Geometric and Gravitational Delays in Lensing

When light from a distant quasar passes near a foreground galaxy, the galaxy's gravity bends the path of the light. This bending produces multiple images that reach us from different directions. Crucially, the travel times of these images are not the same. The combined effect of extra path length and the slowing of light in a gravitational potential introduces a measurable time delay between images. Such delays, observed over weeks or months, allow astronomers to measure the Hubble constant and probe dark matter distributions. Our calculator models the simplest case: a point-mass lens, which despite its simplicity captures the essential physics of more complex lenses.

In the thin-lens approximation, the arrival time of an image at angle $\theta$ relative to the observer–lens line is described by Fermat's principle. The dimensionless Fermat potential combines geometric and gravitational contributions: $\tau =\frac{1}{2}(\theta -\beta )^2-\psi (\theta )$. Here $\beta$ is the angular position the source would have without lensing, and $\psi$ is the scaled lensing potential. For a point mass, $\psi (\theta )=ln|\theta |$. The observable time delay multiplies this potential by a scale factor that depends on the lens mass and cosmological distances.

Rather than require detailed distances, the point-mass lens admits an analytic expression for the relative delay between its two images. Defining the dimensionless impact parameter $y$ as the source position in units of the Einstein angle, the difference in arrival times is

$\mathrm{\Delta t}=\frac{4GM}{c^3}(1+z{l}_{})[y\sqrt{y^2+4}+ln\frac{\sqrt{y^2+4}+y}{\sqrt{y^2+4}-y}]$

This formula neatly encapsulates both the extra distance and the gravitational time dilation. The prefactor $\frac{4\mathrm{GM}}{c^3}$ shows that more massive lenses produce longer delays, while the impact parameter controls how asymmetrical the light paths are. The lens redshift $\mathrm{z\_l}$ accounts for cosmic time dilation: clocks at the lens run slower relative to us by a factor of $1+\mathrm{z\_l}$.

Time delays have become a powerful cosmological probe. By monitoring the variability of lensed quasars and correlating brightness changes among their images, astronomers extract the delay. Combined with a model of the lens potential and the relative distances, these delays yield the so-called $\mathrm{time-delay}\mathrm{distance}$, which scales inversely with the Hubble constant $\mathrm{H\_0}$. Recent efforts like H0LiCOW and TDCOSMO have used a handful of well-characterized lenses to measure $\mathrm{H\_0}$ to a few percent, providing an independent cross-check on cosmic microwave background and supernova determinations.

The simple point-mass model does not capture the richness of real galaxies, which contain stars, gas, dark matter halos, and sometimes multiple mass components. Nevertheless, it offers a useful sandbox. Our calculator asks for the lens mass, the dimensionless impact parameter $y$, and the lens redshift. It outputs the relative time delay between the two images in seconds and days. For a galaxy-mass lens of 10¹¹ solar masses and $y$=0.5 at redshift 0.5, the delay is on the order of tens of days, consistent with observed quasar lenses.

The table below gives representative delays for several impact parameters at a fixed mass of 10¹¹ solar masses and redshift 0.5. As $y$ increases, the images move farther from the Einstein radius, and the delay grows, reflecting the increased asymmetry between the paths. At very small $y$, the images approach symmetric positions and the delay shrinks.

y	Δt (days)
0.1
0.5
1.0

Time delays do more than measure $\mathrm{H\_0}$. They also test general relativity by comparing the lensing mass inferred from dynamics with that required to produce the observed delay. Deviations might signal exotic physics such as modified gravity or additional mass components like dark matter subhalos. In the future, gravitational-wave detectors may observe lensed signals, where time delays could separate multiple waveforms by minutes to weeks. Such events would provide pristine laboratories for cosmology because gravitational waves are immune to dust extinction and have well-understood intrinsic profiles.

The mathematics behind time delays stems from Fermat's principle: light follows paths that extremize travel time. In curved spacetime, the travel time includes contributions from both geometry and the gravitational potential. The derivation of the delay formula involves expressing the potential in scaled coordinates and differentiating the arrival-time surface to find image positions. These stationary points correspond to minima, maxima, or saddle points in the Fermat surface, each with characteristic parities and magnifications. The relative delay is then the difference in $\tau$ multiplied by a scale factor.

While our calculator focuses on a single lens and source, lensing in the universe often involves multiple deflectors, external shear from nearby structures, and complex mass profiles. Realistic modeling employs ray-tracing codes and Bayesian inference to match observed image positions, flux ratios, and delays. Yet the point-mass case retains pedagogical value: it highlights the interplay between gravity and geometry in shaping light paths. By adjusting the inputs, users can explore how lens mass and alignment influence delays, building intuition that transfers to more elaborate scenarios.

To operate the calculator, supply the lens mass in solar masses, choose an impact parameter, and enter the lens redshift. Pressing the button computes the delay with fundamental constants $G$ and $c$ fixed to their CODATA values. The result displays in seconds and days for convenience. If an invalid value is entered, an error message prompts correction. The table populates automatically for three sample $y$ values using the default mass and redshift, demonstrating the sensitivity of the delay to alignment.

Ultimately, gravitational lens time delays connect the smallest of scales—the quantum fluctuations that seed quasar variability—to the largest scales of cosmology. Measuring how long light takes to traverse different cosmic routes opens a window onto the expansion history of the universe and the distribution of matter along the way. Whether you are a student learning lensing fundamentals or a researcher needing a quick estimate, this calculator provides a gateway into the fascinating world where gravity bends not just space, but time itself.