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Rapid Development and Maintenance Language, aka Rapid Development and Maintenance Language, is a historical programming language created in 2005.
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Capitalism has brought us prosperity, certainly. No other economic system has been able to match its energy and innovation. But it has a dark side of exploitation, instability and extremes of inequality.
The answer to these problems however is not to abandon capitalism. Rather, capitalism needs to be led and bounded by values. But which values? What indeed are values anyway and how do we locate and share values strong enough to balance the power of capitalism in society? Relativism has swept away old certainties and we struggle to agree what should lie at the centre of our lives.
We show how with values placed on a strong foundation individual lives can reacquire meaning and purpose. Politics can be transformed from the half-corrupted subject of popular indifference it has become.
Above all, capitalism can and should be a servant rather than a master. Our shared values should be the end it serves.


text:
\section{Introduction}

The gravitational-wave (GW) detectors, Advanced LIGO \cite{AdvancedLIGOREF} and Virgo \cite{TheVirgo:2014hva}, now joined by KAGRA \cite{akutsu2020overview}, have made the observation of compact binary mergers almost routine. In only 6 years, they have reported ${\sim} \mathcal{O}(90)$ such observations that have provided us with unprecedented knowledge on how black holes (BHs) and neutron stars form and how they populate our Universe \cite{LIGOScientific:2018mvr,abbott2021gwtc2,abbott2021gwtc3}. Moreover, these observations have enabled the first tests of General Relativity in the strong-field regime \cite{GWTC3-TGR} and qualitatively new studies of the Universe at a large scale \cite{Abbott:2016blz, LIGOScientific:2019fpa}. All such studies require an accurate identification of the source parameters, which has been possible for most observations owing to a clear initial inspiral stage that allows to identify the parameters of the merging binary. In particular, most of such events have been confidently identified as circular black hole or neutron star mergers (BBHs and BNSs) with negligible orbital eccentricity

The detection of the GW190521 event represented the first departure from such ``canonical'' events \cite{GW190521D, GW190521I}. Owing to the large mass of its source, GW190521 barely displays any pre-merger dynamics, with the vast majority of the signal coming from the final distorted, merged object while it relaxes to its final BH form. In such a situation, there is little information about the parents of the final object, making the inference of their parameters depends strongly on the prior assumptions about them and leading to a variety of interpretations of this event \cite{IAS_GW190521,GW190521_Nitz,Isobel_ecc,Gayahtri_ecc,Proca,Gamba_GW190521}

First, the LIGO-Virgo-KARGA collaboration (LVK) reported a circular BH merger with mild signatures of orbital precession. However, Ref \cite{HOC} showed that, for such short signals, orbital precession can be confused with high eccentricity. Consistently, \cite{Isobel_ecc} and \cite{Gayahtri_ecc} argued that GW190521 could be interpreted as an eccentric merger. Despite their differences, all these interpretations lead to two main conclusions. First, the remnant BH has a mass $M_{\rm f} > 100\,M_\odot$, making it the first observation of a compact object in intermediate-mass BH mass range. Second, the heavier merging BH populates the so-called pair-instability supernova (PISN) gap, a mass-range where no BH formation is expected to occur from stellar collapse \cite{Heger:2002by}. With these two characteristics, GW190521 provides on the one hand an un-valuable clue towards understanding the formation of supermassive BHs via hierarchical merger channels \cite{Volonteri2003,Volonteri2010}. On the other, it poses the challenge of explaining the origin of such merging BHs populating the PISN gap, e.g., invoking hierarchical formation channels~\cite{Kimball2021,Liu2021}. While several other explanations for the origin of such BHs have been proposed \cite{Costa2020,DallAmico2021}, alternative studies have shown that the heavier BH in GW190521 may actually avoid the PISN gap with some probability. For instance, using a population informed prior \cite{straddling} hinted that GW190521 could actually involve one BH above the PISN gap and one below, known as a "straddling binary". Also, using an alternative mass prior, \cite{GW190521_Nitz} showed that GW190521 could be a high-mass ratio binary. Finally, as the departing point of this work, Ref \cite{Proca} showed that GW190521 is consistent with numerically simulated head-on mergers \cite{sanchis2019head} of horizonless compact objects known as Proca stars~\cite{brito2016proca}. While exotic, this interpretation automatically eliminates the presence of a BH populating the PISN gap while still yielding an IMBH remnant.\\

The third observing run of the Advanced LIGO - Virgo network has delivered more short signals similar to GW190521, namely GW190426 and GW200220 albeit with noticeably lower statistical significance of less than 1 per year \cite{abbott2021gwtc3}. In addition, a targeted search for intermediate-mass black holes delivered an intriguing trigger, named S200114f which, while observed with much larger statistical than the former, was not conclusively classified as neither a GW nor noise artefact \cite{o3_imbh}. The morphological characteristics of these signals make them merit further investigation exploring possibilities beyond the BBH paradigm. In this work, we compare all of these events to a catalogue of 759 numerical simulations of Proca-star mergers (PSMs). In particular, we perform model selection on these events between our PSM and a classical BBH model and report the estimated parameters under the PSM model. Finally, we perform a preliminary population study to estimate the fraction of PSMs within the underlying population of compact mergers.  \\

\subsection{Proca stars and dark matter}

Bosonic stars are self-gravitating lumps of bosonic fields, first constructed for massive, complex scalar fields in the late 1960s~\cite{Kaup:1968zz,Ruffini:1969qy} and more recently constructed also for massive, complex vector fields~\cite{brito2016proca}. The latter are also known as Proca fields, and thus the corresponding stars have been dubbed Proca stars. These stars can be either spherical and non-rotating~\cite{Herdeiro:2017fhv} or axially-symmetric and spinning~\cite{Herdeiro:2019mbz}. They can  be rather Newtonian but become compact in regions of the parameter space, to the point that their compactness becomes comparable (albeit smaller) to that of BHs. In this case, bosonic stars are an example of exotic compact objects (ECOs) that can mimic some of the phenomenology attributed to BHs (see e.g.~\cite{Proca, Herdeiro:2021lwl}).

From a macroscopic perspective, the simplest bosonic stars are described by free, complex, massive bosonic fields minimally coupled to gravity. Self-interactions can be introduced in the model and can change their properties~\cite{Schunck:2003kk,liebling2017dynamical,Clough:2022ygm,Coates:2022qia}, but are not mandatory for the existence of solutions (and are absent in the models considered here). From a microscopic perspective, they can be interpreted as many-particle states of ultralight bosons. The ultralightness requirement for the fundamental bosonic particle guarantees (in the simplest models) that  the bosonic stars achieve masses in the astrophysical BH range.  In particular, ultralight bosons with a particle mass $\mu_{\rm B}$ within $10^{-13}\leq\mu_{\rm B}\leq10^{-10}$~eV, yield stars with maximal masses in the interval $\sim$ 1000 and 1 solar masses, respectively.  Such ultralight bosons can be motivated by particle-physics models, from the QCD axion~\cite{Peccei:1977hh}, to the string axiverse~\cite{Arvanitaki2010} and also by simple extensions of the Standard Model of particle physics~\cite{Freitas:2021cfi}. Such ultralight particles could form part, or the whole, of the dark matter budget of the Universe \cite{EuCAPT_astroparticle}, making bosonic stars only detectable via their gravitational signatures.

Unlike other ECO models, bosonic stars have a well-established, field-theoretical description. Their dynamics have been extensively studied (see e.g.~\cite{liebling2017dynamical,bezares2017final,palenzuela2017gravitational,sanchis2017numerical,sanchis2019head}).
The corresponding bosonic fields oscillate at a well-defined frequency $\omega$, which provides a dispersive nature counteracting gravity and determines
the mass and compactness of the star.
Moreover, bosonic stars have a precise formation mechanism, which needs no fine-tuning, known as gravitational cooling~\cite{Seidel1994,DiGiovanni2018}. This is consistent with their dynamical robustness, which has been established for spherical boson stars  both perturbatively and non perturbatively~\cite{liebling2017dynamical}. On the other hand, spinning bosonic stars are more subtle; only recently it was found that in the simplest models they are unstable in the scalar case, but not in the Proca case~\cite{SanchisGual2019,di2020dynamical}.
This motivated considering collisions of spinning Proca stars. In Ref.~\cite{Proca},
it was established that the event GW190521 is consistent with a head-on collision of two Proca stars with $\mu_{\rm B}=8.7\times 10^{-13}$\,eV.\\

We note that alternative searches for signatures of ultra-light bosons in gravitational-wave data have been performed, in particular focusing on the effects that (scalar) boson clouds can produce when surrounding black-holes. On the one hand, these include searches for continuous GW emission arising from super-radiant instability \cite{boson_LVK_followup,palomba2019direct}, which should in principle be detectable by current detectors. On the other hand, such clouds can extract angular momentum from the host black-holes leading to a reduction of its spin, an effect which has also been searched for \cite{Ken_1,Ken_2}. While none of these methods has delivered an actual detection these have been used to place constraints on the possible range of masses of (scalar) ultra-light bosons. Finally, further methods targeting LISA observations have been designed that may establish the existence of ultra-light bosons through a single observation \cite{Hannuksela2019}.

\subsection{Aim and structure of this work}

We perform a systematic analysis of the events GW190521, GW190426 and GW200220 using an expanded catalogue of 759 numerical simulations of head-on mergers of Proca stars (PSMs). In addition, we analyse the trigger S200114f. We compare the incoming detector data to both our catalogue of numerical simulations of and to a state-of-the art waveform models for circular black hole mergers. For the BBH case, we perform a ``canonical'' analysis comparing strain-data to strain-templates. For the case of our numerical simulations, however, we make use of a novel framework that we introduced in~\cite{Psi4_tech} that allows for a comparison of the signal data to the waveform templates for the Newman-Penrose scalar directly outputted by numerical simulations, commonly denoted as $\psi_4$.
The rest of this article is organised as follows.
In section~\ref{sec:setup} we describe our analysis setup, including our waveform models, simulation catalogue and prior choices. In section~\ref{sec:results} we report our parameter estimation and model selection results for all individual events and in section~\ref{sec:pop} we conduct a preliminary population study. Finally, we close with a discussion of the limitations and potential implications of our work.

\section{Analysis set-up}
\label{sec:setup}

For given detector data $d(t)$ and a waveform template model $M_i$ spanning parameters $\theta$, we aim to compute the posterior probability distribution for $\theta_i$
\begin{equation}
    p_{M_i}(\theta\,|\,d) = \frac{\pi(\theta)\,{\cal{L}}_{M_i}(d \,|\, \theta)}{{\cal{Z}}_{M_i}}.
\label{eq:bayesian_prob}
\end{equation}

Here, $\pi(\theta)$ denotes the prior probability for the parameters $\theta$, the term ${\cal{L}}_{M_i}(d\,|\,\theta)$ denotes the likelihood of the data $d$ according to the waveform model $M_i$ given parameters $\theta$. This is given by \cite{Finn1992,Romano2017}
\begin{equation}
    \log{\cal{L}}(d\,|\,\theta) \propto - \frac{(d-h(\theta)|d-h(\theta))}{2},
\label{eq:logl}
\end{equation}
where the operation $(a|b)$ denotes the noise-weighted inner product \cite{Cutler1994}
\begin{equation}
    (a|b) = 4 \times \Re \int_{f_\text{0}}^{f_{\text{m}}} \frac{\tilde{a}(f)\tilde{b}^{*}(f)}{S_n(f)}\, \dd f
\label{eq:logl}
\end{equation}
with $S_n(f)$ the one-sided power-spectral density of the background noise and $(f_\text{0},f_{\text{m}})$ the lower and upper frequency limits. The term ${\cal{Z}}_{M_{i}}$ denotes the Bayesian evidence for the waveform model $M_{i}$. This is equal to the integral of the numerator of Eq.~\eqref{eq:bayesian_prob} over the explored parameter space $\Theta$, given by
\begin{equation}
    {\cal{Z}}_{M_{i}}=\int_{\Theta} \pi(\theta) {\cal{L}}_{M_{i}}(d\,|\,\theta) \,\dd\theta.
\end{equation}
Finally, given two waveform models $M_1$ and $M_2$, the relative probability for the data given the models, or relative Bayes Factor ${\cal{B}}^{M_1}_{M_2}$, is given by
\begin{equation}
{\cal{B}}^{M_1}_{M_2} = \frac{{\cal{Z}}_{M_1}}{{\cal{Z}}_{M_2}}.
\end{equation}

\subsection{Data and Waveform models}

We perform Bayesian parameter estimation and model selection on four seconds of publicly available data \cite{GWOSC} from the two Advanced LIGO and Virgo detectors around the time of GW190521, GW200220, GW190426 and S200114f. We compare the detector data to numerical-relativity simulations of head-on PSMs \cite{Sanchis:catalogue,sanchis2019head} and to the state-of-the art waveform model for circular BBHs \texttt{NRSur7dq4} \cite{NRSur7dq4}. In previous work \cite{Proca} we made use of a catalogue of 96 numerical simulations of PSMs. These were divided in two sets: one is of equal-mass and equal-spin,   therefore equal boson-field frequency $\omega$; and the other is an exploratory unequal-mass family. Here we make use of an expanded catalogue of 759 simulations spanning an uniform grid in the frequencies of the two stars $\omega_1$ and $\omega_2$, which we describe in detail in \cite{Sanchis:catalogue}. These simulations include the dominant GW modes $(\ell,m) = (2,0),(2, \pm 2),(3,\pm 2),(3,\pm 3)$. The \texttt{NRSur7dq4} model is the only existing waveform model directly trained on numerical simulations of circular BBHs including the impact of orbital precession. The model is trained for mass-ratios $q\in[1,4]$ and spin magnitudes $a_1\in[0,0.8]$ but can be extrapolated to values of $q\in[1,6]$  and $a_1\in[0,0.99]$. This model includes all GW modes up to $\ell\leq 4$.\\

\subsection{Data analysis using the Newman-Penrose scalar}

GW data analysis relies on the comparison of the strain data read by the detectors to waveform templates for such strain. We rely on this ``classical'' approach for the case of comparing of the data to the strain model \texttt{NRSur7dq4}. Numerical simulations, however, do not directly output the GW strain but a magnitude known as the  Newman-Penrose scalar, or $\psi_4$, related to the GW strain as $\psi_4(t) = \dv*[2]{h(t)}{t}$. Obtaining the corresponding strain templates therefore requires a double time integration that is subject to well-known potential systematic errors due to spurious low frequencies contaminating the resulting $h(t)$ \cite{Pollney_Reissweig}. These can be especially relevant for highly eccentric mergers for which there is no natural way to diminish these. While we used such strain templates in \cite{Proca}, here we adopt a novel framework presented in \cite{Psi4_tech} that allows for a comparison of the detector data to the $\psi_4$ templates directly extracted from our numerical simulations, therefore avoiding further systematic errors. To do this, given the discrete detector data strain $d[n]$ of duration $T = M\Delta t$ sampled at frequency $1/\Delta t$ and the corresponding PSD $S_n[k]$, we perform the transformation:
\begin{equation}
    \begin{array}{rcl}
       & d[n]  \rightarrow d_{\Psi_4}[n] \equiv (\delta^{2}d)[n]  \\
        & S_n[k] \rightarrow S_{n{\Psi_4}}[k]

