Measuring angular diameter distances of strong gravitational lenses

The distance-redshift relation plays a fundamental role in constraining cosmological models. In this paper, we show that measurements of positions and time delays of strongly lensed images of a background galaxy, as well as those of the velocity dispersion and mass profile of a lens galaxy, can be combined to extract the angular diameter distance of the lens galaxy. Physically, as the velocity dispersion and the time delay give a gravitational potential ($GM/r$) and a mass ($GM$) of the lens, respectively, dividing them gives a physical size ($r$) of the lens. Comparing the physical size with the image positions of a lensed galaxy gives the angular diameter distance to the lens. A mismatch between the exact locations at which these measurements are made can be corrected by measuring a local slope of the mass profile. We expand on the original idea put forward by Paraficz and Hjorth, who analyzed singular isothermal lenses, by allowing for an arbitrary slope of a power-law spherical mass density profile, an external convergence, and an anisotropic velocity dispersion. We find that the effect of external convergence cancels out when dividing the time delays and velocity dispersion measurements. We derive a formula for the uncertainty in the angular diameter distance in terms of the uncertainties in the observables. As an application, we use two existing strong lens systems, B1608+656 ($z_{\rm L}=0.6304$) and RXJ1131$-$1231 ($z_{\rm L}=0.295$), to show that the uncertainty in the inferred angular diameter distances is dominated by that in the velocity dispersion, $\sigma^2$, and its anisotropy. We find that the current data on these systems should yield about 16% uncertainty in $D_A$ per object. This improves to 13% when we measure $\sigma^2$ at the so-called sweet-spot radius. Achieving 7% is possible if we can determine $\sigma^2$ with 5% precision.Comment: Accepted to JCA

Time-delay Cosmography: Increased Leverage with Angular Diameter Distances

Strong lensing time-delay systems constrain cosmological parameters via the so-called time-delay distance and the angular diameter distance to the lens. In previous studies, only the former information was used. In this paper, we show that the cosmological constraints improve significantly when the latter information is also included. Specifically, the angular diameter distance plays a crucial role in breaking the degeneracy between the curvature of the Universe and the time-varying equation of state of dark energy. Using a mock sample of 55 bright quadruple lens systems based on expectations for ongoing/future imaging surveys, we find that adding the angular diameter distance information to the time-delay distance information and the cosmic microwave background data of Planck improves the constraint on the constant equation of state by 30%, on the time variation in the equation of state by a factor of two, and on the Hubble constant in the flat $\Lambda$CDM model by a factor of two. Therefore, previous forecasts for the statistical power of time-delay systems were significantly underestimated, i.e., time-delay systems are more powerful than previously appreciated.Comment: [v2] 18 pages, 12 figures, submitted to JCAP. An error in the fisher matrix for SNIa fixed; conclusions unchange

A Second-order bias model for the Logarithmic Halo Mass Density

We present an analytic model for the local bias of dark matter halos in a LCDM universe. The model uses the halo mass density instead of the halo number density and is searched for various halo mass cuts, smoothing lengths, and redshift epoches. We find that, when the logarithmic density is used, the second-order polynomial can fit the numerical relation between the halo mass distribution and the underlying matter distribution extremely well. In this model the logarithm of the dark matter density is expanded in terms of log halo mass density to the second order. The model remains excellent for all halo mass cuts (from M_{cut}=3\times10^{11}$to$3\times10^{12}h^{-1}M_{\odot}$), smoothing scales (from$R=5h^{-1}$Mpc to$50h^{-1}$Mpc), and redshift ranges (from z=0 to 1.0) considered in this study. The stochastic term in the relation is found not entirely random, but a part of the term can be determined by the magnitude of the shear tensor.Comment: 8 pages, 7 figures, accepted for publication on Ap

