Measuring \( H_0 \)

with strong lenses

Christoph Becker (Durham/ICC)

Nan Li (Nottingham)

2019.06.11

The Value of \( H_0 \)

  • Powerful probe of dark energy
  • neutrino physics
  • Indications of new physics

(e.g. Sekiguchi+10, Linder11, Freedman+12, Weinberg+12)

E.V.Linder 11

The Value of \( H_0 \)

  • Powerful probe of dark energy
  • neutrino physics
  • Indications of new physics

(e.g. Sekiguchi+10, Linder11, Freedman+12, Weinberg+12)

shsuyu.github.io/H0LiCOW/site/, 2019

Independent methods are needed to overcome systematics, especially the unknown unknowns.

Variable sources: e.g. SNe, Quasars

\( H_0 \) from Time-Delays

Advanteges:

  • simple geometry & well-tested physics
  • one-step measurement of cosmological distance
{\color{black}\Delta t} \propto {\color{black}D_{\Delta t}} \times {\color{black}\Delta\phi_{\text{lens}}}

Time Delay

Time Delay Distnace

Lens potential

\( H_0 \) from Time-Delays

\( H_0 \) from Time-Delays

{\color{orange}\Delta t} \propto \frac{D_{l}(1/{\color{green}H_0})D_{s}(1/{\color{green}H_0})}{D_{ls}(1/{\color{green}H_0})} \times {\color{blue}\Delta\phi_{\text{lens}}}

\( D_l \)

\( D_s \)

\( D_{ls} \)

Observable

Goal

Inferable

Effect of LOS-Structures

More l.o.s. structures               larger \(  {\color{blue}\kappa_{\text{ext}}} \)               larger \(  {\color{blue}D_{\Delta t}} \)

{\color{orange}\Delta t} \propto \frac{{\color{green}D^{\text{los}}_{\Delta t}}}{1 - {\color{blue}\kappa_{\text{ext}}}} \times {\color{blue}\Delta\phi_{\text{lens}}}

Observable

Goal

Inferable

Angular diameters are perturbed by large scale structure relative to the homogeneous prediction

\color{orange}{\Delta t} \propto \frac{\color{blue}{D^{\text{model}}_{\Delta t}}(1/\color{green}{H_0})}{1 - \color{blue}{\kappa_{\text{ext}}}} \times \color{blue}{\Delta\phi}
D^{\text{model}}_{\Delta t}(1/{\color{green}H_0}) \propto (1 - {\color{orange}\kappa_{\text{ext}}}) \times \frac{{\color{blue}\Delta t}}{{\color{red}\Delta\phi}}
{\color{green}H_0} \propto \frac{{\color{red}\Delta\phi}}{(1 - {\color{orange}\kappa_{\text{ext}}}){\color{blue}\Delta t}}

Estimating \( \kappa_{\text{ext}} \)

  • Compare relative galaxy number counts to cosmological simulations to calibrate \( \kappa_{\text{ext}} \) (e.g., Fassnacht+2011; Greene+2013; Suyu+2010,2013)
  • Deep multi-band imaging to get photometric redshift and stellar masses to reconstruct line of sight mass distribution (Rusu+2017)
  • Multi-object spectroscopy to characterise nearby galaxies, groups (Sluse+2017)
  • Independent \( \kappa_{\text{ext}} \) constraint using weak lensing data (Tihhonova+2018)

Estimating \( \kappa_{\text{ext}} \)

LSST will find 3000 strong lensed quasars with time-delays

(Oguri&Marshall 10)

Estimating \( \kappa_{\text{ext}} \)

LSST will find 3000 strong lensed quasars with time-delays

Impossible to have follow up wide, deep, spectroscopic imaging for all of them!

