chapter4.tex

\chapter{Event and Population Analysis}\label{ch:analysis}

    Using the population models described in Chapter \ref{ch:synthesis}, one can derive
    the properties of NSBH mergers as seen in the GW and EM regimes.  Specifically, an
    important aspect of NSBH mergers is what fraction of mergers actually produce a
    prompt component which will be detectable by present-day gamma-ray detectors, such
    as \textit{FERMI}. A related aspect is the dependence of this fraction on the priors
    which go into creating these populations. Specifically, different black hole spin
    prior distributions would affect the number of detectable prompt emissions, and so
    this behaviour is worth investigating. Furthermore, this population synthesis
    framework may also help to investigate the rate density of NSBH mergers which lead
    to observable SGRBs, which can then be compared to the rate derived currently from
    purely EM observations.\\
    Additionally, events seen solely in the GW regime without any confident EM
    counterparts may be analysed using the framework set up here. Using the posterior
    distributions for the component masses, spins etc., derived from GW strain data
    analysis, one can derive the corresponding remnant, dynamic and disc masses from
    which the jet energetics can be derived. Under the assumptions of the underlying
    model, this process of computation can then help eliminate or give credence to the
    existence of EM counterparts for events observed via GW only.

\section{Analysis of Population Synthesis Models}

    From the description of the population models given in Chapter \ref{ch:synthesis},
    it is clear that there are three classes of populations, segregated by the black
    hole spin population that is used. Specifically, the three classes of populations
    are:

    \begin{itemize}

        \item \textbf{Class I} -- this uses the standard \textsc{truncated} BH mass
            distribution with a constant NS mass of 1.4 M$_\odot$. The BH spin
            distribution is the beta distribution from \cite{abbott_2020B}, and the NS
            spin is set identically to 0 for all samples\footnote
            {
                See \S\ref{sec:ns_pop} for reasons on why this assumption is valid.
            }
            The remaining NS parameters and their computation (such as the NS
            compactness, tidal deformability etc.) are described in \S\ref{sec:ns_pop},
            and thus omitted for brevity.

        \item \textbf{Class II} -- here, the BH spin distribution is assumed to be
            uniform between [0, 1). The rest of the binary parameter distributions are
            the same as in Class I.

        \item \textbf{Class III} -- here, the BH spin is a "restricted" Gaussian
            distribution between [0, 1), which means that samples between [0, 1) are
            taken as is but samples outside of this range are discarded.  For the
            purposes of being general, samples are drawn from distributions with
            standard deviation, $\sigma = 0.2$ and mean, $\mu = 0.2, 0.5, 0.7$.  The
            rest of the binary parameter distributions are the same as in classes I and
            II.

    \end{itemize}

    For each population, 10$^5$ samples are drawn from the relevant populations. Using
    the samples, the various mass parameters required to compute the energetics of the
    jet are calculated using Eqs. \ref{eq:m_out} -- \ref{eq:constraint}. Then, using
    Eqs.  \ref{eq:e_kin_jet} -- \ref{eq:eiso}, the jet structure is imposed and for each
    event the value of $E_{iso}(\theta_v)$ is computed. These are then converted into
    fluences using the equation:

    \begin{equation}
        \mathcal{F} = \dfrac{E_{iso}(\theta_v)}{4\pi d_L^2}
        \label{eq:fluence}
    \end{equation}