    \end{array}
    \label{eq:transformation}
\end{equation}
Above, $\delta^2 d[n]$ represents the second-order finite difference of $d[n]$, given by
\begin{equation}
   \delta^2 d [n] = \frac{d[n+1] - 2d[n] + d[n-1]}{(\Delta t)^{2}},
\end{equation}
and the transformed PSD $S_{n\Psi_4}(f)$ is obtained through
\begin{equation}
    S_{n\Psi_4}[k] = \frac{1}{(\Delta t)^{4}}
    \left(
    6 -
    8 \cos(\frac{2\pi k}{M}) +
    2\cos(\frac{4\pi k}{M})
    \right)
    S[k].
    \label{eq:noise_psd}
\end{equation}
Finally, we replace the typical strain templates $h[n]$ by the $\psi_4[n]$ templates outputted from numerical-relativity simulations \textit{after applying a correction} that accounts for the difference between second derivative and second-order finite differencing. We denote the resulting template by $\Psi_4[n]$. In particular, expressing waveform templates in the frequency domain, we substitute:
\begin{equation}
       \widetilde{h}[k] \longrightarrow \widetilde{\Psi_4}[k] = K(k\Delta f)\,\widetilde{\psi_4}(k\Delta f),
    \label{eq:transformation2}
\end{equation}
where
\begin{equation}
 K(k\Delta f) = \frac{1-\cos(2\pi \,k\Delta f\,\Delta t)}{2\pi^{2}(k\Delta f\,\Delta t)^{2}}
    \label{eq:transformation2}
\end{equation}
and $\Delta t = 1/(M\Delta f)$.

\subsection{Bayesian Priors}

\subsubsection{Source masses and spins} In GW data analysis, it is a common practice to place uniform priors on the individual masses of the source. Our discrete PSM catalogue, however, prevents us from imposing such prior. Instead, we exploit the fact that each of our PSM simulations -- for given mass-ratio and spins -- scales trivially with the total mass, enabling us to place a uniform prior in the total red-shifted mass of the source. In addition, our simulations cover a uniform grid in the space formed by the two bosonic frequencies $\omega_{1,2}$, enabling us to impose such prior. For consistency, for the BBH case we place an uniform prior on the total mass together with two different priors on the mass ratio: uniform in $Q=m_1/m_2 \geq 1$ and uniform in $q=m_2/m_1 \leq 1$. The motivation to choose both of these priors is that, as shown in e.g.~\cite{GW190521_Nitz,Hector_GW190521,21g_ZTF}, certain prior choices can prevent the exploration of high-likelihood regions of the parameter space strongly down-weighted by them. In order obtain conservative results (i.e., to minimise the evidence in favour of the PSM model) we will always consider the BBH analysis returning the largest evidence. Finally, for the spins, we place priors uniform in spin magnitude and isotropic in spin direction.

\subsubsection{Distance} As in \cite{Proca}, we explore two different distance priors. First, we use a standard ``physically realistic'' prior uniform in co-moving volume. We note, however, that such a prior does favour intrinsically louder sources -- like BBHs -- that can be located at larger distances than weaker sources like our head-on PSMs. In order to gauge this effect and obtain a ballpark estimate we also make use of an nonphysical prior uniform in distance. Also, while work is devoted towards obtaining circular PSM simulations, we also consider the usage of this prior as an attempt to obtain a ballpark value of the evidences that would be obtained with such simulations.

\subsubsection{Source orientation, sky-location and polarisation} We place standard priors in all of these quantities, namely isotropic in source orientation and sky-location and uniform in signal polarisation.

\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.32\textwidth]{S190521g_H1_whitened_final_only_strain.pdf}
\includegraphics[width=0.32\textwidth]{S190521g_L1_whitened_final_only_strain.pdf}
\includegraphics[width=0.32\textwidth]{S190521g_V1_whitened_final_only_strain.pdf}
\caption{\textbf{Whitened $\Psi_4$ time-series around the time of GW190521}, together with the maximum likelihood waveforms returned by the BBH NRSur7dq4 model (red) and our Proca star head-on merger model (green) -- after taking second-order time-differences -- and that returned by our direct analysis of $\Psi_4$ (orange).
}
\label{fig:21g}
\end{center}
\end{figure*}

\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.32\textwidth]{S200114f_H1_whitened_final_only_strain.pdf}
\includegraphics[width=0.32\textwidth]{S200114f_L1_whitened_final_only_strain.pdf}
\includegraphics[width=0.32\textwidth]{S200114f_V1_whitened_final_only_strain.pdf}
\caption{\textbf{Whitened strain time-series around the time of the trigger S200114f}, together with the maximum likelihood waveforms returned by the BBH NRSur7dq4 model (red) and our Proca star head-on merger model (green).
}
\label{fig:14f}
\end{center}
\end{figure*}

\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.32\textwidth]{S200220ad_H1_whitened_final.pdf}
\includegraphics[width=0.32\textwidth]{S200220ad_L1_whitened_final.pdf}
\includegraphics[width=0.32\textwidth]{S200220ad_V1_whitened_final.pdf}
\caption{\textbf{Whitened strain time-series around the time of GW200220}, together with the maximum likelihood waveforms returned by the BBH NRSur7dq4 model (red) and our Proca star head-on merger model (green).
}
\label{fig:20ad}
\end{center}
\end{figure*}

\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.32\textwidth]{S190426l_H1_whitened_final.pdf}
\includegraphics[width=0.32\textwidth]{S190426l_L1_whitened_final.pdf}
\caption{\textbf{Whitened strain time-series around the time of GW190426}, together with the maximum likelihood waveforms returned by the BBH NRSur7dq4 model (red) and our Proca star head-on merger model (green).
}
\label{fig:26l}
\end{center}
\end{figure*}

\section{Results}
\label{sec:results}

Figs.~\ref{fig:21g}-\ref{fig:26l} show the whitened strain and $\Psi_4$ detector data at times around the four analysed events together with the maximum likelihood templates returned by the BBH and the PSM models. Table \ref{tab:logb1} shows the result of our model selection for the events for our two choices of the distance prior. These are labelled by ``V'' (for uniform in co-moving volume) and ``D'' (for uniform in distance) Finally, Table \ref{tab:pe} shows our parameter estimates for these events under the PSM scenario. We report median values and symmetric $90\%$ credible intervals. In the following, we first present the result of model selection for individual events to then proceed with a detailed discussion of the properties of each of them.

\subsection{Model Selection}

Table \ref{tab:logb1} reports natural log Bayes factors, $\log\mathcal{B}$, for the signal $vs.$ noise hypothesis for the for events we consider when these are modelled as either BBHs or PSMs. The bottom row reports the relative probability, or Bayes factor, for PSM $v.s.$ BBH, ${\cal{B}}^{\text{PSM}}_{\text{BBH}}$. As expected, in all cases the weakness of head-on mergers punishes the PSM model when we use the V prior. For this reason, Bayes factors for the PSM case always grow when we use our D prior while those for the much louder BBH scenario remain almost unchanged. Under the former ``physically realistic'' prior, the PSM merger scenario is mildly favoured by GW190521 and S200114f, with ${\cal{B}}^{\text{PSM}}_{\text{BBH}} \simeq 2.5$ and $3.7$ respectively. Next, the PSM hypothesis is weakly rejected by GW200220, with ${\cal{B}}^{\text{PSM}}_{\text{BBH}} \simeq 0.05$ and strongly rejected by GW190426 with ${\cal{B}}^{\text{PSM}}_{\text{BBH}} \simeq 2\times 10^{-4}$. Using our D prior has somewhat significant consequences. For GW190521, the preference for PSM grows to ${\cal{B}}^{\text{PSM}}_{\text{BBH}} \simeq 18$. More spectacularly, for the trigger S200114f we obtain a strong preference for the PSM scenario of ${\cal{B}}^{\text{PSM}}_{\text{BBH}} \simeq 36$ owing to a very small distance estimate of $d_{\rm L} \simeq 150$\,Mpc (see later). Finally, the PSM hypothesis remains strongly rejected for GW190426 with ${\cal{B}}^{\text{PSM}}_{\text{BBH}} \simeq 2\times 10^{-3}$ but very weakly rejected for GW200220, with ${\cal{B}}^{\text{PSM}}_{\text{BBH}} \simeq 0.4$

All in all, for GW190521 we find the same qualitatively preference for the PSM model presented in~\cite{Proca} that has a much smaller catalogue. For the other two catalogued events, GW200220 and GW190426, we find mild and strong preferences for the BBH scenario. Finally the trigger S200114f shows the strongest preference for the PSM scenario. In the following, we analyse in detail these four events, focusing on the parameters we infer under the PSM scenario and, in particular, on potential coincidences in the inferred boson mass $\mu_{\rm B}$ across events.

\begin{table*}
\centering
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{c|ccc|ccc|ccc|ccc}
\rule{0pt}{3ex}%
Event & \multicolumn{3}{c}{GW190521}  &  \multicolumn{3}{c}{GW190426} & \multicolumn{3}{c}{GW200220} & \multicolumn{3}{c}{S200114f}  \\
\hline
    & $\log{\cal{L}}_{\text{max}}$& V & D & $\log{\cal{L}}_{\text{max}}$ & V & D & $\log{\cal{L}}_{\text{max}}$ & V & D & $\log{\cal{L}}_{\text{max}}$ & V & D  \\
Black hole merger & 123.6 & 89.9  & 90.6   & 62.8 & 38.4 & 38.6   & 48.6 & 16.5 & 16.5   & 114.2 & 69.7 & 72.4 \\
Proca star merger & 121.0 & 90.8  & 93.5  & 62.5 & 29.6  & 32.2 & 36.7 & 13.5  & 15.5 & 107.4 & 71.0 & 76.0 \\
$\cal{B}^{\text{PSM}}_{\text{BBH}}$ & & 2.5  & 18.2  & & $2\times 10 ^{-4}$  & $2\times 10 ^{-3}$  &  & 0.05  & 0.4 &  & 3.7 & 36.6 \\
\end{tabular}
\caption{\textbf{Summary of model selection on our selected GW events.} The three columns of each event are, in order, the maximum likelihood values, and the natural log Bayes factors (signal $v.s.$ noise) obtained under a standard prior uniform in co-moving volume (V) and uniform in distance (D). The last row is the corresponding relative Bayes factors. For the BBH model we report  the maximum values among the 4 mass ratio priors we tested using the \texttt{NRSur7dq4} model.}
\label{tab:logb1}
\end{center}
\end{table*}

\begin{table*}[t!]
\centering
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{l|cc|cc}
Parameter  & GW190521 & GW200220 & S200114f & GW190426  \\ \hline
Primary mass $[M_\odot]$ & $125^{+15}_{-12}$ & $121^{+20}_{-19}$ & $118^{+10}_{-12}$  & $129^{+35}_{-17}$  \\
Secondary mass $[M_\odot]$ & $109^{+10}_{-16}$  & $105^{+15}_{-14}$ & $89^{+18}_{-8}$ & $113^{+28}_{-13}$
\\
Total / Final mass $[M_\odot]$ & $232^{+16}_{-16}$ & $227^{+25}_{-30}$ & $208^{+18}_{-14}$ & $243^{+35}_{-27}$
\\
Final spin & $0.69^{+0.04}_{-0.04}$ & $0.67^{+0.09}_{-0.04}$  & $0.65^{+0.05}_{-0.03}$  & $0.71^{+0.07}_{-0.04}$
\\
Inclination $\pi/2-|\iota-\pi/2|$ [rad] & $0.68^{+0.35}_{-0.44}$ & $0.87^{+0.59}_{-0.63}$  & $0.91^{+0.50}_{-0.24}$  & $0.64^{+0.53}_{-0.45}$
\\
Luminosity distance [Mpc] & $577^{+361}_{-261}$  & $874^{+902}_{-433}$ &  $158^{+79}_{-72}$   &   $952^{+580}_{-435}$
\\
Redshift $z$ & $0.12^{+0.07}_{{-0.05}}$ & $0.18^{+0.14}_{{-0.09}}$  & $0.04^{+0.02}_{{-0.02}}$ & $0.19^{+0.10}_{{-0.10}}$
\\
Total / Final redshifted mass $[M_\odot]$  & $260^{+9}_{-8}$ & $267^{+18}_{-17}$ & $215^{+18}_{-15}$ & $289^{+23}_{-17}$
\\
Primary field frequency $\omega/\mu_{\rm B}$  & $0.88^{+0.04}_{-0.06}$ &   $0.87^{+0.05}_{-0.06}$ & $0.81^{+0.11}_{-0.01}$  & $0.90^{+0.03}_{-0.07}$
\\
Secondary field frequency $\omega/\mu_{\rm B}$  & $0.91^{+0.03}_{-0.04}$ & $0.90^{+0.03}_{-0.05}$   & $0.92^{+0.01}_{-0.06}$ & $0.92^{+0.01}_{-0.03}$
\\
Boson mass $\mu_{\rm B}$ [$\times 10^{-13}$eV] & $8.68^{+0.61}_{-0.77}$ & $9.12^{+1.48}_{-1.33}$ & $10.25^{+0.70}_{-0.61}$ &  $7.78^{+0.85}_{-0.85}$
\\
Maximal boson star mass $[M_\odot]$  & $173^{+16}_{-12}$ & $165^{+18}_{-22}$ & $147^{+9}_{-10}$ & $193^{+27}_{-19}$ \\

\end{tabular}
\caption{\textbf{Parameters of the four events discussed in this work under a PSM scenario.}  We quote median values with symmetric $90\%$ credible intervals.}
\label{tab:pe}
\end{center}
\end{table*}

\subsection{Parameter Estimation}

We now discuss the properties we infer from each event individually. We note that our simulation catalogue is built as an expansion of that in~\cite{Proca}, mostly tailored to encompass GW190521. Consequently, for some events, the highest $\mathcal{B}$ (best-fitting) points correspond to corner cases of the catalogue. On the one hand, this can lead to artificially smaller uncertainties in the frequencies $\omega_i/\mu_{\rm B}$ of the star fields and, therefore, to overly constrained boson-mass $\mu_{\rm B}$ estimates. On the other hand, the true best-fitting points may lay beyond the limits of our catalogue, making the evidences discussed above rather conservative.

\subsubsection{GW190521} We find results are fully consistent with those reported in \cite{Proca}. For the final BH, we estimate a red-shifted final mass of $M_{z}=260^{+9}_{-9}\,M_\odot$ and a final spin of $a_f=0.69^{+0.04}_{-0.04}$, compatible with those reported by the LVK \cite{GW190521D}. We note that the final BH mass is essentially equal to the initial mass due to the negligible loss to GWs during head-on mergers, which also leads to a much lower source luminosity. We infer a luminosity distance around ten times closer than that estimated by the LVK at $d_{\rm L}=582^{+447}_{-258}$\,Mpc. Consequently, we obtain a much heavier source-frame mass of $M_{\rm src}=M_{z}/(1+z)=232^{+16}_{-16}\,M_\odot$. The individual source-frame mass estimates are $m_1 = 125^{+15}_{-12}\,M_\odot$ and $m_2 = 109^{+10}_{-16}\,M_\odot$. Remarkably, despite the significant growth of our simulation catalogue these are consistent with those reported in \cite{Proca} even though that study was limited to equal-mass PSMs.\\

As per the Proca-star specific parameters, we estimate star-frequencies $\omega_1=0.89^{+0.03}_{-0.06}$ and $\omega_2=0.91^{+0.02}_{-0.06}$ in agreement with those reported in \cite{Proca}. These, combined with the masses of the individual stars, allow us to estimate the mass of the underlying ultralight boson via

\begin{equation}
    \mu_{\rm B}=1.34\times10^{-10}\,\biggl(\frac{{\cal{M}}_1 + {\cal{M}}_2}{M^{\text{final}}_{\text{BH}}}\biggl)\,\text{eV}.
    \label{eq:boson_mass}
\end{equation}
Here, $\mathcal{M}_{1,2}=\mu_{\rm B}\, m_{1,2}/M^2_{\rm{Pl}}$ is a dimensionless mass characterising each PSM, $m_{1,2}$ denotes the source frame mass of each star, $M_{\rm{Pl}}$ the Planck mass and
$M_{\rm BH}^{\rm final}$ the mass of the final BH expressed in solar masses. We obtain  $\mu_{\rm B}^{\text{GW190521}}=8.68^{+0.61}_{-0.77}\times 10^{-13}$ eV. Finally, we make use of the boson-mass and the individual star masses to infer the maximal mass $M^{\text{Proca}}_{\text{max}}$ that a Proca star could form by such bosons can have before collapsing to a black hole, given by
\begin{equation}
    M^{\rm Proca}_{\rm max} = \qty(\frac{M_\text{max}}{M_{\odot}}) = 1.125\,\biggl(\frac{1.34\times10^{-10}\,\text{eV}}{\mu_{\rm B}}\biggl).
\end{equation}

We obtain $M^{\rm Proca}_{\rm max}=173^{+16}_{-12}\,M_\odot$.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.45\textwidth]{boson_masses.pdf}
\caption{\textbf{Boson-mass estimates for the events analysed in this work.} The grey line denotes the boson-mass prior, which is determined by our priors in the star frequencies, total red-shifted mass and luminosity distance
}
\label{fig:mub}
\end{center}
\end{figure}

\subsubsection{GW200220}

GW200220 was detected by the matched-filter search PyCBC \cite{Usman:2015kfa} during the third observing run of Advanced LIGO and Virgo with an inverse-false-alarm rate (IFAR) of $0.3$\,yr and a probability of astrophysical origin $p_{\text{astro}}= 0.62$, significantly lower than that of GW190521. Despite this low significance, GW200220 outstands as the third-heaviest BBH reported to date, which makes it a promising candidate to fit with our PSM waveforms.\\

We find that GW200220 is essentially a more distant copy of GW190521, with consistent individual masses of $m_1 = 121^{+20}_{-19}M_\odot$ and $m_2 = 105^{+15}_{-14}$ and a similar final spin of $a_f=0.67^{+0.09}_{-0.04}$ but located at a slightly larger $d_{\rm L}=874^{+902}_{-433}$\,Mpc. More interestingly, we obtain very similar field frequencies of  $\omega_1=0.87^{+0.05}_{-0.06}$ and $\omega_2=0.90^{+0.03}_{-0.05}$ and a boson-mass of $\mu_{\rm B}^{\text{GW200220}}=9.12^{+1.48}_{-1.33}\times 10^{-13}$\,eV completely consistent with that of GW190521.\\

Using the formalism in \cite{Greg}, we can test the hypothesis that the two events are sourced by the same ultra-light boson, i.e., that they share the same boson mass. For two events A and B, we can compute the odds-ratio\footnote{The odds-ratio $\mathcal{O}_{C/U} = \mathcal{I}_{\mu_{\rm B}}\,\pi_{C/U}$, where $\pi_{C/U}$ is the prior odds of the two hypotheses, common v.s. uncorrelated, and we have implicitly assume equal probability, i.e.: $\pi_{C/U} = 1$.} for the common v.s. uncorrelated mass through the overlap integral
\begin{equation}
    {\cal{I}}_{\mu_{\rm B}}^{AB} = \int \frac{p(\mu_{\rm B} | \mathrm{A})\,p(\mu_{\rm B} | \mathrm{B})}{\pi(\mu_{\rm B})}\,\dd\mu_{\rm B},
\end{equation}
where $\pi(\mu_{\rm B})$ denotes our prior on the boson mass, represented by the grey curve in Fig.~\ref{eq:boson_mass}. For the pair GW190521-GW200220 we obtain an odds-ratio ${\cal{I}}_{\mu_{\rm B}}=5.5$ favouring a common $\mu_{\rm B}$. This means that, if we consider that the two events share the same boson, the relative evidence for the PSM vs. BBH scenarios rises by a factor of 5.5. Later, we will showcase how to exploit this result in the context of population studies in section \ref{sec:pop}.\\

Finally, we infer a maximal Proca star mass $M^{\text{Proca}}_{\text{max}}=165^{+18}_{-22}\,M_\odot$. On the one hand, this is consistent with the one inferred from GW190521. On the other, the total masses of both events are consistently larger than the  estimated maximal Proca star masses. This implies that, in both cases, the remnant hyper-massive boson star formed at the end of the two mergers has enough mass to collapse to a black-hole and yield the corresponding characteristic ringdown signal expected by current gravitational-wave searches.

\subsubsection{GW190426}

GW190426 was also detected by the matched-filter search PyCBC with an IFAR of $0.5$\,yr and a $p_{\text{astro}}=0.73$, again significantly lower than that of GW190521. Despite this event is strongly discarded as a PSM when compared to our existing simulation catalogue, it is still useful to look at some of the properties that are inferred under such a scenario. First, we note that the secondary field frequency $\omega_2=0.92^{+0.01}_{-0.03}$ clearly trails against the upper limit of our catalogue. This evidences that we need to enlarge our catalogue to correctly encompass this event. Moreover, combining such a low Bayes factor with the fact that both the BBH and PSM models reach similar maximum likelihoods (similarly good fits) to the data, indicates that the PSM templates producing good fits are corner cases within our PSM catalogue, which we have confirmed. At the same time, it is interesting to note that we obtain a boson-mass of $\mu_{\rm B}^{\text{GW190426}}=7.78^{+0.85}_{-0.85}\times 10^{-13}$\,eV lower than (despite consistent with) those inferred from the previous two events. In particular, we find a overlap integrals ${\cal{I}}_{\mu_{\rm B}}=1.9$ and ${\cal{I}}_{\mu_{\rm B}}=3.0$ favouring the common-boson hypothesis when comparing this event with GW190521 and GW200220 respectively.

