Arrival time, geometry, potential, and cosmology
The useful time-delay formula is
\[
\Delta t_{ij}
=
\frac{D_{\Delta t}}{c}
\left[
\phi(\mathbf{\theta}_i,\mathbf{\beta})
-
\phi(\mathbf{\theta}_j,\mathbf{\beta})
\right].
\]
This is not an arbitrary definition. It comes from adding two physical delays: the extra geometrical path length of the bent ray, and the gravitational, or Shapiro, delay caused by the lens potential.
Thin-lens geometry for the geometrical time delay
The observer, lens plane, and source plane are shown with the angular image position theta and source position beta.
lens plane
source plane
O
ray at lens
source
optical axis
ξ = D_l θ
η = D_s β
β
θ
D_l
D_ls
D_s
incoming direction defines image angle θ
The orange broken ray is the observed lensed path. The cyan dashed line is the undeflected reference path to the same source. Expanding the two path lengths to second order in small angles gives the geometrical delay proportional to \(|\mathbf{\theta}-\mathbf{\beta}|^2\).
1. The geometrical delay
Use the thin-lens geometry. The observer is at \(O\), the ray crosses the lens plane at physical transverse position \(\mathbf{\xi}=D_l\mathbf{\theta}\), and the source is at \(\mathbf{\eta}=D_s\mathbf{\beta}\). In the small-angle limit, the bent path length is approximately
\[
\sqrt{D_l^2+|\mathbf{\xi}|^2}
+
\sqrt{D_{ls}^2+|\mathbf{\eta}-\mathbf{\xi}|^2}.
\]
Expand each square root to second order and subtract the undeflected source path,
\[
\sqrt{D_s^2+|\mathbf{\eta}|^2}.
\]
The expansion being used is
\[
\sqrt{D^2+|\mathbf{x}|^2}
=
D\sqrt{1+\frac{|\mathbf{x}|^2}{D^2}}
\simeq
D+\frac{|\mathbf{x}|^2}{2D},
\]
so the path-length difference is
\[
\begin{aligned}
\Delta l_{\rm geom}
&=
\left(
D_l+\frac{|\mathbf{\xi}|^2}{2D_l}
+
D_{ls}+\frac{|\mathbf{\eta}-\mathbf{\xi}|^2}{2D_{ls}}
\right)
-
\left(
D_s+\frac{|\mathbf{\eta}|^2}{2D_s}
\right)\\
&=
\frac{|\mathbf{\xi}|^2}{2D_l}
+
\frac{|\mathbf{\eta}-\mathbf{\xi}|^2}{2D_{ls}}
-
\frac{|\mathbf{\eta}|^2}{2D_s}.
\end{aligned}
\]
In this Euclidean thin-lens sketch the constant terms cancel because \(D_s=D_l+D_{ls}\). Substituting \(\mathbf{\xi}=D_l\mathbf{\theta}\) and \(\mathbf{\eta}=D_s\mathbf{\beta}\) then gives
\[
\begin{aligned}
\Delta l_{\rm geom}
&=
\frac{D_l}{2}|\mathbf{\theta}|^2
+
\frac{|D_s\mathbf{\beta}-D_l\mathbf{\theta}|^2}{2D_{ls}}
-
\frac{D_s}{2}|\mathbf{\beta}|^2\\
&=
\frac{D_lD_s}{2D_{ls}}
\left(
|\mathbf{\theta}|^2
-
2\mathbf{\theta}\cdot\mathbf{\beta}
+
|\mathbf{\beta}|^2
\right)\\
&=
\frac{D_lD_s}{2D_{ls}}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2.
\end{aligned}
\]
In an expanding universe the same standard lensing result is written with angular-diameter distances.
The excess length becomes
\[
\Delta l_{\rm geom}
=
\frac{D_lD_s}{2D_{ls}}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2.
\]
Converting length to time and including the cosmological time dilation at the lens redshift gives
\[
t_{\rm geom}
=
\frac{1+z_l}{c}
\frac{D_lD_s}{D_{ls}}
\frac{1}{2}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2.
