Fermat’s least-time principle and the embedded transparent lens

We present a simplified version of the lowest-order embedded point mass gravitational lens theory and then make the extension of this theory to any embedded transparent lens. Embedding a lens effectively reduces the gravitational potential’s range, i.e., partially shields the lensing potential because the lens mass is made a contributor to the mean mass density of the Universe and not simply superimposed upon it. We give the time-delay function for the embedded point mass lens from which we can derive the simplified lens equation by applying Fermat’s least-time principle. Even though rigorous derivations are only made for the point mass in a flat background, the generalization of the lens equation to lowest order for any distributed lens in any homogeneous background is obvious. We find from this simplified theory that embedding can introduce corrections above the few percent level in weak lensing shears caused by large clusters but only at large impacts. The potential part of the time delay is also affected in strong lensing at the few percent level. Additionally we again confirm that the presence of a cosmological constant alters the gravitational deflection of passing photons.

Figure 1

A photon travels left to right entering a Kottler void at $r=r_1$, $\varphi =\pi -{\tilde{\varphi }}_1$ and returns to the FLRW dust at $r=r_1+\Delta r$, $\varphi ={\tilde{\varphi }}_1+\Delta \varphi$. The photon’s orbit has been chosen to be symmetric in Kottler coordinates about the point of closest approach $r=r_0$, $\varphi =\pi /2$. Because of the cosmological expansion, $\Delta r>0$. The slope of the photon’s comoving trajectory in the $x$-$y$ plane is ${\xi }_1$ when incoming and ${\xi }_1+\alpha$ after exiting. The deflection angle $\alpha$ [see Eq. (4)] is negative by convention and is precisely and uniquely defined by this figure. Expressions for $r_1$, $\Delta r$, ${\xi }_1$, and $\Delta \varphi$ as functions of the two impact parameters, $r_0$ and ${\tilde{\varphi }}_1$, can be found in Refs. 1, 2, 3. The minimum radial coordinate $r_0$ as a function of ${\tilde{\varphi }}_1$ was given in Eq. (A1) of Ref. 4 and allowed $r_1$, $\alpha$, etc., to be given as functions of ${\tilde{\varphi }}_1$ alone.

Figure 2

Lensing geometry of an embedded point mass lens. A photon travels from a source located at angular position ${\theta }_S$, a comoving distance ${\chi }_s$ from the observer, and enters a Kottler void of the comoving radius ${\chi }_b$ centered at the comoving distance ${\chi }_d$ from the observer. The photon is deflected by an angle $\alpha$ ($<0$) and returns to the FLRW dust on its way to the observer, where it appears at angle ${\theta }_I$. The maximum image angle ${\theta }_I$ is ${\theta }_M\equiv {\chi }_b/{\chi }_d$ and occurs when the light ray grazes the Kottler void. The significant simplification of the lens theory given in this paper occurs because we have been able to replace ${\tilde{\varphi }}_1$ of Fig. 1 with the image angle ${\theta }_I$ shown here by using Eq. (3).

Figure 3

(Left) Embedding corrections to the image shear $\gamma$ for a ${10}^{15}{M}_{\odot }$ cluster lens. The source and lens redshifts are $z_s=1.0$ and $z_d=0.5$, respectively. The solid blue curve is the shear, $\gamma$, and the dashed red curve is the fractional difference in the image shear, $\Delta \gamma /\gamma$, caused by embedding. Both are plotted as functions of ${\theta }_I/{\theta }_M$ where the Einstein ring radius is ${\theta }_E={52}^{\prime \prime }.6$, the void radius is ${\theta }_M={32}^{\prime }.8$, and ${\theta }_E/{\theta }_M=0.027$ is the vertical dotted line. (Right) The difference in the embedded time delay and the conventional Schwarzschild time delay, $\Delta T-\Delta T_{\mathrm{Sch}}$, computed using Eq. (7) and the Schwarzschild equivalent for the primary and secondary images, is plotted as a function of source position angle ${\theta }_S/{\theta }_M$. The time-delay difference is given in months for four point mass lenses, $m={10}^{15}{M}_{\odot }$, ${10}^{14}{M}_{\odot }$, ${10}^{13}{M}_{\odot }$, and ${10}^{12}{M}_{\odot }$ (respectively, top to bottom curves). The small vertical ticks on each curve are at respective ${\theta }_E/{\theta }_M$ values.