text:
\section{Introduction} \label{sec:intro}

Current \ac{GW} sources consist of merging \ac{DCO} binaries that have \ac{NS} and/or \ac{BH} components~\citep{abbott_gwtc-1_2019, abbott_gwtc-2_2021, abbott_gwtc-3_2021, abbott_gwtc-21_2022}.
Their progenitors are important markers of binary evolution, as they provide information on the origin of \acp{DCO} and inform which evolutionary scenarios dominate their formation~\citep{podsiadlowski_formation_2003, tauris_formation_2003, kratter_formation_2011, miller_masses_2015, heuvel_high-mass_2018, abbott_astrophysical_2016}.
Such progenitors include \acp{HMXB}, which are binary systems that contain a massive OB-type donor star with an accreting \ac{CO} which can be either a \ac{NS} or a \ac{BH}~\citep[e.g.,][]{verbunt_origin_1993, remillard_x-ray_2006}.
The donor star in \ac{HMXB} systems is massive enough to itself form a \ac{CO}, making \acp{HMXB} prime candidates for the progenitors of \ac{GW} sources observable by current ground-based detectors.

\acp{HMXB} are predominantly wind-fed, meaning the accretion process is fueled by the stellar wind of the donor star~\citep{blondin_wind_1997}.
At its simplest level, accretion can be described by the classical Bondi--Hoyle model~\citep{1944MNRAS.104..273B}, which drives X-ray emission that can be observed with contemporary X-ray surveys~\citep[e.g.,][]{oh_105-month_2018, krivonos_15_2021, pavlinsky_art-xc_2021, predehl_erosita_2021}.
In cases where accretion is Eddington-limited, the apparent X-ray accretion luminosity is also limited to $\sim10^{39}~\mathrm{erg\,s}^{-1}$ for a $10~M_{\odot}$ \ac{BH} accretor.
All observed wind-fed \acp{HMXB} with confirmed \ac{BH} accretors have high Roche lobe filling factors~\citep{orosz_1565-solar-mass_2007, orosz_new_2009, miller-jones_cygnus_2021}, which means that these systems could soon begin Roche lobe overflow mass transfer.

Approximately 80 Galactic \acp{HMXB} have been observed, but not all systems have well-constrained binary properties~\citep{krivonos_integralibis_2012, clavel_nustar_2019, kretschmar_advances_2019}.
Most of these \acp{HMXB} are thought to have \ac{BH} accretors, but the majority have yet to be dynamically confirmed with spectroscopic observations \citep{motta_integral_2021}.
Cyg X-1, a Galactic \ac{HMXB} containing the first-ever dynamically confirmed stellar-mass \ac{BH}~\citep{bolton_identification_1972, webster_cygnus_1972}, is estimated to contain a $21.2^{+2.2}_{-2.2}~M_{\odot}$ \ac{BH} accretor and a ${40.6^{+7.7}_{-7.1}~M_{\odot}}$ main-sequence donor~\citep{miller-jones_cygnus_2021}.
Another well-studied Galactic \ac{HMXB} is Cyg X-3, which is estimated to contain a \ac{CO} accretor of $2.4^{+2.1}_{-1.1}~M_{\odot}$ and a Wolf--Rayet donor of $10.3^{+3.9}_{-2.8}~M_{\odot}$~\citep{zdziarski_cyg_2013}.
While the mass of the accretor falls near the maximum mass allowed by the uncertain \ac{NS} equation of state~\citep[][]{lattimer_neutron_2021}, radio, infrared, and X-ray properties of the system suggest that it is a low-mass \ac{BH}~\citep{zdziarski_cyg_2013}.
Thus, of the few Galactic \acp{HMXB} that have well-constrained binary properties, only Cyg X-1 is thought to contain a \ac{BH} $\gtrsim 20 M_{\odot}$.