\]
2. The potential delay
Light also spends extra coordinate time passing through the gravitational potential of the lens. This is the Shapiro delay. In the weak-field limit, the potential part of the propagation time is proportional to the line-of-sight integral of the Newtonian potential \(\Phi\):
\[
t_{\rm Shapiro}
=
-
\frac{2}{c^3}
\int \Phi(D_l\mathbf{\theta},\ell)\,\mathrm{d}\ell .
\]
The thin-lens approximation packages the same line-of-sight integral into the two-dimensional lensing potential. With the usual lensing normalization,
\[
\psi(\mathbf{\theta})
=
\frac{2}{c^2}
\frac{D_{ls}}{D_lD_s}
\int \Phi(D_l\mathbf{\theta},\ell)\,\mathrm{d}\ell .
\]
Equivalently,
\[
\int \Phi(D_l\mathbf{\theta},\ell)\,\mathrm{d}\ell
=
\frac{c^2}{2}
\frac{D_lD_s}{D_{ls}}
\psi(\mathbf{\theta}).
\]
Substituting this into the Shapiro-delay expression gives the potential part of the arrival time, and the lens-redshift factor \((1+z_l)\) converts it to the observer's clock:
\[
t_{\rm pot}
=
-
\frac{1+z_l}{c}
\frac{D_lD_s}{D_{ls}}
\psi(\mathbf{\theta}).
\]
The important point is that Shapiro delay and lensing potential are not independent ingredients: they are two ways of using the same projected gravitational potential. The minus sign follows the standard Fermat-potential convention; additive constants in \(\psi\) only shift all arrival times by the same amount and do not change observable time-delay differences.
3. Why the product appears
Add the two contributions:
\[
t(\mathbf{\theta},\mathbf{\beta})
=
\frac{1+z_l}{c}
\frac{D_lD_s}{D_{ls}}
\left[
\frac{1}{2}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2
-
\psi(\mathbf{\theta})
\right].
\]
The prefactor has dimensions of distance. The bracket is dimensionless and depends on the lens model and the image/source positions. Therefore we define
\[
D_{\Delta t}
\equiv
(1+z_l)\frac{D_lD_s}{D_{ls}},
\qquad
\phi(\mathbf{\theta},\mathbf{\beta})
\equiv
\frac{1}{2}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2
-
\psi(\mathbf{\theta}).
\]
Then
\[
t(\mathbf{\theta},\mathbf{\beta})
=
\frac{D_{\Delta t}}{c}
\phi(\mathbf{\theta},\mathbf{\beta}).
\]
Only differences between images are observable, because the zero point of arrival time is arbitrary:
\[
\Delta t_{ij}
=
\frac{D_{\Delta t}}{c}
\left[
\phi(\mathbf{\theta}_i,\mathbf{\beta})
-
\phi(\mathbf{\theta}_j,\mathbf{\beta})
\right].
\]
4. Why Fermat potential also gives the lens equation
Images form at stationary points of the arrival-time surface. Since the factor \(D_{\Delta t}/c\) is independent of \(\mathbf{\theta}\), the stationary condition is the same as
\[
\nabla_{\mathbf{\theta}}\phi
=
\mathbf{\theta}-\mathbf{\beta}
-
\nabla_{\mathbf{\theta}}\psi(\mathbf{\theta})
=0.
\]
Using \(\mathbf{\alpha}=\nabla\psi\), this becomes
\[
\mathbf{\beta}
=
\mathbf{\theta}
-
\mathbf{\alpha}(\mathbf{\theta}),
\]
which is the lens equation. So the same Fermat potential both ranks the arrival times of the different images and selects the image positions through its stationary points.
到达时间、几何路径差、引力势延迟与宇宙学距离
强透镜里常用的 time-delay 公式是
\[
\Delta t_{ij}
=
\frac{D_{\Delta t}}{c}
\left[
\phi(\mathbf{\theta}_i,\mathbf{\beta})
-
\phi(\mathbf{\theta}_j,\mathbf{\beta})
\right].
