Tutorial 4: Time Delays

:::info Migrated from Old Wiki This tutorial was created from content in the old GLEE wiki. Content should be double-checked for accuracy. :::

:::tip Prerequisite Before working with time delays, complete Tutorial 2: Image Plane Positions to have a well-converged mass model. :::

Introduction

Gravitationally lensed quasars show time delays between multiple images due to:

1. Geometric delay: Different path lengths through curved spacetime
2. Gravitational potential delay (Shapiro delay): Different potential wells

Measuring these delays enables cosmological constraints, particularly on the Hubble constant $H_0$:

$$\Delta t_{ij} \propto \frac{D_{\Delta t}}{c} \left[ \frac{1}{2}|\vec{\theta}_i - \vec{\theta}_j|^2 - (\psi_i - \psi_j) \right]$$

where:

• $D_{\Delta t} = (1+z_d) \frac{D_d D_s}{D_{ds}}$ is the time-delay distance (Mpc)
• $\vec{\theta}_i$ is the image position
• $\psi_i$ is the lensing potential at image $i$

Time delays are typically measured via light curve monitoring campaigns (e.g., COSMOGRAIL, H0LiCOW).

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Enabling Time-Delay Fitting

To incorporate time-delay constraints, add chi2type tmd (or numerically: chi2type 128):

chi2type img tmd

This combines image-plane positions (img) with time-delay measurements (tmd).

:::note Chi2 Type Combinations

• chi2type 2 → image positions only
• chi2type 128 → time delays only
• chi2type 130 → image positions + time delays (recommended)

The keyword syntax (img, tmd) is preferred over numbers. :::

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Specifying Time Delays

Method 1: Gaussian Uncertainties (Simplest)

For time delays with symmetric Gaussian errors:

sources 1
    Dds/Ds    1.0      exact:
    Dt        1919.0   flat:0,4000   step:100.0
    source weighted
    srcx      3.640    exact:
    srcy      3.706    exact:
    4
    4.670    3.962    error:0.004   t:33.2,1.4
    3.957    4.074    error:0.004   t:33.2,1e+06
    2.479    2.701    error:0.004   t:0,0
    4.589    2.422    error:0.004   t:58.7,2.4

Syntax: t:value,error

• value: Time delay in days (relative to reference image)
• error: 1-sigma uncertainty (days)

Reference image: t:0,0 (zero delay, zero error). Must have exactly ONE reference per system.

:::caution Reference Image Selection

• Set delay to 0 for the reference image
• For other images with unmeasured delays, use large error: t:value,1e+06 (effectively unconstrained)
• GLEE has not been thoroughly tested with non-zero reference delays :::

Time-Delay Distance (Dt)

In single-plane lensing, GLEE needs the time-delay distance to predict delays:

$$D_{\Delta t} = (1+z_d) \frac{D_d D_s}{D_{ds}} \quad [\text{Mpc}]$$

Configuration:

• Fixed cosmology: Dt value exact: (if $H_0$ is known)
• Constrain $H_0$: Dt value flat:min,max step:dS (vary to measure $H_0$)

Example (fixing $D_{\Delta t}$ for $H_0 = 70$ km/s/Mpc):

Dt    1919.0   exact:

Example (varying to measure $H_0$):

Dt    1919.0   flat:1000,3000   step:50.0

:::tip Measuring $H_0$ The posterior distribution of $D_{\Delta t}$ directly constrains $H_0$ via:

$$H_0 = \frac{c}{D_{\Delta t}} (1+z_d) \frac{D_d D_s}{D_{ds}}$$

Use MCMC to sample the $D_{\Delta t}$ posterior, then convert to $H_0$. :::

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Method 2: Correlated Uncertainties

If time-delay errors are correlated (common in monitoring campaigns), provide a covariance matrix:

4.670    3.962    error:0.004   t:33.2,1.4   tcovmatfile:tdcov_inverse.dat
3.957    4.074    error:0.004   t:33.2,1e+06
2.479    2.701    error:0.004   t:0,0
4.589    2.422    error:0.004   t:58.7,2.4

File format: tdcov_inverse.dat is the inverse covariance matrix (precision matrix):

$$\mathbf{C}^{-1} = \mathbf{Cov}[\Delta t]^{-1}$$

Dimensions: (N-1) × (N-1), where N = number of images with delays (excluding reference)

Example (3×3 matrix for 4 images, 1 reference, 3 measured delays):

0.5102040816     0.0000000000     0.0000000000
0.0000000000     1.0000000e-12    0.0000000000
0.0000000000     0.0000000000     0.1736111111

Order: Rows/columns correspond to image order in config file (excluding reference).

:::warning Covariance Notes

• tcovmatfile: can appear on any image line (GLEE will find it), but specify it only once
• When tcovmatfile: is provided, the individual t:value,error uncertainties are ignored (only values are used)
• Matrix must be symmetric and positive definite :::

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Method 3: Non-Analytic Distributions

If time-delay probability distributions are non-Gaussian (e.g., from deconvolution methods), use tfile::

4.670    3.962    error:0.004   tfile:tdelay_im1.dat
3.957    4.074    error:0.004   tfile:tdelay_im2.dat
2.479    2.701    error:0.004   t:0,0
4.589    2.422    error:0.004   tfile:tdelay_im4.dat

File format (tdelay_im1.dat):

• Two columns (NO header)
• Column 1: Time delay values (must increase by constant increment)
• Column 2: Probability density $P(\Delta t)$ (need not be normalized)

Example:

-5.0    0.001
-4.9    0.003
-4.8    0.010
...
33.2    0.150    # Peak at measured value
...
70.0    0.001

Chi-squared: GLEE approximates $\chi^2_{\text{tmd}} \approx -2 \ln P(\Delta t_{\text{pred}})$

:::caution File Requirements

• Time delay column must increment uniformly: $\Delta t_{i+1} - \Delta t_i = \text{const}$ (tolerance: 1e-6)
• One reference image must still use Gaussian format: t:0,0
• Do NOT mix t: and tfile: for the same image :::

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Microlensing Time Delays (Advanced)

Microlensing by stars in the lens galaxy can introduce stochastic time delays (~days to weeks). To marginalize over microlensing:

Step 1: Provide Microlensing Probability Distributions

4.670    3.962    error:0.004   t:33.2,1.4   tmicrofile:tmicro_im1.dat
3.957    4.074    error:0.004   t:33.2,1e+06 tmicrofile:tmicro_im2.dat
2.479    2.701    error:0.004   t:0,0        tmicrofile:tmicro_ref.dat
4.589    2.422    error:0.004   t:58.7,2.4   tmicrofile:tmicro_im4.dat

File format: Same as tfile: (two columns: microlensing delay, probability)

Step 2: Enable Microlensing in Chi2

At the top of your config file:

microlens_time_delay_in_chi2   yes

How it works:

• At each MCMC step, GLEE randomly samples $\Delta t_{\text{micro},i}$ from tmicrofile:
• Predicted delay: $\Delta t_{\text{pred},i} = \Delta t_{\text{geom+pot},i} + \Delta t_{\text{micro},i}$

:::danger Critical Requirement Each image must have a UNIQUE microlensing file. If two images share the same file (name or content), GLEE will draw the same random values → incorrect cancellation of $\Delta t_{\text{micro},i} - \Delta t_{\text{micro},j}$.

Solution: Generate independent realizations for each image, even if distributions are similar. :::

:::note Discretization GLEE samples from the discrete values in column 1 of tmicrofile:. Use fine discretization (small $d\Delta t$) for smooth sampling. :::

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Advanced Options

Predicted Delay Positions

Specify whether to compute predicted delays at observed or predicted image positions:

time_delay_chi2_positions   observed_image_positions

or

time_delay_chi2_positions   predicted_image_positions

Default: observed_image_positions

Use case:

• observed_image_positions → More physically motivated (delays measured at observed locations)
• predicted_image_positions → Useful for testing model-predicted image configurations

:::note Source Position In single-plane lensing, the source position for delay calculation is determined by the source option (weighted, parameter, etc.). In multiplane lensing, GLEE ray-traces from each observed image position to get (potentially different) source positions. :::

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Workflow: Adding Time Delays to a Model

1. Start with image-plane fit (Tutorial 2) → Get baseline mass model
2. Add time-delay constraints:

   chi2type img tmd
   Dt    1919.0   flat:1000,3000   step:50.0

3. Specify delays in sources block (use t:, tcovmatfile:, or tfile:)
4. Run MCMC to sample posterior:

   glee.py mcmc configfile_img_tmd

5. Check convergence:

   glee.py chi2 -c img tmd -- configfile_img_tmd_mcmc_best

6. Extract $H_0$ constraints from $D_{\Delta t}$ posterior distribution

:::tip H0LiCOW Approach For Hubble constant measurements:

• Combine lens model (this tutorial) with external line-of-sight mass distribution
• Marginalize over mass-sheet degeneracy using kinematics
• Report $H_0$ with full systematic error budget

See Suyu et al. (2013), Bonvin et al. (2017), and H0LiCOW collaboration papers. :::

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Checking Time-Delay Chi2

After optimization, verify the time-delay fit quality:

glee.py chi2 -c tmd -- configfile_img_tmd_mcmc_best

Good fit indicators:

• $\chi^2_{\text{tmd}} / \text{dof} \lesssim 1$
• Predicted delays within 1-2$\sigma$ of observed values

Diagnostics:

glee.py analyse configfile_img_tmd_mcmc_best

This outputs predicted vs. observed delays for each image.

:::warning Common Issues

• Poor $\chi^2_{\text{tmd}}$: Mass model may not be well-constrained; check image-plane fit first
• Large $D_{\Delta t}$ posterior: Add external priors (e.g., velocity dispersion) to break mass-sheet degeneracy
• Microlensing mismatch: Ensure tmicrofile: distributions are realistic (consult microlensing simulations) :::

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Summary

Feature	Syntax	Use Case
Gaussian delays	t:value,error	Simple, symmetric uncertainties
Correlated errors	t:value,error tcovmatfile:file.dat	Monitoring campaign covariances
Non-Gaussian	tfile:delay_dist.dat	Asymmetric or multi-modal posteriors
Microlensing	tmicrofile:micro_dist.dat + global flag	Marginalize over stellar microlensing
Time-delay distance	Dt value flat:min,max	Constrain $H_0$
Reference image	t:0,0	Required for all systems with delays

:::tip Next Steps

• Combine with extended source reconstruction (Tutorial 3) for full surface brightness + time-delay modeling
• Explore multiplane lensing (Tutorial 5) for line-of-sight structures
• Use time delays to measure $H_0$ (requires external constraints to break degeneracies) :::