Observational campaigns throughout the past two decades have uncovered a large number of extragalactic \ac{HMXB} candidates~\citep[e.g.,][]{fabbiano_populations_2006, haberl_high-mass_2016, lazzarini_young_2018, rice_x-ray_2021}.
Of these sources, one of the few that has well-resolved binary properties is LMC X-1, the brightest X-ray source in the Large Magellanic Cloud~\citep{mark_detection_1969}.
\citet{orosz_new_2009} estimate it to have a donor mass of $31.8\pm3.5~M_{\odot}$ and a \ac{BH} accretor mass of $10.9\pm1.4~M_{\odot}$.
Another resolved extragalactic \ac{XRB} is M33 X-7, an \ac{HMXB} in the spiral galaxy M33 estimated to have a $38^{+22}_{-10}~M_{\odot}$ donor star and a $11.4^{+3.3}_{-1.7}~M_{\odot}$ \ac{BH}~\citep{ramachandran_phase-resolved_2022}.
Similar to Galactic \acp{XRB}, the population of well-constrained extragalactic sources is small, and in this case none are found to harbor \ac{BH} accretors $>20 M_{\odot}$.

In addition to dynamically-observed \acp{BH} in \acp{XRB}, many stellar-mass \acp{BH} have been discovered with the \ac{LIGO} and Virgo \ac{GW} detectors as \ac{DCO} mergers.
The global \ac{GW} detector network made the first observation of \acp{GW} from a merging \ac{BBH} with component masses of $34.6^{+4.4}_{-2.6} M_{\odot}$ and $30.0^{+2.9}_{-4.6} M_{\odot}$ in 2015~\citep{abbott_astrophysical_2016, abbott_gwtc-21_2022}.
The \ac{LIGO} Scientific, Virgo and KAGRA Collaboration has identified $90$ probable \ac{DCO} merger candidates during their first three observing runs~\citep{abbott_gwtc-1_2019, abbott_gwtc-2_2021, abbott_gwtc-3_2021, abbott_gwtc-21_2022}, including two sources that have component masses consistent with a \ac{NS}--\ac{NS} merger~\citep{abbott_gw170817_2017, abbott_gw190425_2020} and a few sources with component masses consistent with a  \ac{NS}--\ac{BH} merger~\citep{abbott_gw190814_2020, abbott_gwtc-3_2021, abbott_observation_2021, abbott_gwtc-21_2022}.

Using \ac{GW} observations, population analyses are able to constrain aspects of the underlying \ac{BBH} mass distribution, with the most recent analyses identifying substructure in the mass distribution beyond the simplest phenomenological models~\citep{tiwari_emergence_2021, abbott_population_2022, edelman_aint_2022}.
The global peak of the mass distribution is at $\simeq10M_{\odot}$, and there is strong evidence of a secondary peak at $\simeq35M_{\odot}$.
Though there is clear support for masses above what would be allowed for a simple power law with a cutoff, the $99$th percentile of the underlying mass distribution is $44^{+9.2}_{-5.1}M_{\odot}$~\citep{abbott_population_2022}.
Distributions of \ac{BH} source properties, such as their mass, can be used to probe the astrophysics of \ac{BBH} formation and evolution~\citep[][]{belczynski_first_2016, barrett_accuracy_2018, fishbach_most_2020, zevin_one_2021,mandel_merging_2018}.

Since \acp{HMXB} will potentially form bound \ac{DCO} binaries, they are prime candidates for the progenitors of compact binary mergers detected via \acp{GW}.
However, there is uncertainty as to whether \ac{HMXB} progenitors significantly contribute to the merging compact binary population.
For example, the \ac{XRB} donor star and \ac{CO} may merge later in their evolution before becoming a compact binary, get disrupted during the supernova that forms the second \ac{CO}, or have too wide of an orbital separation at \ac{DCO} formation to merge within a Hubble time.

