• detection theory
• gravitational wave background
• non-Gaussian
• optimization
• stochastic processes

The thesis purpose is mainly to show how some common techniques of quantum field theories could be applied to the design of better detection algorithms, in the context of stochastic background of gravitational waves.

The thesis is structured as follows: in Chapter 1 we introduce the framework of propagating wave solutions of Einstein equations. We outline the main features of interferometers, including their coupling to gravitational wave radiation. Then, we summarize the features of our reference cosmological model, a Friedmann-Robertson-Walker-Lemaitre universe. We show the performance of single and multiple interferometers detections.

In Chapter 2 we overview two interesting sources for our study: cosmic strings cusps and binary black hole coalescences. For the former, we outline the spontaneous simmetry breaking mechanisms, and the formation of topological linear defects. We quantify the associated spectrum of gravitational wave, and characterize in time domain the average signal overlap, as seen from a single interferometer. For the latter, we describe two fiducial cosmological population, and define three different regimes for the background, afflicting deeply the best detection strategy.

In Chapter 3 we introduce the basics of stochastic processes, in particular of Campbell processes. We overview two equivalent probabilistic structures behind them. We show how to evaluate their mean values via a generalized concept of probability distributions. We carry out relevant analogies with field theory correlation functions. The reader will be provided here with the necessary tools to manipulate the processes involved in the procedure later explained in Chapter 5.

In Chapter 4 we review the general framework of decision theory. We describe a standard approach to signal detection, and the tools needed to characterize a detector performances. We formulate the criterion for an optimal detection strategy, the Neyman-Pearson lemma. We use some toy signals (both deterministic and stochastic) to split the different aspects of the detection problem into smaller parts. We derive analitically the standard optimal detector for a gaussian background and study its performances.

Then, we switch our attention to the detection of a non Gaussian stochastic background. We show a simple perturbative expansion of the likelihood, assuming the correlators to be hierarchically ordered by a scaling parameter. We show how to construct the relevant statistics of the data. We draw out some considerations on the computational issues of such algorithm.

With all the necessary mathematics in hand, in Chapter 5 we tackle the detection strategy by employing a resummation tecnique borrowed from renormalization group theory. We explain the tecnique in detail with a simpler example of a single variable, then we extend the results for multiple detector timeseries. They will find best confirmations with both numerical and analitical validation to appear in a proposed scientific paper. Throughout the whole text we will underline the analogies and differences with respect to quantum field theories probabilistical structure, considered in many passages as a compass for right direction.

Finally, in Chapter 6 we summarize the results, and draw out some consideration on future developments of the proposed approach.