(Oguri&Marshall 10)

Find \( H_0 \)-\( \kappa_{\text{ext}} \) Relation

1. Construct light-cone with & without LOS-structure

3. Lens mass modelling

2. Ray-tracing through the light-cone

4. \( H_0 \) estimation

Light-Cone

Semi-Analytic model cosmoDC2*:

  • designed for LSST-DESC
  • based on Outer Rim Simulations (dark-matter only)
  • \( \Lambda \)CDM cosmology: \( H_0 = 0.71 \)
  • sky coverage = \( 500 \text{deg}^2 \)
  • \( z = 0 - 3 \)
  • Halo mass = \( 10^{10} - 10^{15} M_{\odot} \)
  • Stellar mass = \( 10^{8} - 10^{13} M_{\odot} \)
  • including positions, redshift, orientations, ellipticity, colors of the galaxies in a given light cone

*github.com/LSSTDESC/cosmodc2

Light-Cone

      \( z_{\text{obs}} \)= 0.0                \( z_{\text{lens}} \)= 0.5                \( z_{\text{src}} \)= 2.0

Light-Cone

Oguri & Marshall 2010

      \( z_{\text{obs}} \)= 0.0                \( z_{\text{lens}} \)= 0.5                \( z_{\text{src}} \)= 2.0

Ray-Tracing

* Nan Li + 16

  • use PICS* to find lensed images
  • 10 lens-planes in redshift space
  • truncated Singular Isothermal Ellipsoid (SIE) lens profile
  • create catalogue for 1000 double and 400 quadruple imaged sources

Lens-Modelling

Gravlens*

SIE

(10 param.)

Glafic**

Softened Power-Law (SPL)

(11 param.)

lens-model:

cross-validate

*Gravlens by Keeton, **Galfic by Oguri

input:

perturbation:        constant shear

image-positions, -time delays, -magnifications, redshifts

errors on input:    assume best observation conditions (sub-%)

\( H_0 \) Estimate

{\color{orange}\Delta t} \propto \frac{D_{l}(1/{\color{green}H_0})D_{s}(1/{\color{green}H_0})}{D_{ls}(1/{\color{green}H_0})} \times {\color{blue}\Delta\phi_{\text{lens}}}

prior on \( H_0 \) = [0.2, 1.2]

\( H_0 \) Estimate

\( H_0 \) Estimate

\( H_0 \) Estimate

Argonne National Laboratory (USA/IL)

Physics: Gravity

Volume: \( 4.225 \text{Gpc}^3 \)

Particle Nr.: \( 10240^3 \)

Dark matter resolution: \( \sim 2.6 10^9 M_{\odot} \)

Light-Cone

Semi-Analytic model named cosmoDC2*:

  • designed for LSST-DESC
  • based on Outer Rim Simulations
  • \( 500 \text{deg}^2 \)
  • \( z = 0 - 3 \)
  • Halo mass = \( 10^{10} - 10^{15} M_{\odot} \)
  • Stellar mass = \( 10^{8} - 10^{13} M_{\odot} \)
  • including positions, redshift, orientations, ellipticity, colors of the galaxies in a given light cone

*github.com/LSSTDESC/cosmodc2

How to measure \( H_0 \)

Observed:

  • Image-positions
  • Magnitudes
  • (Spectra)
  • Time-delay
  • Redshift
  • (Lens mass)

Modelled:

  • mass
  • position
  • ellipticity
  • P.A.
  • shear
  • ...
\Delta t \propto {\color{blue}D_{\Delta t}} \times {\color{blue}\Delta\phi_{\text{lens}}}

S. Suyu + 14

PDF from Millenium Simulation

Estimating \( \kappa_{\text{ext}} \)

Gregory Dobler et al. 2014

Time Delays

How to measure \( H_0 \)

More variables than knows !!!

Options 1:

Collect better data

Options 2:

Collect more data

Fermat Potential difference

{\color{blue}\phi_{\text{lens}}} = \frac{({\color{orange}\theta} - {\color{blue}\beta})^2}{2} - {\color{blue}\Psi_{\text{lens}}}({\color{orange}\theta})

Observable

Goal

Inferable

Observable

Goal

Inferable