Several studies have been conducted to predict the fate of observed \acp{HMXB}.
\citet{neijssel_wind_2021} predict the fate of Cyg X-1 with its updated \ac{BH} and donor star mass estimates from \citet{miller-jones_cygnus_2021} by evolving it forward with the \texttt{COMPAS} population synthesis code~\citep{riley_rapid_2022}.
They find that the system will most likely form a \ac{BH}--\ac{NS} binary that has a $\simeq 7\%$ chance of remaining bound after the \ac{NS} natal kick.
With revised models of mass transfer and natal kicks that produce heavier donor remnant masses, \citet{neijssel_wind_2021} predict that Cyg X-1 could potentially form a \ac{BBH} albeit at a low probability; in these scenarios, they find the probability Cyg X-1 will merge as a \ac{BBH} within a Hubble time is $\simeq$ 4--5\%.
Similar studies have been done for LMC X-1~\citep{belczynski_high_2012} and Cyg X-3~\citep{belczynski_cyg_2013}.
Of these two binaries, only Cyg X-3 is predicted to merge as a \ac{BBH} within a Hubble time, although this outcome heavily relies on where the component masses fall within the measured observational uncertainties as well as the assumed models of binary evolution.
Thus, it is unlikely that any currently observed \acp{HMXB} will form merging \acp{BBH} within a Hubble time.

In addition to the lack of currently observed \acp{HMXB} that are thought to be merging \ac{BBH} progenitors, there are differences in the \ac{BH} masses of the observable \ac{XRB} and merging \ac{BBH} populations.
While the primary \ac{BH} mass distribution predicted from \ac{GW} observations extends to masses higher than the peak near $35M_{\odot}$~\citep{abbott_population_2022}, there are no observed \acp{HMXB} with \ac{BH} accretor masses that fall near or beyond this limit.
Rather than arising from fundamental astrophysical differences, the disparities in the observed \ac{HMXB} and \ac{GW} \ac{BH} population masses may actually be a product of detector selection effects alone.
\citet{fishbach_apples_2022} compare the observed mass distributions of \acp{BH} in \acp{XRB} and \acp{BH} observed with \acp{GW}, and find that when \ac{GW} detector selection effects are accounted for, there are currently no statistically significant mass differences between the \ac{XRB} and \ac{GW} \ac{BH} populations.
However, it is critical that X-ray and \ac{GW} observational selection effects are \textit{jointly} examined, as they may produce differing outcomes in the observed populations.

The spin distributions of X-ray and \ac{GW} \ac{BH} populations can also provide insight on their evolutionary history and the role of selection effects in determining their properties.
\citet{fishbach_apples_2022} examine the \ac{BH} spin distributions of \ac{BH}--\acp{XRB} and \ac{GW}--\acp{BBH} and find them to be inconsistent with one another.
It is possible, however, that differing binary evolutionary channels between the two populations may contribute to this discrepancy.
For example, \citet{gallegos-garcia_high-spin_2022} find that high-spin \acp{XRB} formed through Case-A mass transfer can only form merging \acp{BBH} within a small parameter space, and thus it is not surprising that the observed spin distributions are observed to be different.

In this paper, we show that observational selection effects play an important role in determining the observed \ac{BH} populations and the evolutionary predictions of detected \acp{HMXB}.
We compare detectable \ac{HMXB} populations with detectable \ac{GW} populations using simulated astrophysical samples of binaries, and we quantify the probability that observable \acp{HMXB} will form \ac{BH}--\ac{BH} binaries that will merge in a Hubble time, accounting for X-ray selection effects.
We do not attempt to reproduce \ac{XRB} or \ac{GW} observations in detail; rather, we seek to understand how observational selection effects may produce the aforementioned differences in mass ranges and evolutionary predictions of \acp{HMXB}.

In Section~\ref{sec:method}, we discuss our methods for sampling populations of binaries distributed throughout the universe, as well as how we model their X-ray and \ac{GW} emission properties.
We develop a formalism for quantifying the probability of obtaining a given source in our sampled population, taking into account observational selection effects.
In Section~\ref{sec:results}, we discuss parameter distribution properties for detectable \acp{HMXB} and detectable merging \acp{BBH} in our sample, as well as evolutionary probabilities for these sources.
Last, in Section~\ref{sec:discussion}, we discuss our main results, caveats in our analysis, and the implications of our findings in the context of X-ray and \ac{GW} observations.
Throughout this work, we assume \textit{Planck 2018} cosmological parameters~\citep{planck_collaboration_planck_2020}, including values for the Hubble constant ($H_0 = 67.7 \, \mathrm{km} \, \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$) and the mass and dark energy density parameters ($\Omega_\mathrm{m}=0.31$ and $\Omega_{\Lambda}=0.69$, respectively).