H_0 from Lensed Quasars

Strong gravitational lens systems with time delays between the multiple images are a powerful probe of cosmology, particularly of the Hubble constant (H0). The H0 Lenses In COSMOGRAIL's Wellspring (H0LiCOW) project has measured H0 from lensed quasars using deep Hubble Space Telescope and AO imaging, precise time delay measurements from the COSMOGRAIL monitoring project, a measurement of the velocity dispersion of the lens galaxies, and a characterization of the mass distribution along the line of sight. Our latest results from a total of six lenses constrains H0 to be 73.3(-1.8,+1.7) km/s/Mpc for a flat Lambda CDM cosmology, which is a measurement to 2.4% precision. These results are consistent with independent determinations of H0 using type Ia supernovae calibrated by the distance ladder method, and are in 3.1-sigma tension with the results of Planck CMB measurements. Combined with the latest distance ladder results from the SH0ES project, we find a 5.3-sigma tension between Planck and late-Universe probes, hinting at possible new physics beyond the standard LCDM model and highlighting the importance of this independent probe

Measuring angular diameter distances in the universe by Baryon Acoustic Oscillation and strong gravitational lensing

textWe discuss two ways of measuring angular diameter distances in the Universe: (i) Baryon Acoustic Oscillation (BAO) , and (ii) strong gravitational lensing. For (i), we study the effects of survey geometry and selection functions on the 2-point correlation function of Lyman- alpha emitters in 1.9 < z < 3.5 for Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). We develop a method to extract the BAO scale (hence a volume-averaged angular diameter distance D_V, which is a combination of the angular diameter distance and the Hubble expansion rate, i.e., [cz〖(1+z)〗^2 〖D_A〗^2 H^(-1) ]^(1/3)) from a spherically averaged 1-d correlation function. We quantify the statistical errors on such measurements. By using log-normal realizations of the HETDEX dataset, we show that we can determine DV from HETDEX at 2% accuracy using the 2-point correlation function. This study is complementary to the on-going effort to characterize the power spectrum using HETDEX. For (ii), a previous study (Para ficz and Hjorth 2009) looked at the case of a spherical lens following a singular isothermal distribution of matter and an isotropic velocity distribution, and found that combining measurements of the Einstein ring radius with the time delay of a strong lens system directly leads to a measurement of the angular diameter distance, D_A. Since this is a very new method, it requires more careful investigations of various real-world eff ects such as a realistic matter density pro file, an anisotropic velocity distribution, and external convergence. In more realistic lens confi gurations we find that the velocity dispersion is the dominant source of the uncertainty ; in order for this method to achieve competitive precision on measurements of DA, we need to constrain the velocity dispersion, down to the percent level. On the other hand, external convergence and velocity dispersion anisotropy have negligible e ect on our result. However, we also claim that the dominant source of the uncertainty depends largely on the image con figuration of the system, which leads us to the conclusion that studying the angular dependence of the lens mass distribution is a necessary component.Astronom

A measurement of the Hubble constant from angular diameter distances to two gravitational lenses

The local expansion rate of the Universe is parametrized by the Hubble constant, $H_0$, the ratio between recession velocity and distance. Different techniques lead to inconsistent estimates of $H_0$. Observations of Type Ia supernovae (SNe) can be used to measure $H_0$, but this requires an external calibrator to convert relative distances to absolute ones. We use the angular diameter distance to strong gravitational lenses as a suitable calibrator, which is only weakly sensitive to cosmological assumptions. We determine the angular diameter distances to two gravitational lenses, $810^{+160}_{-130}$ and $1230^{+180}_{-150}$~Mpc, at redshifts of $z=0.295$ and $0.6304$. Using these absolute distances to calibrate 740 previously-measured relative distances to SNe, we measure the Hubble constant to be $H_0=82.4^{+8.4}_{-8.3} ~{\rm km\,s^{-1}\,Mpc^{-1}}$.Comment: This paper presents the measurements of angular diameter distances to two time-delay lenses, and the Hubble constant derived only from these two distances and the JLA supernova sample. One of the distance measurements is further used for the cosmological inference in the H0LiCOW XIII paper (arxiv:1907.04869). Published in Scienc

