\chapter{The low frequency window}
\label{LF}
The scientific research exposed in this thesis focusses on the improvement of ground-based gravitational-wave detectors at low frequency. This chapter intends to frame the work done in this context and highlight why the lower frequency window is so important. The discussion around this topic is relatively recent and it has been widely debated during dedicated workshops which the author of this thesis attended since 2018.

\section{Sources of gravitational waves}
Fig. \ref{spec} summarizes the possible objects that can be gravitational waves sources, their frequency emission and what kind of instrument can detect them. The terrestrial interferometric detectors are the most involved at present times, but the efforts of the scientific community are going towards the development of new detectors both ground- and space-based in order to widen the frequency window of observation.

\begin{figure}[h!]
\centering
\includegraphics[scale=1.6]{images/spectrum}
\caption[Sources of gravitational waves]{Spectrum of emission of sources of gravitational waves (adapted from https://lisa.nasa.gov).}
\label{spec}
\end{figure}

\noindent
The best modelled sources are binary systems, typically Neutron Stars (NS), White Dwarfs and Black Holes (BH), orbiting each other. Fig. \ref{binary} shows the main phases of the evolution of the systems, emitting gravitational waves at different frequencies, depending on the phase.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/bin.png}
\caption[Phases of gravitational waves emission by a binary system]{The three phases of a BH-BH binary system emitting gravitational waves (amplitude vs time) \cite{first}. \textbf{Inspiral phase}: the orbits shrink, velocity increases and frequency of the waves emitted increases as $f_{gw} = 2f_{orbital}$. \textbf{Merging phase}: the objects merge and the signal is maximum. \textbf{Ring-down phase}: a new BH is formed and the signal emitted decreases in frequency as a damped sinusoid.}
\label{binary}
\end{figure}

\noindent
Gravitational waves from binary systems can provide several information about the equation of state of Neutron stars, masses and spin of Black holes, test of General Relativity theory.\\
Currently, the ground-based observatories are tuned to detect binary systems sources: interferometers are the instruments that have been able to detect gravitational waves from binary systems.\\

\noindent
The first detection of gravitational waves happened on the 14th September 2015 and confirmed the Theory General Relativity, opening a new window on the Universe: the signal from a merger of two black holes have been observed thanks to the emission of gravitational waves, confirming the existence of these objects, still mostly unknown \cite{first}. The detector responsible of the new discovery is based in the USA and it is one of the terrestrial interferometers currently in use for gravitational waves detection \footnote{The working principles of the interferometers and details about the US instrument are exposed in Chapter \ref{LIGO}}.

\section{Opening the low frequency window}
As we will see in the next chapter, the grund-based detectors involved in the search of gravitational waves are cover a wide range of frequencies, but they are affected by some noises which make them unable to detect waves from sources emitting below 30 Hz. We will see later the nature of these noises. The reason why it is important to open the lower frequency window is that it can give access to the detection of gravitational waves emitted by sources which physical structure and astrophysical features are still unknown.\\
This is the effort towards which a huge part of the scientific collaboration is involved.

\subsection{Frequencies of emission}
We saw in the previous chapter that the emitted amplitude depends on the masses and the orbital frequency involved: the emitted frequency is also linked to these parameters. For mergers of binary systems, the frequency of a gravitational wave is twice the orbital frequency of its source \cite{mag} and hence it can be used to know the relation between the masses and the time to coalescence, i.e. the time to the merger \footnote{A simple example based on point-like masses in circular orbits is explained in details in \cite{mag}.}. For masses in circular orbits, this is given by:

\begin{equation}
\centering
\tau \simeq 2.18 s \left(\dfrac{1.21 M_{\odot}}{M_c}\right)^{5/3} \left(\dfrac{100 Hz}{f_{gw}}\right)^{8/3}.
\end{equation}

\noindent
This equation is particularly useful if we want to know information about the radiation emitted my a certain mass, at a certain frequency at a certain time before the merger. Predictions about this time and the frequency where it is possible to detect the radiation is essential for several reasons, from efficiency of the detector in terms of variety of sources to Multimessenger astronomy, in which timing is important to assure a correct localization of the source CITA MULTI.\\
In our case of interest, if we apply the lowest range of frequency available by ground-based detectors ($\sim$ 10 Hz in order of magnitude and consider M$_c$ = 1.21 M$_{\odot}$, it is possible to observe the radiation emitted at $\tau$ = 17 minutes to coalescence. This equation says that the larger is the time to coalescence, the smaller is the masses involved \footnote{A useful exercise to prove this is by applying the Kepler's law for different emitting frequencies and masses. Some interesting examples are given in \cite{mag}.}.\\
Recalling Fig. \ref{spec}, the range of the frequencies of emission below 10 Hz lies almost all in the space-based detectors dominion. Opening this frequency window would allow the ground-based detectors to access to a frequency bandwidth which is still not investigated and would allow the detection from sources whose physics is still unknown. 

\subsection{Duty cycle of the detector}


\section{The goals of the collaboration}