    This is done in order to ascertain the detectability of the prompt emission with a
    reference gamma-ray detector, such as the \textit{FERMI-GBM}. Additionally, a
    fluence cutoff of $2 \times 10^{-7}$ erg/cm$^2$ is assumed, which is a typical
    number observed for the fluence of cosmological SGRBs. Any event with a fluence
    lower than this value will be considered as a non-detection. Albeit this is a tight
    and somewhat ephemeral restriction, it helps to ascertain the rough number of
    detections versus non-detections in the absence of more rigorous cut-offs for such
    detectors (see \cite{bhat_2016}).\\
    Similarly, with the binary parameters set by the sampling process, the GW network
    SNR is computed with the RWF as the GW template, using Eqs. \ref{eq:rho} --
    \ref{eq:freq_integral}. The detector network configuration consisting of Advanced
    LIGO (LIGO-Livingston, L1 and LIGO-Hanford, H1) and VIRGO (V1) detectors was set up
    using PyCBC's \texttt{detector} module, which builds up the detector locations,
    antenna pattern functions, location phase factors etc. For all three detectors, the
    Advanced LIGO Design sensitivity was used as an approximation for their respective
    future configurations. This was done, since with the current network configuration,
    there have been no confident detections of NSBH mergers which have \textit{also} had
    an EM counterpart.\\
    The results of the simulations are given below, with each population producing a
    particular number of NSBH binary merger events seen in:

    \begin{enumerate}

        \item The GW regime alone, $\boxed{\mathcal{N}_{GW}}$ -- these are events whose
            GW network SNR is calculated to be higher than the network SNR threshold of
            10.

        \item The EM regime alone, $\boxed{\mathcal{N}_{EM}}$ -- these are events whose
            simulated gamma-ray fluence is calculated to be above the fluence
            limit of $2 \times 10^{-7}$ erg.

        \item The GW \textit{and} EM regimes both, $\boxed{\mathcal{N}_{EM+GW}}$ --
            these are events which satisfy both the GW and EM `cut-offs', and thus
            represent joint detections.

    \end{enumerate}

    These numbers are collected together in Table \ref{tab:popln_numbers} for each
    population class. From this table it is evident that nature of the spin distribution
    heavily affects the number of EM-only events, and by extension the number of joint
    events.\\
    This can be explained by looking at the dependence of $M_{disc}$ on the BH
    spin, from Eq. \ref{eq:disc_mass}, for a fixed mass ratio. Since a highly spinning
    black hole produces more tidally disrupted material for a given mass ratio than a
    low spinning black hole, binaries with a higher black hole spin are more likely to
    produce a more massive disc (assuming not much mass is lost as dynamical mass).
    This behaviour of $M_{disc}$ is also shown below in Fig. \ref{fig:m_disc}.

    \begin{table}[H]
        \centering
        \caption{Number and Kind of Detections across Population Classes}
        \begin{tabular}{cccc}
            \toprule

                &
            $\mathcal{N}_{EM}$ &
            $\mathcal{N}_{GW}$ &
            $\mathcal{N}_{EM+GW}$ \\

            Class I & 28 & 12283 & 9 \\

            Class II & 666 & 11274 & 241 \\

            Class III$_{\mu = 0.2}$ &
            25 & 11250 & 6 \\
            Class III$_{\mu = 0.5}$ &
            290 & 11690 & 93 \\
            Class III$_{\mu = 0.7}$ &
            932 & 11468 & 330 \\
            \bottomrule
        \end{tabular}
        \label{tab:popln_numbers}
    \end{table}


    \begin{figure}[H]
        \centering
        \includegraphics[width=0.8\linewidth]{m_disc}
        \caption[Variation of $M_{\mathrm{disc}}$ with $\chi_{BH}$ and $\mathcal{Q}$]{
            Variation of the disc mass with the mass ratio, $\mathcal{Q}$ and the black
            hole spin, $\chi_{BH}$. Note that the higher disc mass values are achieved
            when $\chi_{BH}$ is high and $\mathcal{Q}$ is low. Reproduced from
            \cite{barbieri_2019b}.
        }
        \label{fig:m_disc}
    \end{figure}

    This then means that because populations with `high spin' distributions (such as
    Class I or Class III$_{\mu = 0.7}$) have a higher proportion of high spin binaries,
    they have a higher number of EM and joint detections, since a more massive disc
    produces a more energetic jet from Eq. \ref{eq:eiso}, and thus increases the
    possibility of detections.

\section{Event Analysis}\label{sec:event_analysis}

    \subsection{GW190426\_152155}\label{ssec:nsbh_190426}

        This event, as described in GWTC-2 and in \S\ref{sec:190426}, has an
        astronomical source probability of 42\% and a terrestrial event possibility of
        $\sim$52\%. Even though this is the case, there is still speculation on whether
        this event could be an NS merger event.  This is still possible since the priors
        used for the classification mechanism depend sensitively on knowledge about the
        stochastic GW background and the source classification may very well change in
        the future, as more analysis is done on the GW noise observed as the LIGO/VIRGO
        interferometers operate.\\
        With this uncertainty in mind, asking the question of whether GW190426\_152155
        could be an NSBH event is pertinent, and the current framework makes for a
        suitable test bed to test this hypothesis with. Using the LSC data release for
        the binary system parameters associated with GW190426\_152155, which gives
        samples from the posterior distributions for $M_{BH}, M_{NS}$ etc., the disc
        mass was computed for each set of binary parameters. Computing the disc mass is
        expeditious since it directly gives an estimate of how many sets of samples will
        be able to \textit{at least launch} a jet, since if $M_{disc} = 0$ for a
        particular set of samples, there is no possibility of detecting a jet from this
        particular binary.\\
        Fig. \ref{fig:mdisc_q_190426} is the result of this analysis in the
        $\chi_{BH}-\mathcal{Q}$ plane. From this figure, it is seen that a minor but
        non-negligible fraction of posterior samples ($\sim$32\%) actually falls within
        the section of the parameter space where $M_{disc} \neq 0$. Furthermore, using
        the posterior samples for the inclination angle $\iota$ and the luminosity
        distance $D_L$, the fluence for each sample set is computed using Eq.
        \ref{eq:fluence}.  This distribution is shown in Fig.
        \ref{fig:fluence_190426}.\\

        \begin{figure}[ht]
            \centering
            \includegraphics[width=0.8\linewidth]{fluence_190426}
            \caption[Fluence distribution for GW190426, from LSC Posteriors]
            {
                The distribution of fluence computed from the posterior samples released
                by the LSC for GW190426, using Eq. \ref{eq:fluence}.
            }
            \label{fig:fluence_190426}
        \end{figure}

        From this analysis, it is seen that out of a total of 4796 samples, 1580 samples
        formed a disc with non-zero mass. Out of these 1580 samples, 1344 samples formed
        a jet with a fluence lesser than $1.7 \times 10^{-7}$ erg/cm$^2$, which is what
        was observed by \textit{INTEGRAL} around the time of merger. So overall,
        operating under the assumption that the \emph{NSBH} event GW190426 launched a
        SGRB jet, roughly $\sim 30$\% (= 1344/4796) agree with the observations.  This
        means correspondingly that the scenario where the \emph{NSBH} event GW190426
        launches a SGRB jet is disfavoured, but non-negligible.\\
        This can be interpreted trivially to conclude that the event was of terrestrial
        origin (as is evidenced by the large classification probability assigned to
        $\mathcal{P}$(\textbf{Terr.}) in \ref{tab:p_astro}), and thus it follows that it
        would not be compatible with the launching of an astrophysical jet. But
        non-trivially, one can also conclude that the event:
        \begin{itemize}
            \item  Could have been an BNS merger event which had a jet that was launched
                via mechanisms other than the Blandford-Znajek mechanism.

            \item Could have been an NSBH merger event whose jet energetics was decided
                by equations different to the ones used here. For example, the value of
                $\epsilon$ used to calculate the kinetic energy of the jet (see Eq.
                \ref{eq:e_kin_jet}) is not constrained well enough, and a large enough
                variation in its value could affect the simulated fluences to increase
                the odds of GW190426 being an NSBH event.