\section{Methods} \label{sec:method}

We examine populations of \acp{HMXB} with the rapid binary population synthesis code \texttt{COSMIC}~\citep[version 3.4;][]{breivik_cosmic_2020}.
\texttt{COSMIC} is based on the single-stellar evolution formulae from \citet{hurley_comprehensive_2000} and the binary evolution prescriptions from \citet{hurley_evolution_2002}.
\texttt{COSMIC} includes many updates to these prescriptions, such as those for OB stellar winds~\citep{vink_mass-loss_2001}, Wolf--Rayet star winds~\citep{vink_metallicity_2005}, the initiation of unstable mass transfer~\citep{belczynski_compact_2008, claeys_theoretical_2014, neijssel_effect_2019}, remnant formation \citep{fryer_compact_2012}, and pair instability supernovae~\citep{woosley_pulsational_2017, marchant_pulsational_2019, spera_merging_2019}.
We generate binary populations with \texttt{COSMIC} across a grid of metallicities, and then sample systems from these populations and distribute them across redshifts.

\subsection{Binary Population Models}

We use \texttt{COSMIC} to generate populations of \acp{HMXB}.
To do this, we allow the formation of \ac{DCO} binaries where the first-born object is a \ac{BH} or \ac{NS} and the second-born object is a \ac{WD}, \ac{NS}, or \ac{BH}.
We only keep binaries that remain bound after the first \ac{SN} event and have stellar companion masses $>5M_{\odot}$ at the formation time of the first-born \ac{CO}.
We make these cuts because we define \acp{HMXB} to be systems that are concurrently bound and contain one accreting \ac{CO} and one stellar object $\geq$ 5~$M_{\odot}$ at some point in their evolution.
These cuts ensure that our initial sample space allows for \ac{HMXB} formation but is not restricted to only merging \ac{BBH} progenitors.

We simulate 16 \ac{DCO} populations across a discrete log-spaced metallicity grid that spans $(1/200) Z_{\odot}$ to $(7/4) Z_{\odot}$, as this metallicity range is approximately that of the grids used for stellar evolution within \texttt{COSMIC}~\citep{hurley_comprehensive_2000}.
Our population models include a set of key assumptions.
We sample initial conditions according to the correlated multidimensional distributions from \citet{moe_mind_2017}.
All of our binaries are drawn from the same initial mass function, and we assume a solar metallicity of $Z_{\odot} = 0.017$~\citep{grevesse_standard_1998} for all calculations.
To calculate \ac{CO} remnant properties from the pre-supernova properties of a star, we employ the Delayed remnant mass prescription from \citet{fryer_compact_2012} that allows for compact object formation in the lower mass gap~\citep[e.g.,][]{zevin_exploring_2020}.
We assume a conservative upper-limit for the maximum \ac{NS} mass of $3~M_\odot$ \citep{rhoades_maximum_1974, kalogera_maximum_1996}.
To determine respective remnant masses for pulsational pair-instability supernovae and pair-instability supernovae, we use fits to Table 1 of \citet{marchant_pulsational_2019}.
For common-envelope evolution, we use the prescription from \cite{belczynski_compact_2008} to determine the critical mass ratio for the onset of unstable mass transfer.
We set $\alpha = 1$ for the common-envelope efficiency parameter~\citep{livio_common_1988, ivanova_common_2013}, and use a variable prescription for the envelope binding energy factor $\lambda$~\citep{claeys_theoretical_2014}.
Last, we set the wind-accretion efficiency factor to $0.5$, and enforce Eddington-limited accretion in all binaries.

These assumptions represent a single fiducial point in a high-dimensional parameter space of binary evolution uncertainties that can have a significant impact on the population properties of compact binaries and their progenitors~\citep[e.g.,][]{barrett_accuracy_2018, belczynski_uncertain_2021, broekgaarden_impact_2022}.
Because we do not attempt to reproduce observations in detail and rather seek to understand how selection effects impact \ac{HMXB} and merging \ac{BBH} populations, a fiducial model is satisfactory for our purposes.
We reserve a more systematic exploration of this parameter space for future work, and we comment on possible impacts to our results in Section \ref{sec:discussion}.

\subsection{Population Sampling} \label{sec:pop_samp}

From our set of \texttt{COSMIC} models run at discrete metallicities, we sample two populations of binaries: the first is a local population sampled out to redshift 0.05 ($z_{<0.05}$), and the second is a population sampled out to redshift 20 ($z_{<20}$).
Because our \texttt{COSMIC} populations assume a single burst of star formation and evolve all systems for a Hubble time, we must assign binary redshifts and metallicities in postprocessing.
We sample a $z_{<0.05}$ population to acquire robust observable \ac{HMXB} population statistics that cannot be acquired in a $z_{<20}$ sample, as the effective sample size of detectable \acp{HMXB} in the $z_{<20}$ population is small (there are $\simeq 100$ detectable \acp{HMXB} in the $z_{<20}$ sample compared to $\simeq 7 \times10^3$ in the $z_{<0.05}$ sample).
Our choice to use a locally-sampled population of \acp{HMXB} in our analysis is discussed in more detail in Section \ref{sec:xrb_results}.

We draw redshifts and metallicites jointly from a two-dimensional redshift--metallicity grid using the weights
\begin{equation}\label{eq:redshift_metallicity_weights}
    \mathcal{W}(z,Z) = \zeta(Z) P(Z|z) \psi[z(t)]\ \frac{\mathrm{d}t}{\mathrm{d}z}.
\end{equation}
Here, $\zeta(Z)$ is the formation efficiency of the target population at metallicity $Z$, which we define as the number of \acp{HMXB} formed in our sample divided by the total sampled stellar mass of the population; $P(Z|z)$ is the probability of drawing metallicity $Z$ at redshift $z$; $\psi[z(t)]$ is the star formation rate at redshift $z$ attained by marginalizing over metallicity in a grid of stellar mass formed per unit lookback time and metallicity; and the term $\mathrm{d}t/\mathrm{d}z$ is a change of variables from time to redshift space for $\psi[z(t)]$,
\begin{equation}
    \frac{\mathrm{d}t}{\mathrm{d}z} = \frac{1}{(1+z)E(z)},
\end{equation}
where $E(z)$ is the cosmological factor for a flat universe~\citep{ryden_introduction_2002},
\begin{equation} \label{eqn:cosmo_factor}
    E(z) = \sqrt{\Omega_\mathrm{m}(1+z)^3 + \Omega_{\nu}(1+z)^4 + \Omega_{\Lambda}}.
\end{equation}

We determine $P(Z|z)$ using publicly-available data from the \texttt{Illustris-TNG} simulation~\citep{nelson_illustristng_2019}.
This simulation provides stellar mass formed per unit lookback time and metallicity in a $100~\mathrm{Mpc}^3$ comoving box \citep[][Section 2.3]{zevin_suspicious_2022}.
The bounds of the distribution for $P(Z|z)$ are $Z/Z_{\odot}=5\times10^{-5}$ and $Z/Z_{\odot}=0.04$, which approximately correspond to the lower and upper bounds of our \texttt{COSMIC} population metallicity grid.
Though the redshift--metallicity evolution $P(Z|z)$ is highly uncertain, especially at high redshifts, the star formation rate density evolution predicted from \texttt{Illustris-TNG} falls within the range of uncertainty \citep[e.g.,][]{chruslinska_chemical_2022}, and the evolution of the metallicity distribution is more astrophysically motivated than analytic approaches, such as truncated log-normal distributions.

We jointly draw $N_\mathrm{total}=10^6$ redshifts and metallicities using the relative weights from Eq.~\eqref{eq:redshift_metallicity_weights}, and randomly sample binaries from the discrete \texttt{COSMIC} metallicity models based on which discrete metallicity model is closest to a drawn metallicity in log-space.
This provides a \ac{DCO} population distributed across metallicity and redshift that is representative of a population of binaries in the universe.