\subsubsection{S200114f}
S200114f is a short-duration transient observed during the first half of the third observing run of Advanced LIGO and Virgo. This intriguing trigger was missed by matched-filter searches targeting black hole mergers (which neglect orbital precession \cite{Harry:2016ijz,Williamson:2017evr} and higher-order harmonics \cite{Capano:2013raa,Harry:2017weg,CalderonBustillo:2018zuq}) but was observed by the model-agnostic search coherent Wave Burst \cite{Klimenko:2015ypf} with an IFAR of 30\,yr \cite{o3_imbh}. Due to the lack of detection by matched-filter searches S200114f has not been labelled as a confirmed detection but nevertheless it was neither conclusively labelled as background noise. Remarkably, parameter estimation was performed on this trigger with three different state of the art waveform models \cite{NRSur7dq4,SEOBNRv4PHM,PhenomPv3HM}, with all results across different models returning values for the individual masses. Rather than revealing that this trigger is not a black hole merger, or not of astrophysical origin, these results showcase the inconsistencies between these BBH approximants at the regions of the parameter space that best fit the signal (see, e.g.~\cite{21g_ZTF}). We therefore consider it very interesting to analyse this event from the perspective of further waveform models and, in particular, under our PSM catalogue.

In terms of its masses, we find that S200114f is essentially a lighter and more nearby version of GW190521 with a much larger inclination. We estimate a final total red-shifted mass of $M_{z}=215^{+18}_{-15}\ M_\odot$ and a distance of $d_{\rm L}=158^{+79}_{-72}$\,Mpc. Owing to the standard distance prior, the louder and  BBH scenario should be mildly favoured by our analysis. Despite this, the ${\cal{B}}^{\text{PSM}}_{\text{BBH}}$ is still ${\sim} 3.7$, slightly preferring the PSM scenario. Moreover, removing the effect of such prior yields a ${\cal{B}}^{\text{PSM}}_{\text{BBH}}\simeq 37$ rather strongly preferring the PSM model.

The above combination of red-shifted mass and distance results in a source-frame mass of $M_{\rm src}=208^{+18}_{-14}\ M_\odot$. The final black hole would have a spin of $a_{\rm f}=0.65^{+0.05}_{-0.03}$. The main difference in the intrinsic properties of S200114f w.r.t. GW190521 arises from the frequency of their bosonic fields. We estimate $\omega_1=0.81^{+0.11}_{-0.01}$ and $\omega_2=0.92^{+0.01}_{-0.06}$ for this event. We note that the extremely small uncertainties of $\delta\omega_i=0.01$ in the lower and upper ends of the respective frequency ranges are solely due to the fact that this event lies on the edge of our simulation catalogue, which makes our posterior distributions trail against such limits. On the one hand, this means that all the provided results are over-constrained even within the head-on paradigm. On the other hand, this reveals that there is room for improvement in fitting this event within the scenario we propose. Altogether, we obtain a value for the boson mass $\mu_{\rm B}^{\text{S200114f}}=10.25^{+0.70}_{-0.61}\times 10^{-13}$\,eV, larger than for the previous events. While we find that the common-boson hypothesis is favoured with ${\cal{I}}_{\mu_{\rm B}}=3.7$ w.r.t. GW200220, it is rejected w.r.t. GW190521 with ${\cal{I}}_{\mu_{\rm B}}=0.1$. Finally, the boson masses of S200114f and GW190426 are highly inconsistent with ${\cal{I}}_{\mu_{\rm B}}=0.01$.\\

Even though some of the studied pairs of events yield rather inconsistent boson masses, we stress that we are imposing the very restrictive scenario of a head-on merger. Recall that the frequency of the bosonic field -- which determines the boson masses -- fixes the spins of the individual stars and consequently the spin of the final BH. Therefore, the preferred star frequencies for the merging stars are those that can lead to the correct final BH spin. Expanding our numerical relativity catalogue to less eccentric configurations would provide an extra contribution from the orbital angular momentum to the final spin, therefore allowing for a wider range of star frequencies and, consequently, boson masses. The expectation is that without the head-on restriction the true boson mass posteriors should be significantly broader, which would lead to a much better consistency for the mass across events. For these reasons, we think it is quite remarkable that the analysed events yield the slightest consistency.

\begin{table}
\centering
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{l | c c c c }
        & GW190521 & GW200220 & GW190526 &  S200114f  \\[4pt]
Triplet & 0.08 (0.17)    & 0.02 (0.05)    & 0.82  (1.22)   &  12.0 (7.3)  \\
\hline
GW190521             &   --      &    --     &   --        &  --   \\
GW200220             & 5.5 (6.2) &    --     &   --        &  --   \\
GW190526             & 3.0 (1.5) & 1.9 (1.2) &   --        &  --   \\
S200114f             & 0.1 (0.2) & 3.7 (2.8) & 0.01 (0.03) &  --   \\
\hline
\end{tabular}
\label{tab:overlap}
\caption{\textbf{Mass-overlap integrals for pairs and triplets.} The three bottom rows show the overlap integrals $\mathcal{I}^{AB}_{\mu_{\rm B}}$ for each pair of the events we study. The top row shows the overlap integral for each of the possible triplets, excluding the event on the top of the corresponding column. Values within parentheses correspond to analyses using a uniform distance prior while the rest correspond to a standard prior uniform in co-moving volume. Values larger than one favour the common-boson hypothesis over the uncorrelated one.}
\end{table}

\section{Population}
\label{sec:pop}

The existence of multiple events that can be compared with our PSM model invites the question of their population and by what fraction $\zeta$ do they occupy among all compact binary mergers. Starting from our observational data set of four events $\{ d_i\}$, we consider a population of compact objects consisting on a fraction $\zeta$ of PSMs and a fraction $1-\zeta$ of BBHs. With this, we can compute the likelihood of our data set given $\zeta$ as
\begin{equation}
    \begin{aligned}
        p(\{d_i\}|\zeta) &= \prod_{i=1}^{N=4} p(d_i|\text{PSM})\,\zeta + p(d_i|\text{BBH})\,(1-\zeta) \\
                         &\propto \prod_{i=1}^{N=4} {\cal{B}}^{\text{PSM}}_{\text{BBH}}\,\zeta + (1-\zeta),
    \end{aligned}
    \label{eq:pop}
\end{equation}
where ${\cal{B}}^{\text{PSM}}_{\text{BBH}} = {\cal{B}}^{\text{PSM}}_{\text{N}}/{\cal{B}}^{\text{BBH}}_{\text{N}} $ denotes the relative Bayes factor between PSM and BBH models of the $i$-th event.\\

\subsection{Boson-mass agnostic calculation}

The left panel of Fig. \ref{fig:pop} shows the posterior distribution of $\zeta$, where we use the Bayes factors reported in Table \ref{tab:logb1}. Solid curves include S200114f as a real event while dashed ones exclude it. Blue curves correspond to a uniform in co-moving volume prior. In these cases, we see that ignoring S200114f returns a posterior that peaks at $\zeta=0$ but that shows support all the way to $\zeta=1$. In particular, we obtain $\zeta = 0.27^{+0.45}_{-0.25}$, with $\zeta > 0.04$ at the $90\%$ credible level. The inclusion of S200114f as a true event raises this to $\zeta = 0.42^{+0.41}_{-0.34}$ with $\zeta > 0.08$ at the $90\%$ credible level, with a peak at $\zeta \simeq 0.25$. Red curves correspond to our uniform distance prior. In this case, ignoring S200114f we obtain a posterior peaking at $\zeta \simeq 0.4$ with a $90\%$ lower bound of $\zeta > 0.13$. Including S200114f as a real event raises the latter to $\zeta > 0.26$.\\

\subsection{Exploiting boson-mass consistencies}

The fact that some of the events show consistent boson masses further invites the question of whether they can be analysed assuming a common value. In such a case, the evidence for the PSM model would rise due to the reduction of the number of parameters and the consequent reduction of the Occam penalty (see e.g., \cite{Bustillo2021,Thrane2019}). Given the original prior for the boson-mass $\pi(\mu_{\rm B})$, the original posterior for each event $p_i(\mu_{\rm B})$ and a new prior for the boson-mass $\pi^{*}(\mu_{\rm B})$, the new value of the Bayesian evidence ${\cal{Z}}^*_i$ for each event can be obtained through
\begin{equation}
    {\cal{Z}}^{*}_i = \mathcal{Z}_i\int \pi^{*}(\mu_{\rm B}) \frac{p(\mu_{\rm B})}{\pi(\mu_{\rm B})} \,\dd\mu_{\rm B} = {\cal{I}}^{\pi^{*}}_{\mu_{\rm B}}{\cal{Z}}_i,
\end{equation}
where ${\cal{Z}}_i$ denotes the Bayesian evidence obtained under the original prior $\pi(\mu_{\rm B})$. While in principle a mass-prior assuming an unique ultra-light boson should be given by a delta function centred at a given mass, here we set a prior equal to the posterior for the most significant of our four events. This is, we choose $\pi^{*}(\mu_{\rm B}) = p^{\text{GW190521}}(\mu_{\rm B})$. The updated evidence for the remaining events under the PSM hypothesis is therefore given by
\begin{equation}
    {\cal{Z}}^{*}_i = \mathcal{Z}_i \int p^{\text{GW190521}}(\mu_{\rm B})\frac{p(\mu_{\rm B})}{\pi(\mu_{\rm B})}\, \dd\mu_{\rm B} = {\cal{I}}^{\text{GW190521},i}_{\mu_{\rm B}}{\cal{Z}}_{i}.
\end{equation}
The factor ${\cal{I}}^{\text{GW190521},i}_{\mu_{\rm B}}$ is known as the overlap integral as previously reported in \cite{Greg} which is equivalent to the relative Bayes factor between the common-source v.s. uncorrelated source hypotheses for the two compared events. In Table~\ref{tab:overlap}, we display these values for all signal pairs. The new PSM v.s. noise Bayes factor is then given by ${\cal{B}}_i^{*}={\cal{I}}^{\text{GW190521}, i}_{\mu_{\rm B}}\,{\cal{B}}_i$. Finally, by replacing $\mathcal{B}^{\rm PSM}_{\rm N}$ in Eq.~\eqref{eq:pop} with these, we can recompute the posterior distribution the fraction of PSMs $\zeta$ under the assumption that all events share the same boson as GW190521.\\

The right panel of Fig.~\ref{fig:pop} shows the new posteriors of $\zeta$. Exploiting common masses has dramatic consequences when S200114f is not considered as a true event. This is expected as the overlap integrals of the remaining two events support the common boson hypothesis, therefore increasing their evidence as PSMs. In particular, for each of our two distance priors, we now obtain a posteriors peaked at $\zeta=0.4$ and $\zeta=0.7$ and $90\%$ lower bounds of $\zeta=0.06$ and $\zeta=0.21$. While a similar qualitative effect is observed when including S200114f, this is quantitatively less dramatic. The reason is that the raise $\cal{B}^{\text{PSM}}_{\text{BBH}}$ for the other events is now accompanied by a reduction of that for S200114f due to its highly inconsistent boson mass with respect to GW190521.\\

These should be considered proof-of-principle calculations with relevant shortcomings that can artificially favour each of the PSM and BBH hypotheses. First, we have ignored the prior on the relative abundance of BHs and Proca stars in the Universe, as well as ruling out other kinds of possible compact binaries, for instance, binary neutron stars. Second, because at the moment no simulations for circular PSM exist, we ignore the fact that highly eccentric (let alone head-on) mergers are highly astrophysically suppressed. Finally, we also note that the black-hole merger model \cite{NRSur7dq4} is limited to non-eccentric binaries with mass-ratio $q\leq 6$ and that some of these events may be better reproduced when adding the effect of orbital eccentricity, as it is the case for GW190521 \cite{HeadOnArxiv,Isobel_ecc,Gayahtri_ecc}, or even by mass-ratios larger than those allowed by the model. On the other hand, we also note that our PSM model is also incomplete and constrained to a narrow number of cases, which makes some of the analysed events end up reside in the edges of our parameter space. Increasing our parameter coverage would most likely lead to improved fits and, therefore, increased evidences of these events.

\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{Pop_Agnostic_prior.pdf}
\includegraphics[width=0.49\textwidth]{Pop_GW190521_prior.pdf}
\caption{\textbf{Population fractions of boson-star mergers $\zeta$ within our data set for varying priors and events considered}. In the left panel we ignore any correlations between the boson-mass obtained for our events. In the right panel, we impose a boson-mass prior given by the posterior for GW190521. Blue curves make use of a distance prior uniform in co-moving volume while red ones impose a uniform-in-distance prior. Finally, solid (dashed) curves include (exclude) S200114f as a real gravitational-wave event.
}
\label{fig:pop}
\end{center}
\end{figure*}

\section{Discussion \\}

Despite their canonical interpretation as black-hole mergers, short GW transients displaying barely any pre-merger emission merit further exploration of their possible origin. We have compared four such events to a catalogue of 759 numerical-relativity simulations of PSMs. Performing model selection with respect to vanilla quasi-circular BBH mergers, we find that the most significant of them (GW190521) and the loud trigger S200114f favour the PSM hypothesis. The weaker events GW200220 and GW190426 respectively weakly and strongly reject such hypothesis. Remarkably, we find that the \textit{two confirmed GW events} which are not strongly discarded as PSMs, namely GW190521 and GW200220, yield consistent boson masses around $9\times 10^{-13}$\,eV. Next, we have performed the first population study of compact binaries considering a mixed Black Hole-Proca star mergers population and showcased a preliminary strategy to exploit boson-mass consistencies across events. \\

This is the first extensive and systematic analysis of GW events under an exotic compact-merger scenario alternative to BBHs. Although our new simulation catalogue has been significantly expanded since our initial study \cite{Proca}, it still suffers from important limitations. These are mainly the range of parameters covered by our numerical simulations and the fact that all of these correspond to the unrealistic astrophysical configuration of a head-on merger. The latter limits the type of morphologies we can possibly fit due to the shortness of the templates, significantly over-constrains our parameter estimates; and also over-punishes the PSM model due to its weak luminosity. On the other hand, our limited range of field frequencies coverage may prevent us from correctly fitting some of the events we analyse. For instance, we know that the numerical simulations best-fitting S200114f lay in the edges of our catalogue.

While progress is made towards numerical simulations of more realistic and less eccentric configurations, we highlight that our results are highly promising and should strongly motivate pursue of such extended catalogues. First, these simple configurations suffice to fit the data as good as the most developed BBH models, if not better. Second, even though the standard prior in typical GW parameter estimation is by-default designed to prefer loud circular configurations for which GW detectors have a much larger reach, our analysis shows that in some cases the Proca scenario is marginally preferred. In fact, when removing such ``bias'' to foresee what results would be obtained considering louder and circular configurations, two events show a comparable preference to both scenarios and other two, GW190521 and S200114f, show stronger preferences for PSM.