        \end{itemize}

        The actual non-detection of such a jet (if launched under these alternative
        conditions) could be either due to the inefficiency of the internal engine for
        such jet launching mechanisms, or due to a viewing angle large
        enough that the jet was relativistically deboosted to below the thresholds of
        current-day gamma-ray observatories, which were observing 100\% of the
        localization region.

        \begin{figure}[H]
            \centering
            \def\svgwidth{\linewidth}
            \input{figures/190426_in_q-chi_high.pdf_tex}
            \caption[GW190426 in the $\chi_{BH}-\mathcal{Q}$ plane]
            {
                GW190426\_152155 in the $\chi_{BH}-\mathcal{Q}$ plane. The colour coding
                at a point in the plane indicates the value of $M_{disc}$ corresponding
                to that particular $\chi_{BH}$ and $\mathcal{Q}$.  Unfilled circles with
                a black outline correspond to samples from the posterior distributions
                for GW190426\_152155, which have $M_{disc} = 0$, and so cannot support
                jet launching, and filled black circles are those with $M_{disc} \neq
                0$.
            }
            \label{fig:mdisc_q_190426}
        \end{figure}

    \subsection{GW170817}

        For GW170817, as described in Chapter \ref{ch:introduction}, a lot of questions
        remain regarding the specifics of the jet launching mechanisms. Recent studies
        also show that there is a non-negligible possibility that the event was an NSBH
        merger, with a low mass BH acting as the other, non-NS component (see for
        example, \cite{hinderer_2019}). This would be compatible with the outflows
        observed from GW170817, specifically the SGRB jet GRB170817A, optical Kilonova
        AT2017gfo and associated afterglows (see \cite{abbott_2017}) since NSBH mergers
        also can produce similar outflows as shown below.

        \begin{figure}[H]
            \centering
            \includegraphics[width=\linewidth]{nsbh_outflows}
            \caption[EM outflows from NSBH mergers, from \cite{barbieri_2019a}]
            {
                Schematic diagram showing the various possible outflows from a suitable
                NSBH merger, along with their windows of visibility in the EM spectrum,
                their physical properties and their relevant launching/energy production
                mechanisms. From \cite{barbieri_2019a}.
            }
            \label{fig:nsbh_outflows}
        \end{figure}

        In order to better understand this event, an analysis similar to that in
        \S\S\ref{ssec:nsbh_190426} was carried out. The LSC data release for GW170817
        (part of the GWTC-1 catalogue data release; see \cite{gwtc1_DR}) was used from
        which the binary system parameters were extracted. Note that for this data
        release, there are two `flavours' of posterior distributions which are supplied
        by the LSC data product, classified by the spin priors using in the Bayesian
        parameter estimation process.  The first is termed
        \texttt{IMRPhenomPv2NRT\_highSpin\_posterior}\footnote
        {
            \texttt{IMRPhenomPv2NRT} is the LAL C routine that models the
            phenomenological inspiral-merger-ringdown gravitational waveform for a
            spinning, precessing binary with numerical relativity-tuned tidal effects
            (see \cite{lalsuite}, and \cite{dietrich_2019}).
        }
        whereas the second is termed \texttt{IMRPhenomPv2NRT\_lowSpin\_posterior}, and
        only the first is considered in the current analysis, since the latter assigns
        to both binary components a low spin distribution (since the assumption there
        was that both components are galactic NSs, and thus will have low spins).\\
        With the assumption that the distribution with the larger median mass is that of
        the BH and the other is that for the NS, the mass ratios, corresponding spins
        and tidal deformabilities are extracted from the posterior samples. These
        samples were used to compute the corresponding $M_{disc}$ and the results for
        each flavour of the posterior distribution is shown in Fig.
        \ref{fig:170817_high} below.

        \begin{figure}[ht]
            \centering
            \def\svgwidth{\linewidth}
            \input{figures/170817_in_q-chi-high.pdf_tex}
            \caption[$M_{disc}$ for GW170817's High Spin Posterior Distribution]
            {
                Samples from the high spin posterior distribution for GW170817, in the
                $\mathcal{Q}-\chi_{BH}$ plane. The colour coding for a point in the
                plane is the $M_{disc}$ computed for that $\mathcal{Q}$ and $\chi_{BH}$,
                and unfilled (white) circles represent posterior samples which have
                $M_{disc} = 0$.
            }
            \label{fig:170817_high}
        \end{figure}