A SHARP view of H0LiCOW: H[SUB]0[/SUB] from three time-delay gravitational lens systems with adaptive optics imaging

peer reviewedWe present the measurement of the Hubble constant, H[SUB]0[/SUB], with three strong gravitational lens systems. We describe a blind analysis of both PG 1115+080 and HE 0435-1223 as well as an extension of our previous analysis of RXJ 1131-1231. For each lens, we combine new adaptive optics (AO) imaging from the Keck Telescope, obtained as part of the SHARP (Strong-lensing High Angular Resolution Programme) AO effort, with Hubble Space Telescope (HST) imaging, velocity dispersion measurements, and a description of the line-of-sight mass distribution to build an accurate and precise lens mass model. This mass model is then combined with the COSMOGRAIL-measured time delays in these systems to determine H[SUB]0[/SUB]. We do both an AO-only and an AO + HST analysis of the systems and find that AO and HST results are consistent. After unblinding, the AO-only analysis gives H[SUB]0[/SUB]=82.8^{+9.4}_{-8.3} km s^{-1} Mpc^{-1} for PG 1115+080, H[SUB]0[/SUB]=70.1^{+5.3}_{-4.5} km s^{-1} Mpc^{-1} for HE 0435-1223, and H[SUB]0[/SUB]=77.0^{+4.0}_{-4.6} km s^{-1} Mpc^{-1} for RXJ 1131-1231. The joint AO-only result for the three lenses is H[SUB]0[/SUB]=75.6^{+3.2}_{-3.3} km s^{-1} Mpc^{-1}. The joint result of the AO + HST analysis for the three lenses is H[SUB]0[/SUB]=76.8^{+2.6}_{-2.6} km s^{-1} Mpc^{-1}. All of these results assume a flat Λ cold dark matter cosmology with a uniform prior on Ω[SUB]m[/SUB] in [0.05, 0.5] and H[SUB]0[/SUB] in [0, 150] km s^{-1} Mpc^{-1}. This work is a collaboration of the SHARP and H0LiCOW teams, and shows that AO data can be used as the high-resolution imaging component in lens-based measurements of H[SUB]0[/SUB]. The full time- delay cosmography results from a total of six strongly lensed systems are presented in a companion paper. <P /

H0LiCOW-XIII. A 2.4 per cent measurement of H0from lensed quasars: 5.3σ tension between early-and late-Universe probes

We present a measurement of the Hubble constant (H0) and other cosmological parameters from a joint analysis of six gravitationally lensed quasars with measured time delays. All lenses except the first are analysed blindly with respect to the cosmological parameters. In a flat Λ cold dark matter (ΛCDM) cosmology, we find a precision measurement, in agreement with local measurements of H0 from type Ia supernovae calibrated by the distance ladder, but in 3.1σ tension with Planck observations of the cosmic microwave background (CMB). This method is completely independent of both the supernovae and CMB analyses. A combination of time-delay cosmography and the distance ladder results is in 5.3σ tension with Planck CMB determinations of H0 in flat ΛCDM. We compute Bayes factors to verify that all lenses give statistically consistent results, showing that we are not underestimating our uncertainties and are able to control our systematics. We explore extensions to flat ΛCDM using constraints from time-delay cosmography alone, as well as combinations with other cosmological probes, including CMB observations from Planck, baryon acoustic oscillations, and type Ia supernovae. Time-delay cosmography improves the precision of the other probes, demonstrating the strong complementarity. Allowing for spatial curvature does not resolve the tension with Planck. Using the distance constraints from time-delay cosmography to anchor the type Ia supernova distance scale, we reduce the sensitivity of our H0 inference to cosmological model assumptions. For six different cosmological models, our combined inference on H0 ranges from ∼73 to 78 km s-1 Mpc-1, which is consistent with the local distance ladder constraints