# $$H_0$$ from strong lensing time delay

(Based on Wong et al. 2019 and Kochanek 2020)

Wei Zhu (祝伟)

2020-04-30

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

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

## How to measure $$H_0$$ from strong lensing (1)

$$D_S$$

$$D_L$$

$$D_{LS}$$

$$\beta$$

$$\alpha$$

$$\hat{\alpha}$$

$$r_{\perp}$$

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 $$\kappa \equiv \frac{\Sigma}{\Sigma_{\rm crit}}$$

$$\theta$$

Lensing geometry alone does not provide a distance scale (L=VT).

## How to measure $$H_0$$ from strong lensing (2)

$$D_S$$

$$D_L$$

$$D_{LS}$$

$$\beta$$

$$\alpha$$

$$\hat{\alpha}$$

$$r_{\perp}$$

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

$$\theta$$

Without information about the lens density distribution ($$\kappa$$), time delay itself does not measure $$H_0$$.

## H0LiCOW

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

If uncertainty is dominated by kinematics, $$\sigma(H_0) = \sigma(\kappa) \approx \sigma(\sigma_\star^2) =2\sigma(\sigma_\star)$$.

## Why H0LiCOW $$H_0$$ is so "precise"

$$D_S$$

$$D_L$$

$$D_{LS}$$

$$\beta$$

$$\alpha$$

$$\hat{\alpha}$$

$$r_{\perp}$$

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 $$\kappa_E = (3-n)/2$$.

For each pair of images

• Lens equation$$\theta_1 - \alpha_1 = \theta_2 - \alpha_2$$
• $$\theta_E^{n-1} = \frac{\theta_1+\theta_2}{\theta_1^{2-n} + \theta_2^{2-n}}$$
• With $$>2$$ images, the problem is over constrained.

$$\theta_1$$

$$\theta_2$$

## Simulations

Lensing data alone

Lensing data & kinematics (10% uncertainty)

## Simulations

No less than ~10% uncertainty on $$H_0$$. Adding more lenses does not reduce the uncertainty.

## 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."

By Wei Zhu（祝伟）

# H0 from lensing

A paper discussion of Kochanek (2020) on group meeting

• 215