    As can be seen from Fig. \ref{fig:170817_high}, more than 50\% of the posterior
    sample sets produce a disc mass (i.e. $M_{disc} \neq 0$), and thus can support jet
    launching given the right black hole spin and energy extraction conditions.
    Specifically for this high-spin posterior, around 57.1\% of the samples produce a
    disc.\\
    Furthermore, from EM follow-up observations and analysis, estimates have been
    obtained for the viewing angle, $\theta_v$ at which GW170817 was seen (see
    \cite{finstad_2018}). This value has been constrained to be $\theta_v \sim
    20^{\circ}$. Using this, the value of the apparent isotropic equivalent energies for
    the posterior samples is calculated, which show a median value of
    $E_{iso}(20^{\circ}) \approx 1.21 \times 10^{47}$ erg for the high-spin posterior.
    This number is comparable to that derived from the gamma-ray observations of
    GRB170817A by \textit{FERMI-GBM}. Although this argument suffices to show what was
    intended, more work is being done to take into account the actual constraints
    derived for the viewing angle.

    \subsection{GW190425}\label{ssec:nsbh_190425}

        For completeness, the analysis carried out for GW170817 and GW190426\_152155 is
        also carried out for GW190425. A similar process is performed and the
        posterior samples released for the component masses, component spins etc., are
        used to compute the disc mass using the current framework. The result of doing
        this analysis is shown in Fig. \ref{fig:190425_high}.\\
        As can be seen from this figure, most of the samples support the formation of a
        disc around the remnant black hole. In fact, for the high-spin posterior, around
        72.1\% of the samples have a $M_{disc} > 0$. Additionally, the fluences were
        computed for the posterior sample sets for this event. Out of a total of 226598
        posterior samples, of which 163443 samples form a disc, only 2072 samples launch
        a jet with a fluence that is compatible with the fluence measured by
        \textit{INTEGRAL} at the time of the merger (which is $(1.6 \pm 0.4) \times
        10^{-7}$ erg/cm$^2$). Thus, less than 1\% of the total samples are compatible
        with the observations. This can be interpreted as indicating the fact that
        GW190425 was in fact not an NSBH event.\\

        \begin{figure}[ht]
            % The SVG for this is too big!
            \centering
            \includegraphics[width=\linewidth]{190425_in_q-chi-high}
            \caption[$M_{disc}$ for GW190425's High Spin Posterior Distribution]
            {
                Samples from the high spin posterior distribution for GW190425, in the
                $\mathcal{Q}-\chi_{BH}$ plane. The colour coding for a point in the
                plane is the $M_{disc}$ computed for that $\mathcal{Q}$ and $\chi_{BH}$,
                and unfilled (white) circles represent posterior samples which have
                $M_{disc} = 0$.
            }
            \label{fig:190425_high}
        \end{figure}

\section{Summary}

    In this chapter, the populations synthesized using codes described in Chapter
    \ref{ch:synthesis} and events of interest introduced in Chapter \ref{ch:candidates}
    are analysed more carefully. Specifically, in the case of GW190426\_152155, the
    hypothesis that it could have been an NSBH merger event is carefully considered
    under the current framework. Since it did not fit in well with the current
    framework, alternatives hypotheses are proposed, although more work needs to be done
    in terms of verifying these alternative hypotheses.\\
    Similar analyses was also carried out for the events GW170817 and GW190425, which
    are currently classified as BNS mergers to a high degree of confidence, but still
    admit non-negligible possibility to launching a jet within the current framework.\\
    As for the population analysis, it was ascertained that the EM outflows from NSBH
    mergers with the currently assumed population parameters strongly depends on the
    black hole spin prior distribution. In the absence of any strong indication to
    favour one spin distribution over another in the case of black holes in an NSBH
    binary, all of these spin distributions must be considered likely. Further work
    needs to be done in order to take these assumptions about the spin distributions to
    their logical ends, and in doing so, it is hoped that multiple spin distributions
    would be ruled out due to incompatibility with the observational data. This is what
    is planned to be carried out, via the computation of the rate density of SGRBs as
    calculated from NSBH merger rates. This number will be an independent estimate of
    the SGRB rate density as calculated by EM observers, and will serve to differentiate
    physical from non-physical priors.