updates

main/GW.tex

@@ -7,7 +7,7 @@ A detailed structure of the thesis is then following.
 
 \section{Gravitational waves and their detection}
 
-Gravitational waves are an astrophysical event that takes place when massive objects move and deform the fabric of the spacetime \cite{mag} \footnote{An in-depth source about how gravitational waves have been computed and their features is \cite{mag}.}. They have been theorized by Einstein in 1915 and discovered a hundred years later by a joint collaboration of three detectors \cite{nar} \cite{first}, which was worth of the Nobel Prize for Physics in 2017 \footnote{See Appendix C for some information about the first detection of gravitational waves.}.\\
+Gravitational waves are an astrophysical event that takes place when massive objects move and deform the fabric of the spacetime \cite{mag} \footnote{An in-depth source about how gravitational waves have been computed and their features is \cite{mag}.}. They have been theorized by Einstein in 1915 and discovered a hundred years later by a joint collaboration of two detectors \cite{nar} \cite{first}, which was worth of the Nobel Prize for Physics in 2017 \footnote{See Appendix C for some information about the first detection of gravitational waves.}.\\
 
 \noindent
 The effect of the gravitational waves when they pass through an object is to produce a deformation on the physical lengths (L). This effect is very small ($\Delta$L/L $\sim$ 10$^{-21}$): masses able to deform the fabric of the spacetime and generate gravitational waves are of the order of more than the solar mass $M_{\odot}$, so they need to be looked for in the Universe.\\

main/LF.tex

@@ -46,9 +46,9 @@ The emitted frequency from a source of gravitational waves depends on the masses
 \end{equation}
 
 \noindent
-where $M_c$ is the combination of the two involved masses m$_1$ and m$_2$, defined as \textit{chirp mass} = (m$_1$ m$_2$)$^{3/5}$/(m$_1$ + m$_2$)$^{1/5}$.\\
+where $M_c$ is the combination of the two involved masses $m_1$ and $m_2$, defined as \textit{chirp mass} $M_c$ = ($m_1$ $\cdot$ $m_2$)$^{3/5}$/($m_1$ + $m_2$)$^{1/5}$.\\
 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 are essential for several reasons, going from efficiency of the detector in detecting different 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 hence 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}.}.\\
+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 hence 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}