\]
这不是凭空定义出来的。它来自两部分真实的传播时间：弯折光路比直线路径多走的几何路径差，以及光经过透镜引力势时产生的 Shapiro delay。
薄透镜近似下的几何时间延迟
图中展示 observer、lens plane 和 source plane，以及像角位置 theta 和源角位置 beta。
lens plane
source plane
O
光线过透镜处
source
optical axis
ξ = D_l θ
η = D_s β
β
θ
D_l
D_ls
D_s
入射方向给出 image angle θ
橙色折线是观测到的弯折光路，蓝色虚线是到同一个源的未弯折参考路径。对小角度路径长度做二阶展开，就会得到与 \(|\mathbf{\theta}-\mathbf{\beta}|^2\) 成正比的几何时间延迟。
1. 几何路径差从哪里来
采用薄透镜近似。观测者在 \(O\)，光线穿过透镜平面的位置是物理横向坐标 \(\mathbf{\xi}=D_l\mathbf{\theta}\)，源的位置是 \(\mathbf{\eta}=D_s\mathbf{\beta}\)。在小角度近似下，弯折光路的长度近似为
\[
\sqrt{D_l^2+|\mathbf{\xi}|^2}
+
\sqrt{D_{ls}^2+|\mathbf{\eta}-\mathbf{\xi}|^2}.
\]
对每个平方根展开到二阶，并减去未弯折参考路径
\[
\sqrt{D_s^2+|\mathbf{\eta}|^2}.
\]
这里用到的展开是
\[
\sqrt{D^2+|\mathbf{x}|^2}
=
D\sqrt{1+\frac{|\mathbf{x}|^2}{D^2}}
\simeq
D+\frac{|\mathbf{x}|^2}{2D}.
\]
所以几何路径差可以先写成
\[
\begin{aligned}
\Delta l_{\rm geom}
&=
\left(
D_l+\frac{|\mathbf{\xi}|^2}{2D_l}
+
D_{ls}+\frac{|\mathbf{\eta}-\mathbf{\xi}|^2}{2D_{ls}}
\right)
-
\left(
D_s+\frac{|\mathbf{\eta}|^2}{2D_s}
\right)\\
&=
\frac{|\mathbf{\xi}|^2}{2D_l}
+
\frac{|\mathbf{\eta}-\mathbf{\xi}|^2}{2D_{ls}}
-
\frac{|\mathbf{\eta}|^2}{2D_s}.
\end{aligned}
\]
在这个平直薄透镜示意图里，常数项因为 \(D_s=D_l+D_{ls}\) 相互抵消。再代入 \(\mathbf{\xi}=D_l\mathbf{\theta}\) 和 \(\mathbf{\eta}=D_s\mathbf{\beta}\)，得到
\[
\begin{aligned}
\Delta l_{\rm geom}
&=
\frac{D_l}{2}|\mathbf{\theta}|^2
+
\frac{|D_s\mathbf{\beta}-D_l\mathbf{\theta}|^2}{2D_{ls}}
-
\frac{D_s}{2}|\mathbf{\beta}|^2\\
&=
\frac{D_lD_s}{2D_{ls}}
\left(
|\mathbf{\theta}|^2
-
2\mathbf{\theta}\cdot\mathbf{\beta}
+
|\mathbf{\beta}|^2
\right)\\
&=
\frac{D_lD_s}{2D_{ls}}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2.
\end{aligned}
\]
在膨胀宇宙里的标准强透镜公式中，同一个结果写成角直径距离的组合。
整理以后，多出来的几何路径长度是
\[
\Delta l_{\rm geom}
=
\frac{D_lD_s}{2D_{ls}}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2.
\]
再把路径长度除以光速，并乘上透镜红移处的宇宙学时间膨胀因子 \((1+z_l)\)，得到
\[
t_{\rm geom}
=
\frac{1+z_l}{c}
\frac{D_lD_s}{D_{ls}}
\frac{1}{2}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2.
