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Register a new userThe Hubble Constant from Gravitational Lens Time Delays
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Authors
C.S. Kochanek
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There are now 10 firm time delay measurements in gravitational lenses. The physics of time delays is well understood, and the only important variable for interpreting the time delays to determine H_0 is the mean surface mass density <k> (in units of the critical density for gravitational lensing) of the lens galaxy at the radius of the lensed images. More centrally concentrated mass distributions with lower <k> predict higher Hubble constants, with H_0~1-<k> to lowest order. While we cannot determine <k> directly given the available data on the current time delay lenses, we find H_0=48+/-3 km/s/Mpc for the isothermal (flat rotation curve) models, which are our best present estimate for the mass distributions of the lens galaxies. Only if we eliminate the dark matter halo of the lenses and use a constant mass-to-light ratio (M/L) model to find H_0=71+/-3 km/s/Mpc is the result consistent with local estimates. Measurements of time delays in better-constrained systems or observations to obtain new constraints on the current systems provide a clear path to eliminating the <k> degeneracy and making estimates of H_0 with smaller uncertainties than are possible locally. Independent of the value of H_0, the time delay lenses provide a new and unique probe of the dark matter distributions of galaxies and clusters because they measure the total (light + dark) matter surface density.
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Present day estimates of the Hubble constant based on Cepheids and on the cosmic microwave background radiation are uncertain by roughly 10% (on the conservative assumption that the universe may not be PERFECTLY flat). Gravitational lens time delay measurements can produce estimates that are less uncertain, but only if a variety of major difficulties are overcome. These include a paucity of constraints on the lensing potential, the degeneracies associated with mass sheets and the central concentration of the lensing galaxy, multiple lenses, microlensing by stars, and the small variability amplitude typical of most quasars. To date only one lens meets all of these challenges. Several suffer only from the central concentration degeneracy, which may be lifted if one is willing to assume that systems with time delays are either like better constrained systems with non-variable sources, or alternatively, like nearby galaxies.
We present a refined gravitational lens model of the four-image lens system B1608+656 based on new and improved observational constraints: (i) the three independent time-delays and flux-ratios from VLA observations, (ii) the radio-image positions from VLBA observations, (iii) the shape of the deconvolved Einstein Ring from optical and infrared HST images, (iv) the extinction-corrected lens-galaxy centroids and structural parameters, and (v) a stellar velocity dispersion, sigma_ap=247+-35 km/s, of the primary lens galaxy (G1), obtained from an echelle spectrum taken with the Keck--II telescope. The lens mass model consists of two elliptical mass distributions with power-law density profiles and an external shear, totaling 22 free parameters, including the density slopes which are the key parameters to determine the value of H_0 from lens time delays. This has required the development of a new lens code that is highly optimized for speed. The minimum-chi^2 model reproduces all observations very well, including the stellar velocity dispersion and the shape of the Einstein Ring. A combined gravitational-lens and stellar dynamical analysis leads to a value of the Hubble Constant of H_0=75(+7/-6) km/s/Mpc (68 percent CL; Omega_m=0.3, Omega_Lambda=0.7. The non-linear error analysis includes correlations between all free parameters, in particular the density slopes of G1 and G2, yielding an accurate determination of the random error on H_0. The lens galaxy G1 is ~5 times more massive than the secondary lens galaxy (G2), and has a mass density slope of gamma_G1=2.03(+0.14/-0.14) +- 0.03 (68 percent CL) for rho~r^-gamma, very close to isothermal (gamma=2). (Abridged)
An attempt to measure the Hubble constant with gravitational lens time delays is often limited by the strong degeneracy between radial mass profiles of lens galaxies and the Hubble constant. We show that strong gravitational lensing of type Ia supernovae breaks this degeneracy; the standard candle nature of type Ia supernova luminosity function allows one to measure the magnification factor directly, and this information is essential to constrain radial mass profiles and the Hubble constant separately. Our numerical simulation demonstrates that the Hubble constant can be determined with sim 5% accuracy from only several lens events if magnification factors are used as constraints. Therefore, distant supernova survey is a promising way to measure the global Hubble constant independently with the local estimates.
Strong lensing gravitational time delays are a powerful and cost effective probe of dark energy. Recent studies have shown that a single lens can provide a distance measurement with 6-7 % accuracy (including random and systematic uncertainties), provided sufficient data are available to determine the time delay and reconstruct the gravitational potential of the deflector. Gravitational-time delays are a low redshift (z~0-2) probe and thus allow one to break degeneracies in the interpretation of data from higher-redshift probes like the cosmic microwave background in terms of the dark energy equation of state. Current studies are limited by the size of the sample of known lensed quasars, but this situation is about to change. Even in this decade, wide field imaging surveys are likely to discover thousands of lensed quasars, enabling the targeted study of ~100 of these systems and resulting in substantial gains in the dark energy figure of merit. In the next decade, a further order of magnitude improvement will be possible with the 10000 systems expected to be detected and measured with LSST and Euclid. To fully exploit these gains, we identify three priorities. First, support for the development of software required for the analysis of the data. Second, in this decade, small robotic telescopes (1-4m in diameter) dedicated to monitoring of lensed quasars will transform the field by delivering accurate time delays for ~100 systems. Third, in the 2020s, LSST will deliver 1000s of time delays; the bottleneck will instead be the aquisition and analysis of high resolution imaging follow-up. Thus, the top priority for the next decade is to support fast high resolution imaging capabilities, such as those enabled by the James Webb Space Telescope and next generation adaptive optics systems on large ground based telescopes.
It is well known that measurements of H0 from gravitational lens time delays scale as H0~1-k_E where k_E is the mean convergence at the Einstein radius R_E but that all available lens data other than the delays provide no direct constraints on k_E. The properties of the radial mass distribution constrained by lens data are R_E and the dimensionless quantity x=R_E a(R_E)/(1-k_E)$ where a(R_E) is the second derivative of the deflection profile at R_E. Lens models with too few degrees of freedom, like power law models with densities ~r^(-n), have a one-to-one correspondence between x and k_E (for a power law model, x=2(n-2) and k_E=(3-n)/2=(2-x)/4). This means that highly constrained lens models with few parameters quickly lead to very precise but inaccurate estimates of k_E and hence H0. Based on experiments with a broad range of plausible dark matter halo models, it is unlikely that any current estimates of H0 from gravitational lens time delays are more accurate than ~10%, regardless of the reported precision.
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