\subsection{X-Ray Binary Detection Flux} \label{sec:xrb_calc}

As we are interested in binaries that can potentially lead to \ac{BBH} mergers, we consider \ac{BH}--\acp{HMXB} in our sample to be systems that are concurrently bound and contain one \ac{BH} object and one stellar object $\geq$ 5~$M_{\odot}$ at some point in their evolution.
The \ac{HMXB} phase begins at first \ac{BH} formation and ends at second \ac{CO} formation or when the donor mass falls below 5~$M_{\odot}$.

Since the time series resolution needed to calculate the X-ray emission of \acp{HMXB} is much finer than what \texttt{COSMIC} can handle for large binary populations, we re-evolve \acp{HMXB} in our sample with a small timestep resolution of $10^3$ years.
We only apply this fine resolution during the HMXB phase to ensure computational efficiency.
This is possible due to the ability of \texttt{COSMIC} to restart binaries in the middle of their evolution and evolve them forward~\citep{breivik_cosmic_2020}.\footnote{Updated \texttt{COSMIC} functionality is described at  \href{https://cosmic-popsynth.github.io/docs/stable/examples/index.html}{cosmic-popsynth.github.io/docs/stable/examples/index.html}.}

We adopt the method from \cite{podsiadlowski_formation_2003} to calculate the \ac{HMXB} accretion luminosity,
\begin{equation} \label{lum}
L_\mathrm{acc} = \eta \dot{M}_\mathrm{acc}c^2,
\end{equation}
where $\dot{M}_\mathrm{acc}$ is the Eddington-limited accretion rate of matter onto the \ac{BH} and $\eta$ is an efficiency factor for the \ac{BH} conversion of rest mass into radiative energy.
For simplicity, we calculate $\eta$ assuming zero initial \ac{BH} spin and that it is determined entirely by the last stable particle orbit. For $M_\mathrm{BH} < \sqrt{6}M_{\mathrm{BH},0}$, the efficiency $\eta$ is
\begin{equation}
    \eta = 1 - \sqrt{1 - \bigg(\frac{M_{\mathrm{BH}}}{3M_{\mathrm{BH},0}}\bigg)^2},
\end{equation}
where $M_{\mathrm{BH}}$ is the \ac{BH} mass at a given time and $M_{\mathrm{BH},0}$ is the initial \ac{BH} gravitating mass--energy at formation according to \cite{bardeen_kerr_1970}.
If $M_{\mathrm{BH}}$ exceeds $\sqrt{6}M_{\mathrm{BH},0}$, then $\eta$ is taken to be 0.42.
The accretion rate is related to the change in \ac{BH} mass by
\begin{equation}
    \dot{M}_{\mathrm{BH}} = (1-\eta)\dot{M}_{\mathrm{acc}},
\end{equation}
as energy released as radiation will not contribute to the BH mass.
\texttt{COSMIC} includes wind prescriptions for mass loss in its evolution, so we do not need to explicitly calculate wind accretion.
All changes in \ac{BH} mass from both Roche lobe overflow and wind mass transfer are included in $\dot{M}_{\mathrm{BH}}$.

To determine if a given \ac{HMXB} is detectable, we impose a flux limit for observations.
The X-ray flux is
\begin{equation}
    F = \frac{L_{\mathrm{acc}}}{4\pi D_\mathrm{L}(z)^2},
\end{equation}
where $D_\mathrm{L}(z)$ is the luminosity distance of the binary at the start of the \ac{HMXB} phase.
An \ac{HMXB} is considered detectable when its flux exceeds $5 \times 10^{-15}$~erg s$^{-1}$ cm$^{-2}$, which is an observational threshold commonly used in X-ray surveys \citep[e.g.,][]{chandra_x-ray_center_chandra_2021}.

