$H_0$ from strong lensing time delay

$H_0$ Lenses in COSMOGRAIL's Wellspring (H0LiCOW )

Wong et al. (2019)

• 2.4% error on $H_0$;
• 5.3$\sigma$ tension with CMB $H_0$.

$D_S$

$D_L$

$D_{LS}$

$\beta$

$\alpha$

$\hat{\alpha}$

$r_{\perp}$

Source

Image

Lens

• Lens equation

$$\vec{\beta} =\vec{\theta} - \vec{\alpha} = \vec{\theta} - \frac{D_{LS}}{D_S} \vec{\hat{\alpha}}$$

• Deflection angle

$$\hat{\alpha} = \frac{4GM(<r_\perp)}{c^2 r_\perp};\quad r_\perp=D_L\theta$$

• Perfect alignment $\vec{\beta}=\vec{\theta_E} - \vec{\alpha}(\vec{\theta_E})=0$
  • Einstein radius $\theta_E$
  • Critical surface density $\Sigma_{\rm crit} \equiv \frac{M(<r_\perp)}{\pi r_\perp^2}$
  • Convergence $\kappa \equiv \frac{\Sigma}{\Sigma_{\rm crit}}$

$\theta$

$D_S$

$D_L$

$D_{LS}$

$\beta$

$\alpha$

$\hat{\alpha}$

$r_{\perp}$

Source

Image

Lens

• Lensing time delay

• Time delay distance

$$D_{\Delta t} \equiv (1+z_L) \frac{D_L D_S}{D_{LS}}$$

• Lensing potential

$$\nabla_{\theta} \psi(\theta) = \vec{\alpha}$$

$$\nabla^2_{\theta} \psi(\theta) = 2\kappa(\theta)$$

• Time delay between two images

$$\Delta t \propto H_0^{-1} (1-\langle \kappa \rangle )$$

• Usually, $\langle \kappa\rangle \approx \kappa (\theta_E) \equiv \kappa_E$.

$\theta$

H0LiCOW

• Spectroscopy to get kinematics of the lens galaxy.
• Assuming certain density profile, extrapolate to the lensing region (i.e., Einstein radius).

$D_S$

$D_L$

$D_{LS}$

$\beta$

$\alpha$

$\hat{\alpha}$

$r_{\perp}$

Source

Image

Lens

$\theta$

• Lens has a simple power-law density profile $\rho \propto r^{-n}$.
• Image offset $\alpha = \theta_E^{n-1} \theta^{2-n}$.
• Surface density $\Sigma(r_\perp) \propto r_\perp^{1-n}$.
• Convergence $\kappa_E = (3-n)/2$.

$\theta_1$

$\theta_2$

Simulations

Lensing data alone

Simulations

Summary

• H0LiCOW reported a 2.4% $H_0$ measurement.

• Combining lensing information and lens galaxy kinematics, in order to break the $H_0-\kappa$ degeneracy.

• Kochanek (2020) argues the lens model is overly simplified, leading to very precise and inaccurate $H_0$.

• Kochanek (2020): "It is unlikely that any current estimates of $H_0$ from gravitational lens time delays are more accurate than ∼10 per cent, regardless of the reported precision."

• Measuring the time derivative of the time delay? (e.g., Wucknitz et al. 2020)