\]
2. 引力势延迟从哪里来
光经过透镜的引力势时，还会产生额外的传播时间，也就是 Shapiro delay。在弱场近似下，引力势对传播时间的贡献正比于 Newtonian potential \(\Phi\) 沿视线的积分：
\[
t_{\rm Shapiro}
=
-
\frac{2}{c^3}
\int \Phi(D_l\mathbf{\theta},\ell)\,\mathrm{d}\ell .
\]
薄透镜近似做的事情，就是把同一个沿视线积分打包成二维 lensing potential。按强透镜里的常用归一化，
\[
\psi(\mathbf{\theta})
=
\frac{2}{c^2}
\frac{D_{ls}}{D_lD_s}
\int \Phi(D_l\mathbf{\theta},\ell)\,\mathrm{d}\ell .
\]
也就是说，
\[
\int \Phi(D_l\mathbf{\theta},\ell)\,\mathrm{d}\ell
=
\frac{c^2}{2}
\frac{D_lD_s}{D_{ls}}
\psi(\mathbf{\theta}).
\]
把这个关系代回 Shapiro delay，再乘上透镜红移处的时间膨胀因子 \((1+z_l)\)，就得到到达时间里的引力势项：
\[
t_{\rm pot}
=
-
\frac{1+z_l}{c}
\frac{D_lD_s}{D_{ls}}
\psi(\mathbf{\theta}).
\]
重点是：Shapiro delay 和 lensing potential 不是两个独立东西，它们都来自同一个投影后的引力势。这里的负号来自 Fermat potential 的标准写法；\(\psi\) 里加一个常数只会让所有像的到达时间一起平移，不影响可观测的时间差。
3. 为什么会变成 distance 乘 potential
把两部分加起来：
\[
t(\mathbf{\theta},\mathbf{\beta})
=
\frac{1+z_l}{c}
\frac{D_lD_s}{D_{ls}}
\left[
\frac{1}{2}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2
-
\psi(\mathbf{\theta})
\right].
\]
前面的因子有距离量纲，控制把角度尺度转换成真实时间的宇宙学尺度；中括号里的量是无量纲的，控制由像位置、源位置和透镜质量分布决定的相对延迟形状。因此定义
\[
D_{\Delta t}
\equiv
(1+z_l)\frac{D_lD_s}{D_{ls}},
\qquad
\phi(\mathbf{\theta},\mathbf{\beta})
\equiv
\frac{1}{2}
\left|\mathbf{\theta}-\mathbf{\beta}\right|^2
-
\psi(\mathbf{\theta}).
\]
于是到达时间可以写成
\[
t(\mathbf{\theta},\mathbf{\beta})
=
\frac{D_{\Delta t}}{c}
\phi(\mathbf{\theta},\mathbf{\beta}).
\]
真正可观测的是两个像之间的时间差，因为绝对到达时间的零点可以任意平移：
\[
\Delta t_{ij}
=
\frac{D_{\Delta t}}{c}
\left[
\phi(\mathbf{\theta}_i,\mathbf{\beta})
-
\phi(\mathbf{\theta}_j,\mathbf{\beta})
\right].
\]
4. 为什么 Fermat potential 还给出 lens equation
成像位置对应到达时间面的驻点。因为 \(D_{\Delta t}/c\) 不依赖于 \(\mathbf{\theta}\)，驻点条件等价于
\[
\nabla_{\mathbf{\theta}}\phi
=
\mathbf{\theta}-\mathbf{\beta}
-
\nabla_{\mathbf{\theta}}\psi(\mathbf{\theta})
=0.
\]
使用 \(\mathbf{\alpha}=\nabla\psi\)，就得到
\[
\mathbf{\beta}
=
\mathbf{\theta}
-
\mathbf{\alpha}(\mathbf{\theta}),
\]
这正是 lens equation。所以 Fermat potential 同时做两件事：它的驻点给出实际成像位置，而不同驻点的 Fermat potential 差决定不同像之间的 time delay。