f(R)f(R) gravity theories in the Palatini Formalism constrained from strong lensing

$f(R)$ gravity, capable of driving the late-time acceleration of the universe, is emerging as a promising alternative to dark energy. Various $f(R)$ gravity models have been intensively tested against probes of the expansion history, including type Ia supernovae (SNIa), the cosmic microwave background (CMB) and baryon acoustic oscillations (BAO). In this paper we propose to use the statistical lens sample from Sloan Digital Sky Survey Quasar Lens Search Data Release 3 (SQLS DR3) to constrain $f(R)$ gravity models. This sample can probe the expansion history up to $z\sim2.2$, higher than what probed by current SNIa and BAO data. We adopt a typical parameterization of the form $f(R)=R-\alpha H^2_0(-\frac{R}{H^2_0})^\beta$ with $\alpha$ and $\beta$ constants. For $\beta=0$ ($\Lambda$CDM), we obtain the best-fit value of the parameter $\alpha=-4.193$, for which the 95% confidence interval that is [-4.633, -3.754]. This best-fit value of $\alpha$ corresponds to the matter density parameter $\Omega_{m0}=0.301$, consistent with constraints from other probes. Allowing $\beta$ to be free, the best-fit parameters are $(\alpha, \beta)=(-3.777, 0.06195)$. Consequently, we give $\Omega_{m0}=0.285$ and the deceleration parameter $q_0=-0.544$. At the 95% confidence level, $\alpha$ and $\beta$ are constrained to [-4.67, -2.89] and [-0.078, 0.202] respectively. Clearly, given the currently limited sample size, we can only constrain $\beta$ within the accuracy of $\Delta\beta\sim 0.1$ and thus can not distinguish between $\Lambda$CDM and $f(R)$ gravity with high significance, and actually, the former lies in the 68% confidence contour. We expect that the extension of the SQLS DR3 lens sample to the SDSS DR5 and SDSS-II will make constraints on the model more stringent.Comment: 10 pages, 7 figures. Accepted for publication in MNRA

Statistical Inference with Stochastic Gradient Methods under ϕ\phi-mixing Data

Stochastic gradient descent (SGD) is a scalable and memory-efficient optimization algorithm for large datasets and stream data, which has drawn a great deal of attention and popularity. The applications of SGD-based estimators to statistical inference such as interval estimation have also achieved great success. However, most of the related works are based on i.i.d. observations or Markov chains. When the observations come from a mixing time series, how to conduct valid statistical inference remains unexplored. As a matter of fact, the general correlation among observations imposes a challenge on interval estimation. Most existing methods may ignore this correlation and lead to invalid confidence intervals. In this paper, we propose a mini-batch SGD estimator for statistical inference when the data is $\phi$-mixing. The confidence intervals are constructed using an associated mini-batch bootstrap SGD procedure. Using ``independent block'' trick from \cite{yu1994rates}, we show that the proposed estimator is asymptotically normal, and its limiting distribution can be effectively approximated by the bootstrap procedure. The proposed method is memory-efficient and easy to implement in practice. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory