tangential critical curve. This growth is just a simple consequence of the increasing mass at the origin.

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More interesting are the radial critical curves. For low values of κ_P Equation (39) has two solutions. The larger one (x_PR1) starts at x_R, the radius of the NFW radial critical curve, and decreases with κ_P. The smaller one (x_PR2) starts at 0 and increases with κ_P. As κ_P grows, the two curves approach each other until they merge at a critical value κ_P = κ_PC ≈ 2.714 × 10⁻⁴ (for our choice of κ_s). For supercritical κ_P > κ_PC Equation (39) has no solution, i.e., there are no radial critical curves.

The two corresponding radial components of the caustic reflect the behavior of the critical curves, with y_PR1 starting at y_R, the radius of the NFW radial caustic, and y_PR2 starting at 0. Even though the larger radial caustic grows with κ_P, the two caustic circles approach each other faster than the corresponding critical-curve circles and vanish beyond κ_PC.

The critical curves and caustics in the two regimes are illustrated by the panels in the bottom row of Figure 5. The two left columns show the Jacobian plot with critical curves and the total magnification map with caustics in the subcritical case (for κ_P = 10⁻⁴), and the two right columns show the corresponding plots in the supercritical case (for κ_P = 10⁻³). The notation is the same as that in Figure 2, with the additional cyan circle indicating the position and Einstein ring of the point mass in units of the angular scale radius, θ_EP D_l/r_s.

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View figure set (20 panels)

Tarball of figure set images

For the lower-mass case with κ_P = 10⁻⁴, the Jacobian plot resembles the plot from Figure 2, except the region near the origin. In this example the additional radial critical curve is marginally larger than the point-mass Einstein radius. Within it the point mass dominates and det J < 0. Between the two radial critical curves det J > 0, between the outer radial and tangential critical curves det J < 0, and outside the tangential critical curve det J > 0. Instead of the single radial caustic of the NFW halo seen in Figure 2, there are two circular radial caustics, with the outer and inner ones corresponding to the outer and inner radial critical curves, respectively. The total magnification diverges at both caustics from the side of the annular region between them.

As κ_P increases, so does the negative-Jacobian region around the origin dominated by the point mass, while the outer radial critical curve and the enclosed positive-Jacobian annulus shrink. The outer radial caustic grows, but the inner one grows faster. At κ_P = κ_PC the two radial critical curves (as well as the two radial caustics) merge, causing the positive-Jacobian annulus to vanish. This situation is illustrated in the central column of the online Figure 5.2. The peculiar properties of the single merged radial critical curve and caustic are described in Appendix B. For larger κ_P the radial critical curves and caustics disappear.

For the higher-mass case with κ_P = 10⁻³, only the tangential critical curve remains and the negative-Jacobian region around the origin extends all the way to it, as shown in Figure 5. The caustic is reduced to the single point at the origin, while the magnification shows a brighter central region roughly the size of the radial caustic that has vanished.

Comparison with Figure 2 illustrates how dramatically a single added object may change the lensing properties of the NFW halo. The critical value κ_PC ≈ 2.714 × 10⁻⁴ for our halo parameter choice given by Equation (26) can be used in Equation (34) to compute the corresponding critical mass ratio M_P/M_NFW ≈ 3.91 × 10⁻⁴. Even though such a relative mass may seem low, its gravitational field is strong enough to destroy the radial critical curve and caustic of the NFW halo in which it is embedded.

Similar behavior has been found previously for a central point mass embedded in different axially symmetric mass distributions, such as in a cored isothermal sphere (Mao et al. 2001) or in a Plummer model (Werner & Evans 2006). In either of these models there is a critical mass of the central point, above which the lens has no radial critical curves. Nevertheless, this behavior is not ubiquitous: a central point mass embedded in a singular isothermal sphere has no radial critical curves, irrespective of its mass (Mao & Witt 2012).

When positioning the point mass at increasing distances x_P from the halo center, the critical curve undergoes a sequence of transitions in which its loops connect, disconnect, appear, or vanish. These transitions are accompanied by underlying metamorphoses of the caustic (Schneider et al. 1992). The curves start from the central configurations described in Section 3.3 and end with separate critical curves and caustics of an NFW halo and a distant point-mass lens. In Section 3.4.1 we describe the overall changes of the critical curves and caustics. In Section 3.4.2 we explore the parameter space of the point mass and identify boundaries at which the critical-curve transitions occur. We illustrate the details of several transitions and transition sequences in Section 3.4.3.

