Harnessing Unresolved Lensed Quasars: The Mathematical Foundation of the Fluctuation Curve

TL;DR

This paper explores the mathematical basis of a fluctuation-based method for identifying unresolved gravitationally lensed quasars, demonstrating its effectiveness and robustness against noise by analyzing the auto-correlation of the light curve derivatives.

Contribution

It provides a detailed mathematical explanation of the fluctuation curve technique, showing its reliance on the auto-correlation of the derivative of the joint light curve for detecting lensed QSOs.

Findings

Fluctuation signal is dominated by the auto-correlation of the derivative of the joint light curve.

The derivative's ACF is more reliable than the light curve's ACF due to quasar variability.

Minimization of fluctuation yields better detection accuracy under noisy conditions.

Abstract

Strong gravitational lensed quasars (QSOs) have emerged as powerful and novel cosmic probes as they can deliver crucial cosmological information, such as a measurement of the Hubble constant, independent of other probes. Although the upcoming LSST survey is expected to discover $10^3-10^4$ lensed QSOs, a large fraction will remain unresolved due to seeing. The stochastic nature of the quasar intrinsic flux makes it challenging to identify lensed ones and measure the time delays using unresolved light curve data only. In this regard, Bag et al (2022) introduced a data-driven technique based on the minimization of the fluctuation in the reconstructed image light curves. In this article, we delve deeper into the mathematical foundation of this approach. We show that the lensing signal in the fluctuation curve is dominated by the auto-correlation function (ACF) of the derivative of the joint light curve. This explains why the fluctuation curve enables the detection of the lensed QSOs only using the joint light curve, without making assumptions about QSO flux variability, nor requiring any additional information. We show that the ACF of the derivative of the joint light curve is more reliable than the ACF of the joint light curve itself because intrinsic quasar flux variability shows significant auto-correlation up to a few hundred days (as they follow a red power spectrum). In addition, we show that the minimization of fluctuation approach provides even better precision and recall as compared to the ACF of the derivative of the joint light curve when the data have significant observational noise.