The existence of an ultralight bosonic field would have profound implications. It could at least account for part of dark matter, since it would give rise to a remarkable energy extraction mechanism from astrophysical spinning BHs, which eventually form new sorts of  ``hairy" BHs~\cite{herdeiro2014kerr,herdeiro2016kerr}. In addition, such field could serve as a guide toward beyond-standard-model physics, possibly pointing to the stringy axiverse. From an astrophysical perspective, having massive bosonic stars could also have an impact in black hole populations, if these objects merge and collapse frequently, contributing to the formation of intermediate-mass black holes.

\section*{Acknowledgements \\}

We thank Tom Callister and Kaze Wong for enlightening discussions about population studies. The analysed data and the corresponding power spectral densities are publicly available at the online Gravitational Wave Open Science Center \cite{GWOSC}. LVK parameter estimation results quoted throughout the paper and the corresponding histograms and contours in Fig. 2 have made use of the publicly available sample release in \texttt{https://dcc.ligo.org/P2000158-v4}. JCB is supported by the Australian Research Council Discovery Project DP180103155 and by the Direct Grant from the CUHK Research Committee with Project ID: 4053406. The project that gave rise to these results also received the support of a fellowship from ``la Caixa'' Foundation (ID 100010434) and from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 847648.
The fellowship code is LCF/BQ/PI20/11760016. JAF is supported by the Spanish Agencia Estatal de Investigaci\'on  (PGC2018-095984-B-I00) and by the  Generalitat  Valenciana  (PROMETEO/2019/071). This work is supported by the Center for Research and Development in Mathematics and Applications (CIDMA)
through the Portuguese Foundation for Science and Technology (FCT - Funda\c {c}\~ao para a Ci\^encia e a Tecnologia), reference UIDB/04106/2020, and by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19.
We also acknowledge support  from  the  projects  PTDC/FIS-OUT/28407/2017,   CERN/FIS-PAR/0027/2019, PTDC/FIS-AST/3041/2020 and CERN/FIS-PAR/0024/2021. NSG was also supported by the Spanish Ministerio de Universidades, reference UP2021-044, within the European Union-Next Generation EU.
This work has further been supported by the European Union's Horizon 2020 research and innovation (RISE) programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740.  The authors would like to acknowledge networking support by the COST Action CA16104.
We acknowledge the use of IUCAA LDG cluster Sarathi for the computational/numerical work. The authors acknowledge computational resources provided by the CIT cluster of the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY0823459; and the support of the NSF CIT cluster for the provision of computational resources for our parameter inference runs. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. This manuscript has LIGO DCC number P2200169.

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text:
Q: Show that mapping θ(x):a → log a is an isomorphism from R* under multiplication to R under addition. An exercise in a book states that "Show that mapping θ(x):a → log a is an isomorphism from R* under multiplication to R under addition."
What should be done first?

Edit (interpreted question):

Show that the mapping $\theta : (\mathbb{R}_{>0}, \cdot) \to (\mathbb{R}, +)$, defined by $\theta(x) = \log(x)$ for all $x \in \mathbb{R}_{>0}$, is an isomorphism.

A: I'm assuming that $(\mathbb R_{> 0},*)$ is the domain. First of all, it is a common rule that $\log(ab)=\log(a)+\log(b)$, so it is a homomorphism.
If you don't believe the last statement, you can prove the claim and this last statement by noting that it has an inverse, namely $\exp$, which shows that is a bijective homomorphism.

A: First, show that $\theta$ is a homomorphism, namely, that $\theta(xy) = \theta(x) + \theta(y)$. Next, show that $\theta$ is surjective, i.e., for all $y \in \mathbb{R}$ there exists $x \in \mathbb{R}^+$ such that $\theta(x) = y$. Finally, show that $\theta$ is injective. You can either show that if $\theta(x) = \theta(y)$ then $x = y$, or you can show that if $\theta(x) = 0$, then $x = 1$. These are the three properties of an isomorphism between two groups.
These can be done in any order, though if you prove injectivity by the second method I suggested, it would be best practice to show $\theta$ is a homomorphism first.

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WELCOME TO 'KIDZ IN THE MIX' a Dance -Yoga - Acrobatics program for 6-11 year old, created and taught by Tanja Pazou.
'Kidz in the Mix' performed at UIC Chicago, Veganmania in Chicago, Six Flags, Senior homes in the Du Page area, Community Events in the Chicago area and countless Birthday parties.
In 2011 and in 2013 Pazou Productions created two television pilots to demonstrate the significance of the program, and to educate children about the deeper meaning of this artistic and educational program.
In 2013 'Kidz in the Mix" auditioned for America's Got Talent and were filmed by the Production Team of America's Got Talent for that season.
I developed 'Kidz in the Mix' in 2009. I have been teaching a variety of dance programs to adults and children for many years. During the course of my teaching experience,I have noticed, that children are getting easily bored with technical jargon, and when introduced to challenging choreography, they lose interest. I was trying out different combinations of dance, yoga, and acrobatic classes to make it more interesting for 6-11 year old, and keep their attention at least for 60-minutes. After a while it worked, but still it wasn't as fun as I hoped. Then I came up with with the concept of 'Kidz in the Mix', a mix of games, dance, yoga, acrobatics, knowledge based information, and a healthy approach to competition, and it worked! 'Kidz in the Mix' has been fun, educational, challenging and informative to over 3000 students so far. Thank you for your interest in our programs!
Tanja Pazou - Educator, Choreographer, Artistic Director for over 20 years.
'Kidz in the Mix' is a program dedicated to educate, inspire and challenge young upcoming dancers, to become the best version of themselves and to share their gift with the world.
 In our program 'Kidz in the Mix', children unite through music and movement. It becomes a common goal to express a feeling or a mission, and to create a show, choreography or a story around it.
Our current group: Annie, Katie, Josie, Olivia, Chloe, Isabella, Lila, Lexi and Ava are training to become the next top performers in Chicago. Thanks for your hard work!
Dance Sequences are great coordination exercises for children and adults. For children especially, it has a great affect on their brain development. When we create and develop new choreography, students are constantly switching back and forth between right and left brain activity; dancing is a supreme artistic activity to develop skills like coordination, balance and flexibility.
Developing Acrobatic Skills at "Kidz in the Mix" is part of the weekly curriculum. For children it is very important overcoming fears of the "unknown", once they master a particular challenge, it opens up a new horizon of possibilities!
"Kidz in the Mix" is a Dance - Yoga - Acrobatics program for children ages 6-11 with a focus on environmental and nutritional education.
To look to the future, to strain after an ideal, indicates sluggishness of mind and desire to avoid the present.
When we train our children according to a system of thought or a particular discipline, when we teach them to think within departmental divisions, we prevent them from growing into integrated men and women, and therefore they are incapable of thinking intelligently, which is to meet life as a whole.
The right kind of education consists in understanding the child as he is without imposing upon him an ideal of what we think he should be. To enclose him in the framework of an ideal is to encourage him to conform, which breeds fear and produces in him a constant conflict between what he is and what he should be; and all inward conflicts have their outward manifestations in society. The pursuit of an ideal excludes love, and without love no human problem can be solved.
For political and industrial reasons, discipline has become an important factor in the present social structure, and it is because if our desires to be psychologically secure that we accept and practice various forms of discipline. Discipline guarantees a result, and to us the end is more important than the means, but the means determine the end.
One of the dangers of discipline is that the system becomes more important than the human beings who enclosed in it. Discipline then becomes a substitute for love, and it is because our hearts are empty that we cling to discipline. Freedom can never come through discipline, through resistance; freedom is not a goal, an end to be achieved. Freedom is at the beginning, not at the end, it is not to be found in some distant ideal.
"Kidz in the Mix" having fun in class!!
PRESS about "KIDZ IN THE MIX"