3.4.1. Critical-curve and Caustic Gallery

3.4.2. Boundaries in Point-mass Parameter Space

We track the changing structure of critical curves and caustics in the point-mass parameter space region defined by the intervals κ_P ∈ [0, 0.0035] and x_P ∈ [0, 0.4]. Within this space we identify boundaries at which the overall topology of the critical curve changes in transitions such as those seen in Section 3.4.1. These involve situations in which critical-curve loops connect/disconnect, or appear/vanish. In addition, we identify situations in which loops shrink to a point, or touch without connecting. We choose the upper limits on the parameter intervals empirically: the x_P limit lies above the final transition in the studied κ_P interval; for values above the κ_P limit there are no further changes to the structure of the boundaries.

We note that straightforward use of the first expression for ${ \mathcal F }(x)$ from Equation (6) leads to numerical instabilities for x ≪ 1, close to the origin of the image plane. For tracking the changing structure of critical curves in this region it is necessary to use the exact and stable expressions given by Equations (A5) and (A6), as shown in Appendix A.1.

Finding the boundaries involves several steps. We start by computing the critical curves and caustics on a rough parameter grid, inspecting the results to identify pairs of neighboring grid points with different topologies of the critical curve and different caustic structures. For each such pair we proceed by interval halving to pinpoint the intersection of the boundary between the points with the respective grid line. Where necessary, we add more points between these grid line intersections to obtain smoother boundaries. In the emerging boundary plot we check all mass parameter intervals with different vertical sequences of boundaries to make sure the changes across the boundaries agree with the characteristics of the corresponding metamorphoses (such as characteristic changes in the numbers of loops, or changes in the numbers of cusps on caustic loops). Finally, we check the continuity of the critical curves and caustics close to the axes, which correspond to analytically studied axisymmetric lenses: in the zero-mass limit close to the vertical axis (comparison with Section 2.3 and Figure 2), and in the zero-displacement limit close to the horizontal axis (comparison with Section 3.3 and the online Figure 5.2).

The mapped boundaries are presented in an x_P versus κ_P plot in Figure 6. The left panel shows the full explored parameter space; the right panels show two expanded regions in more detail: the bottom right for low masses close to the origin, and the top right for intermediate positions. For better interpretation of the results, the red crosses indicate the parameter combinations for the critical curves and caustics shown in Figure 5; the critical curves and caustics for the full set of red and black crosses are shown in the figure set available with the online version of Figure 5. For additional orientation, we mark the NFW halo radial and tangential critical curve radii (x_R and x_T, respectively) on the vertical axis, and the critical mass parameter κ_PC on the horizontal axis.

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The colors of the boundaries indicate the type and multiplicity of the associated caustic metamorphosis. Single metamorphoses occur in the source plane at a point on the symmetry axis passing through the point mass and the halo center, while two same simultaneous metamorphoses occur at points symmetrically offset from the axis. Cyan indicates a single beak-to-beak metamorphosis, occurring here at two boundaries: from x_R on the vertical axis downward to κ_PC on the horizontal axis and from x_T on the vertical axis upward. Blue indicates two simultaneous beak-to-beak metamorphoses, occurring here at three boundaries: from the origin up to the boundary spike just under κ_P = 0.001 (plotted by a dashed line to reveal the closely adjacent orange lips boundary), from x_R on the vertical axis upward to the point marked by the circle, and from x_T on the vertical axis first downward, then upward to the boundary spike just under κ_P = 0.003.

Orange indicates a single lips metamorphosis, occurring here at three boundaries: from the origin along the entire horizontal axis (corresponding to x_P = 0), from the origin upward to the boundary spike just under κ_P = 0.001 (just above the closely adjacent dashed blue boundary), and from κ_PC on the horizontal axis upward to the same boundary spike. Dark orange indicates two simultaneous lips metamorphoses, occurring here at one boundary: from the point marked by the circle upward to the boundary spike just under κ_P = 0.003. Green indicates two simultaneous elliptic umbilic metamorphoses, occurring here at one boundary: from the unit convergence radius x₀ ≈ 0.0936 on the vertical axis upward to the point marked by the circle. Violet indicates two simultaneous hyperbolic umbilic metamorphoses, occurring here at one boundary: from the point marked by the circle upward. Finally, the point marked by the black circle indicates two simultaneous parabolic umbilics. Four boundaries meet at this point, all of them corresponding to two simultaneous metamorphoses. In clockwise order from lower left these are beak-to-beak, elliptic umbilic, hyperbolic umbilic, and lips boundaries.

In addition to the metamorphoses for which we plot the parameter space boundaries in Figure 6, the caustic also undergoes the swallow-tail metamorphosis. In it the caustic develops a local "twist" with a self-crossing point and two additional cusps. However, unlike the plotted metamorphoses, the swallow tail has no effect on the critical-curve topology. It occurs here always simultaneously in twos, always closely adjacent to simultaneous beak-to-beak boundaries. One such simultaneous swallow-tail pair occurs between the beak-to-beak and lips boundaries extending in Figure 6 from the origin upward to the boundary spike just under κ_P = 0.001; two such simultaneous pairs occur in close succession just beneath the beak-to-beak boundary extending from x_R on the vertical axis upward to the point marked by the circle.

Before exploring individual transition sequences, metamorphoses, and boundaries in more detail in Section 3.4.3, it is worth pointing out that there is an astounding richness of structures, transitions, and lensing regimes between them, which is quite unexpected in such a simple lens model.

3.4.3. Details of Specific Transitions and Transition Sequences

The first interesting transition occurs along the horizontal axis of the parameter space plot in Figure 6, i.e., for an arbitrarily small displacement of the point mass from the halo center. In such a situation the two opposite Jacobian divergences described in Section 3.2 no longer coincide, and the Jacobian must cross zero at some point between them. This results in a small new critical-curve loop encircling the weaker $\det \,J({\boldsymbol{x}})\to \infty$ divergence at the origin of the image plane. The caustic undergoes a lips metamorphosis, in which a new two-cusped loop forms beyond the point-mass position. This particular lips metamorphosis is unusual in its appearance at infinity rather than at a finite distance from the origin of the source plane. A further increase in the point-mass displacement brings the new caustic loop rapidly toward the origin.

Inspecting the changes along the left column of crosses in Figure 6 corresponding to the subcritical κ_P = 10⁻⁴ example, we see that the critical curve undergoes a total of eight transitions as we vary the point-mass position from the halo center to an asymptotic distance. Only a few of these can be identified in the gallery in Figure 5 and the figure set available with its online version, as described in Section 3.4.1. In order to provide a more comprehensive overview we present the details of three transition sequences below.

After displacing the point mass away from the halo center, the first encountered boundaries are the adjacent dashed blue and orange lines corresponding to the beak-to-beak and lips metamorphoses, respectively. These are illustrated by the critical-curve and caustic details in Figure 7, with the entire sequence occurring between x_P = 0 and x_P = 0.01 (online Figures 5.2 and 5.3, respectively). The additional third column shows the caustic detail from the second column with the horizontal scale expanded 10× to reveal the caustic structure.

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The critical-curve details from bottom to top reveal the small positive-Jacobian loop around the origin expanding and approaching the perturbed inner radial NFW critical curve, connecting with it symmetrically at two points in the third row for x_P ≈ 0.00617325, leaving a small negative-Jacobian loop, which shrinks and vanishes between the top two rows. The caustic plot in the bottom row shows the small two-cusped loop approaching the perturbed inner radial NFW caustic from outside. The two caustic loops overlap in the second row. In the third row two simultaneous beak-to-beak metamorphoses occur, in which the outer part of the two-cusped loop touches the perturbed inner radial NFW caustic at two points lying symmetrically above and below the horizontal axis. This pairwise metamorphosis leads to a thin two-cusped crescent detaching from the caustic on the outer side. The small self-crossing features on the larger caustic in the fourth row vanish by the fifth row in two simultaneous swallow-tail metamorphoses. By the sixth row, the two-cusped crescent vanishes in a lips metamorphosis, leaving a barely noticeable higher-magnification trace in the magnification map.

The detachment of the two small critical-curve loops from the perturbed radial NFW critical curve that occurs between the second and the third row of Figure 5 corresponds to a more complicated sequence of metamorphoses on the caustic, as shown in Figure 8. The caustic detail presented in the right panels corresponds to the critical-curve detail below the point mass in the left panels. In the bottom row the caustic detail has a single cusp. By the second row it has undergone a swallow-tail metamorphosis, which adds a self-intersection and two cusps. A similar swallow-tail metamorphosis occurs on the caustic above the original cusp before the third row, adding a similar smaller feature. In the third row at x_P ≈ 0.08075 two simultaneous beak-to-beak metamorphoses occur, in which the facing cusps of the two swallow-tail features touch and reconnect. In the fourth row the caustic detail consists of a smooth fold and a detached three-cusped loop.

