Galaxy mass profiles from strong lensing I: The circular power-law model

In this series of papers we develop a formalism for constraining mass profiles in strong gravitational lenses with extended images, using fluxes in addition to positional information. We start in this paper with a circular power-law profile and show that the slope $\gamma$ is uniquely determined by only two observables: the flux ratio $f_1/f_2$ and the image position ratio $\theta_1/\theta_2$ of the two images. We derive an analytic expression relating these two observables to the slope, a result which does not depend on the Einstein angle or the structure or brightness of the source. We then find an expression for the uncertainty on the slope $\sigma_\gamma$ that depends only on the position ratio $\theta_1/\theta_2$ and the total S/N in the images. For example, in a system with position ratio $\theta_1/\theta_2=0.5$, S/N $=100$ and $\gamma=2$ we find that $\gamma$ is constrained to a precision of $\pm0.03$. We then test these results against a series of mock observations. We invert the images and fit an 11 parameter model, including ellipticity and position angle for both lens and source and measure the uncertainty on $\gamma$. We find agreement with the theoretical estimate for all mock observations. In future papers we will examine the radial range of the galaxy over which the constraint on the slope applies, and extend the analysis to elliptical lenses.