\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 ground-based detectors involved in the search of gravitational waves 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.\\

\noindent
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 by 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 \cite{branchesi}.\\
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 are 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{Redshifted frequencies}
When dealing with cosmological objects, we need to take into account the contribution of the redshift: in the case of gravitational waves, the redshift acts on the observed frequency. In a cosmological context, the time-scale is redshifted, and so it is the frequency observed $f_{obs}$ with respect to the emitted one $f_{em}$ by:

\begin{equation}
\centering
f_{obs} = f_{em}/(1+z).
\end{equation}

\noindent
The implication of this effect lies in a factor (1+z) multiplied to the masses involved.\\
An important consequence is that if the instrument could be able to detect in a broader range of lower frequencies, it could be possible to identify objects located at higher redshifts, i.e. more ancient, or apparent high masses increased by the cosmological distance \cite{yu}. Examples of these objects are Intermediate Mass Black Holes (IMBH) or stellar-mass BHs, whose nature and physics are still unknown.

\subsection{Multi-messenger astronomy and low frequencies}
Multi-messenger astronomy is a branch of astronomy born with the discovery of the first gravitational wave. It has been seen that the signal of a gravitational wave can be followed up by observatories operating in other frequency bands to localize and study the source under several other points of view \footnote{A general overview about multi-messenger astronomy can be found in \cite{branchesi}. An interesting paper about a multi-messenger GW-source detection and its implications is \cite{multi}.}.\\
It is then important that the communication between these observatories is the best of the efficiency: the joint-collaboration is  determinant to provide a precise localization of the source in the sky and a complete set of data to study the object in all its details \cite{bird}.\\
The main challenge when an electromagnetic observatory tries to follow up a signal from a gravitational-wave detector is the time spent in the communication of the signal, and in the adjustments of the instrument towards the right position in the sky. This can be achieved if the gravitational-wave detector is able to provide a location quickly and accurately.\\
A significant contribution to this goal could be added by the opening of the lower frequency window of ground-based gravitational-wave detectors. As see in the previous section, the time to coalescence scales with frequency as $f^{-8/3}$. Lowering the frequency of observation would increase the time of observation before the coalescence. This would give more time for the electromagnetic detectors to adjust the position once received the coordinates. Moreover, the more the two inspirilling objects are far from coalescence, the more are they far from each other, increasing the volume of observation in the sky.


\subsection{Duty cycle of the detector}
The ground-based instruments are currently tuned to detect inspiraling binaries: the duty cycle of the detector is then very important for assessing its sensitivity.  This quantity represents the number of cycles spent by the instrument in the frequency bandwidth of interest. As we will see in Chapter \ref{CPSdiff}, this depends on how much time the detector can maintain resonance, i.e. on its stability, and it is given by \cite{mag}:

\begin{equation}
\centering
N_c = \int f_{gw}(t) dt.
\end{equation}

\noindent
This quantity defines for how many cycles (and hence how much time) the detector can follow the evolution of a signal in a given frequency band. The ground-based detectors are sensitive to operate for thousands of cycles \footnote{Interesting examples on typical duty cycles for ground- and space-based detectors can be found in \cite{mag}.}. Lowering the frequency band and increasing the sensitivity would increase the number of cycles, allowing to follow a signal for more time \cite{mag}. The consequent advantage is more precise waveform predictions based on these observations, besides to the detection of objects still unknown.

\section{The goals of the gravitational-wave collaboration}
The efforts of the scientific collaboration, towards the opening of the low frequency window, are devoted to the development of new technologies for active control of the noise sources, responsible of the lack of sensitivity below 30 Hz \cite{lf1} \cite{lf2} \cite{lf3}. This has been the target focussed on during the workshops dedicated to the low frequency band, which the author attended between 2018 and 2021.\\
The goal of these meetings is to update the state-of-the-art of the topic and work together on new ideas and possible new solutions.\\
We will see in the next chapters that one of the most important noise sources, affecting the detectors in the low frequency range, is the seismic noise. The strategy investigated is based on the subtraction of this noise source: in particular, modelling, controls and reduction of the noise of seismic platforms are currently under exam for increasing the sensitivity below 30 Hz. Besides this, the study for  the development of lower-noise sensors is also an up-to-date topic of discussion.\\

\noindent
The importance of the improvement of seismic motion has been widely outlined and highlighted \cite{lantztalk}: the final goal is to reduce the noise coupling into the gravitational-wave signal, and an important contribution could be provided by the efforts of the people working on the seismic noise suppression. \\

\noindent
It is in this frame that the work exposed in this thesis finds place. The experiments carried on cover both the studies for noise suppression of seismic platforms on gravitational-wave detectors, and the development of new devices for sensing and reduce seismic motion.