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The transition following the detachment of the two loops also occurs between the second and the third row of Figure 5. The sequence shown in Figure 9 corresponds to the green elliptic umbilic boundary in Figure 6. The two small critical-curve loops shrink to two points at x_P ≈ 0.095844 in the second row, before expanding again to two loops. A similar effect can be seen in the caustic plots. The two small three-cusped loops shrink to two points at the elliptic umbilic metamorphosis in the second row, before expanding again to three-cusped loops.

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Along the right column of crosses in Figure 6, which corresponds to the supercritical κ_P = 10⁻³ example, the sequence is simpler, with the critical curve undergoing only five transitions as the point-mass position varies from the halo center to an asymptotic distance. All the transitions occurring from the second row to the top of Figure 5 are the same as those for the subcritical example. The development from the first to the second row is much simpler than in the subcritical case, with the small positive-Jacobian critical-curve loop around the origin directly expanding to form the perturbed NFW radial critical curve. The two-cusped caustic loop approaches the origin from the right, forming the crescent-like caustic seen in the first row, and extending further to form the perturbed NFW radial caustic.

However, for supercritical mass parameters κ_P ≲ 9.86 × 10⁻⁴ the sequence of transitions close to the halo center differs markedly from either of the previous examples. For illustration, we demonstrate in Figure 10 the transition across the orange, dashed blue, and orange boundaries along the κ_P = 8 × 10⁻⁴ grid line in the bottom right panel of Figure 6. These correspond to a sequence of lips, two simultaneous beak-to-beak, and lips metamorphoses. The additional third column in Figure 10 shows the caustic detail from the second column with the horizontal scale expanded 6× to reveal the caustic structure.

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Between the first and the second row, a second small positive-Jacobian critical-curve loop appears to the left of the first one. The two loops expand and connect together symmetrically at two points in the fourth row for x_P ≈ 0.0213954. The detached inner small negative-Jacobian loop shrinks and vanishes between the top two rows. At larger x_P the remaining positive-Jacobian loop forms the perturbed radial NFW critical curve. In the bottom row the caustic detail shows the small two-cusped loop that arrived from the right to the brighter ring in the magnification map. In the second row the caustic has a second thin two-cusped loop inside the first loop that appeared along the bright ring in a lips metamorphosis. The two caustic loops overlap in the third row. In the fourth row two beak-to-beak metamorphoses occur, in which the outer part of the first loop reconnects symmetrically at two points with the inner part of the second loop. In the fifth row the caustic displays a disconnected thin inner two-cusped loop and small self-crossing features on the outer caustic loop. The features vanish in simultaneous swallow-tail metamorphoses, followed by the vanishing of the small two-cusped loop in a lips metamorphosis. At larger x_P, the two-cusped caustic loop seen in the top row forms the perturbed radial NFW caustic.

For mass parameters κ_P ≳ 1.323 × 10⁻³, higher than those in the presented examples, the green boundary in Figure 6 corresponding to the elliptic umbilic continues as the violet hyperbolic umbilic boundary. We illustrate in Figure 11 the transition across this boundary along the κ_P = 2 × 10⁻³ grid line in Figure 6. The two small critical-curve loops touch the perturbed NFW radial critical curve at x_P ≈ 0.1317 in the second row, before detaching again, as seen in the third row. In the bottom row the caustic detail shows a single cusp on the larger caustic loop and a closely adjacent small two-cusped loop. At the hyperbolic umbilic metamorphosis in the second row, both parts of the caustic touch at a blunt angular point. In this metamorphosis a cusp gets transferred from one part of the caustic to another. This can be seen in the third row, where the detached smaller loop has three cusps while there is no cusp on the larger perturbed NFW radial caustic.

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For completeness, we mention the parabolic umbilic that occurs for κ_P ≈ 1.323 × 10⁻³ at x_P ≈ 0.1202. At this point, which is marked by the black circle in Figure 6, boundaries corresponding to four different pairwise metamorphoses meet, namely, elliptic umbilic, hyperbolic umbilic, lips, and beak-to-beak. When passing through this point in the parameter plot in the sense of increasing x_P, two small loops detach from cusp-like points on the perturbed radial NFW critical curve, starting as points and expanding as in the elliptic umbilic. During the underlying caustic metamorphosis the two cusps on the perturbed radial NFW caustic disappear as two small three-cusped caustic loops detach from them. These caustic loops also start as points and increase in size. Varying the point-mass parameters in the vicinity of the parabolic umbilic point leads to a variety of changes to the local structure of the caustic (e.g., Godwin 1971; Poston & Stewart 1978).

