Time domain astrophysics with transient sources

Delay estimate via Cross Correlation Function techniques

¹ Department of Physics, University of Trento, Via Sommarive 14, 38122 Povo (TN), Italy
² Dipartimento di Fisica, Università degli Studi di Cagliari, SP Monserrato-Sestu, km 0.7, I-09042 Monserrato, Italy
³ Università degli Studi di Palermo, Dipartimento di Fisica e Chimica, Via Archirafi 36, 90123 Palermo, Italy
⁴ INAF–Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I–34143 Trieste, Italy
⁵ Istituto Nazionale di Fisica Nucleare (INFN), sezione di Trieste, Trieste, Italy
⁶ Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Brno, Czech Republic
⁷ Ioffe Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia
⁸ INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
⁹ INAF/IASF Palermo, Via Ugo La Malfa 153, I-90146 Palermo, Italy
¹⁰ IRAP, Université de Toulouse, CNRS, UPS, CNES, 9, Avenue du Colonel Roche BP 44346, F-31028 Toulouse, Cedex 4, France

^⋆ Corresponding author: wladimiro.leone@unitn.it

Received: 18 November 2024
Accepted: 25 June 2025

Abstract

The timing analysis of transient events allows for investigating numerous still open areas of modern astrophysics. The article explores all the mathematical and physical tools required to estimate delays and associated errors between two Times of Arrival (ToA) lists, by exploiting Cross Correlation Function (CCF) techniques. The CCF permits the establishment of the delay between two observed signals and is defined on two continuous functions. A detector does not directly measure the intensity of the electromagnetic signal (interacting with its material) but rather detects each photon ToA through a probabilistic process. Since the CCF is defined on continuous functions, the crucial step is to obtain a continuous rate curve from a list of ToA. This step is treated in the article and the constructed rate functions are light curves that are continuous functions. This allows, in principle, the estimation of delays with any desired resolution. Due to the statistical nature of the measurement process, two independent detections of the same signal yield different photon times. Consequently, light curves derived from these lists differ due to Poisson fluctuations, leading the CCF between them to fluctuate around the true theoretical delay. This article describes a Monte Carlo technique that enables reliable delay estimation by providing a robust measure of the uncertainties induced by Poissonian fluctuations. GRB data are considered as they offer optimal test cases for the proposed techniques. The developed techniques provide a significant computational advantage and are useful analysis of data characterized by low-count statistics (i.e., low photon count rates in c/s), as they allow overcoming the limitations associated with traditional fixed bin-size methods.