In Section 3.3 we study the structure of critical curves and caustics for centrally positioned point masses, and demonstrate the distinction between the sub- and supercritical cases. If we look at the transition sequences for different values of κ_P in Figure 6, we see that the boundary intersections and limiting points form a finer subdivision into κ_P intervals, each with a specific transition sequence. The approximate values separating these intervals are

$$\begin{eqnarray}&&\begin{array}{r}\begin{array}{c}\begin{array}{l}{\kappa }_{{\rm{P}}}\in \left\{0,1.62\times {10}^{-4},2.71\times {10}^{-4},9.86\times {10}^{-4}\right.,\\ \,\,\,\left.1.32\times {10}^{-3},2.26\times {10}^{-3},2.97\times {10}^{-3}\right\}.\end{array}\end{array}\end{array}\end{eqnarray} \tag{ 42 }$$

To be precise, the first nonzero value actually should be replaced by two very close κ_P values, because the cyan boundary in the bottom right panel of Figure 6 intersects the orange before the adjacent dashed blue boundary. The eight different sequences of critical-curve transitions can be read off Figure 6, aided by the gallery in Figure 5 and the figure set available with its online version, as well as the transition sequences described in this section.

Note that the κ_P = 10⁻⁴ subcritical sequence with all the described transitions lies in the first interval defined by the values in Equation (42). Hence, any lower-mass point will have the same sequence of critical-curve transitions. Similarly, all point masses with κ_P ≳ 2.97 × 10⁻³ share the same simple sequence, in which the only transition between the appearance of the small critical-curve loop around the origin and the final detachment of the point-mass loop from the perturbed tangential NFW critical curve is the hyperbolic umbilic. In this umbilic transition the perturbed radial and tangential NFW critical curves merely touch at two points. In the underlying caustic metamorphosis, the central four-cusped loop (similar to the caustic in the third row of Figure 5) extends and touches the two cusps of the crescent-like loop, resulting in a six-cusped central caustic and a smooth perturbed radial NFW caustic.

We point out that the interpretation of the parameter space structure in Figure 6 presented above focuses on vertical transitions across the boundaries, which corresponds to placing a fixed-mass point at different positions in the halo. One may interpret the figure just as well by following transitions in the horizontal sense, corresponding to placing points of different masses at a fixed position in the halo. Such explorations reveal that the structure of the critical curve is more sensitive to the mass of the point when it is placed in certain regions of the halo, e.g., close to its center or in a part of the annular region between the radial and tangential NFW critical curves.

For x ≪ 1 the four leading orders of the expansion of ${ \mathcal F }(x)$ can be written as

$$\begin{eqnarray}&&{ \mathcal F }(x)=-\mathrm{ln}\displaystyle \frac{x}{2}-\displaystyle \frac{{x}^{2}}{2}\,\mathrm{ln}\displaystyle \frac{x}{2}-\displaystyle \frac{{x}^{2}}{4}+{ \mathcal O }({x}^{4}\,\mathrm{ln}x),\end{eqnarray} \tag{ A1 }$$

which yields the two leading orders of the expression

$$\begin{eqnarray}&&\mathrm{ln}\displaystyle \frac{x}{2}+{ \mathcal F }(x)=-\displaystyle \frac{{x}^{2}}{2}\,\mathrm{ln}\displaystyle \frac{x}{2}-\displaystyle \frac{{x}^{2}}{4}+{ \mathcal O }({x}^{4}\,\mathrm{ln}x),\end{eqnarray} \tag{ A2 }$$

in which the logarithmic divergence of ${ \mathcal F }(x)$ is canceled and the combination shrinks to zero as ${x}^{2}\,\mathrm{ln}x$. By substituting Equation (A1) in Equation (5) we get the four leading orders of the NFW convergence expansion,

$$\begin{eqnarray}\begin{array}{r}\begin{array}{c}\begin{array}{rcl}\kappa (x) & = & -2\,{\kappa }_{{\rm{s}}}\phantom{\rule{thickmathspace}{0ex}}\left(\,{\rm{l}}{\rm{n}}\displaystyle \frac{x}{2}+1+\displaystyle \frac{3}{2}\,{x}^{2}\,{\rm{l}}{\rm{n}}x+\displaystyle \frac{5}{4}\,{x}^{2}\,\right)+{ \mathcal O }({x}^{4}\,{\rm{l}}{\rm{n}}x),\end{array}\end{array}\end{array}\end{eqnarray} \tag{ A3 }$$

showing its logarithmic divergence. The Jacobian behavior close to the origin can be obtained by substituting Equation (A1) in Equation (17), yielding the three leading orders of its expansion:

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}det\,J({\boldsymbol{x}}) & = & 4\,{\kappa }_{{\rm{s}}}^{2}\,{{\rm{l}}{\rm{n}}}^{2}\displaystyle \frac{x}{2}+4\,{\kappa }_{{\rm{s}}}(1+2\,{\kappa }_{{\rm{s}}}){\rm{l}}{\rm{n}}\displaystyle \frac{x}{2}\\ & & \,+\,1+4\,{\kappa }_{{\rm{s}}}+3\,{\kappa }_{{\rm{s}}}^{2}+{ \mathcal O }({x}^{2}\,{\rm{l}}{\rm{n}}x).\end{array}\end{array}\end{eqnarray} \tag{ A4 }$$

Getting higher-order terms would require a higher-order expansion in Equation (A1). This result reveals the ${\mathrm{ln}}^{2}x$ divergence of the NFW Jacobian at the origin.

For computing the critical curves and caustics we need exact analytic expressions rather than series expansions. We first express the top row of Equation (6) in an equivalent form,

$$\begin{eqnarray}&&{ \mathcal F }(x)=\displaystyle \frac{1}{\sqrt{1-{x}^{2}}}\,\mathrm{ln}\displaystyle \frac{1+\sqrt{1-{x}^{2}}}{x},\end{eqnarray} \tag{ A5 }$$

which can then be combined with $\mathrm{ln}(x/2)$ to cancel the divergence at the origin,

$$\begin{eqnarray}\begin{array}{c}\begin{array}{lll}{\rm{l}}{\rm{n}}\displaystyle \frac{x}{2}+{ \mathcal F }(x) & = & \displaystyle \frac{-{x}^{2}}{1-{x}^{2}+\sqrt{1-{x}^{2}}}\,{\rm{l}}{\rm{n}}\displaystyle \frac{x}{2}\\ & & +\,\displaystyle \frac{1}{\sqrt{1-{x}^{2}}}\,{\rm{l}}{\rm{n}}\displaystyle \frac{1+\sqrt{1-{x}^{2}}}{2}.\end{array}\end{array}\end{eqnarray} \tag{ A6 }$$

The first term reveals the leading order of the expansion at the origin as seen in Equation (A2), while the second term contributes to higher orders starting from ${ \mathcal O }({x}^{2})$. Both Equations (A5) and (A6) are valid for any x < 1. These numerically stable expressions can be used in Equation (35) to compute the Jacobian and the critical curve, and in Equation (33) to compute the caustic.

At the halo scale radius the x < 1 expression in Equation (6) transitions smoothly to the x > 1 expression. For illustration, we provide the three leading terms of the expansion valid on both sides of x = 1:

$$\begin{eqnarray}&&{ \mathcal F }(x)=1-\displaystyle \frac{2}{3}(x-1)+\displaystyle \frac{7}{15}{\left(x-1\right)}^{2}+{ \mathcal O }({\left(x-1\right)}^{3}).\end{eqnarray} \tag{ A7 }$$

Using this result, we may expand the lens equation combination:

$$\begin{eqnarray}&&\begin{array}{r}{\rm{l}}{\rm{n}}\displaystyle \frac{x}{2}+{ \mathcal F }(x)=1-{\rm{l}}{\rm{n}}\,2+\displaystyle \frac{x-1}{3}-\displaystyle \frac{{\left(x-1\right)}^{2}}{30}+{ \mathcal O }({\left(x-1\right)}^{3}).\end{array}\end{eqnarray} \tag{ A8 }$$

Substituting Equation (A7) in Equation (5) yields the two leading orders of the NFW convergence expansion:

$$\begin{eqnarray}&&\kappa (x)={\kappa }_{{\rm{s}}}\left[\,\displaystyle \frac{2}{3}-\displaystyle \frac{4}{5}(x-1)\right]+{ \mathcal O }({\left(x-1\right)}^{2}).\end{eqnarray} \tag{ A9 }$$

Getting higher-order terms would require a higher-order expansion in Equation (A7).