\subsection{Gravitational-Wave Detection Probability} \label{sec:gw_pdet}

We calculate the \ac{GW} detection probabilities $p_{\mathrm{det}}$ of \ac{DCO} binaries in our sample that have merged by today ($z=0$).
We evaluate the source signal-to-noise ratio (S/N) $\rho$ as
\begin{equation}
    \rho = \rho_0 \omega,
\end{equation}
where we set $\rho_0$ to be the optimal S/N of a source with component masses $m_1$ and $m_2$ at redshift $z$, and $\omega$ is a  projection  factor  that  depends  on  the relative  angular  orientation  of  the  source  and  the detector~\citep{dominik_double_2015}.
We approximate $\rho_0$ following \cite{fishbach_does_2018} as
\begin{equation}
     \rho_0 = 8\left[\frac{\mathcal{M}(1+z_\mathrm{m})}{\mathcal{M}_8}\right]^{5/6}\frac{D_{\mathrm{L},8}}{D_\mathrm{L}},
\end{equation}
where $\mathcal{M} =  (m_1 m_2)^{3/5} / (m_1 + m_2)^{1/5}$ is the source chirp mass, $D_\mathrm{L}$ is the luminosity distance of the source, $z_\mathrm{m}$ is the merger redshift, and the constants $\mathcal{M}_8$ and $D_{\mathrm{L},8}$ are set to $10 \;M_{\odot}$ and $1~\mathrm{Gpc}$, respectively, such that they represent the typical distances at which sources are detectable by Advanced LIGO at design sensitivity~\citep{chen_distance_2021}.
This scaling approximates the amplitude of a \ac{GW} coalescence signal to first order.
For the projection factor $\omega$, we use the analytical expression for a single-detector network from \cite{dominik_double_2015}, which is equivalent to $\Theta/4$ in \cite{finn_observing_1993}.
We calculate the detection probability $p_{\mathrm{det}}$ by Monte Carlo sampling $\omega$ and taking the fraction of the resulting source S/N that exceed a detection threshold of $\rho = 8$~\citep{thorne_gravitational_1997, chen_distance_2021}.

\subsection{Population Probabilities} \label{sec:probs}

We are interested in calculating the probability of obtaining sources in our sampled populations.
In the most general form, we define the probability of finding a given source in state $X$ in our sample as
\begin{equation} \label{eqn:probs_general}
    P({X}) = \frac{N_{X}}{N_{\mathrm{total}}},
\end{equation}
where $N_X$ is the number of sources in state $X$ and $N_{\mathrm{total}}$ is the number of sources in the full sample of interest.
Because we need to count sources with different lifetimes distributed across space and time relative to an observer, a more involved framework is necessary to obtain values for $N_X$ and $N_{\mathrm{total}}$.

We first need to count the number of binaries of interest $N_X$ in a given spatial comoving volume $V_\mathrm{c}$ with spatial points $\vec{x}$ surveyed from comoving times $t_1=t_1(\vec{x})$ to $t_2=t_2(\vec{x})$ in the source frame.
Each source in a small spatial density of sources $n(\vec{x},t)$ has its own worldline along which it may be in some state $X$ for some time, which we track with an indicator function $I(\vec{x},t)$ that is nonzero when in state $X$ and zero otherwise, and normalized such that for a single source $\int \int I(\vec{x},t) \,\mathrm{d}t \mathrm{d}\vec{x} = 1$.
Thus, we count the number of systems in state $X$ as
\begin{equation} \label{eqn:definition}
    N_X = \int_{V_\mathrm{c}} \int_{t_1(\vec{x})}^{t_2(\vec{x})} {n(\vec{x}, t) I(\vec{x},t)}\,\mathrm{d}t \mathrm{d}\vec{x}.
\end{equation}

\subsubsection{Long-lived Sources}\label{sec:longlived}

For any type of source whose lifetime is significantly longer than the length of the observing window ${T = (t_2-t_1)}$, the number of counted sources $N_{\mathrm{long}}$ does not change during the observing period.
This implies that the spatial density is only a function of position $n(\vec{x})$.
The indicator function takes on a form $I(x)/T$, and all observing-time dependence disappears after integrating over time.
One instance where this is the case is with detectable \acp{HMXB}.
As detailed in Section~\ref{sec:xrb_results}, even \acp{HMXB} that emit observably for a comparatively short amount of time emit on the order of thousands of years, making it valid to count them as long-lived sources compared to the timescale of astronomical surveys.
This long-lived limit also holds for binaries that exist in some evolutionary state for a significant length of time, e.g., bound \acp{BBH} that have not merged.

With this simplification considered, we now count the number of long-lived sources as
\begin{equation}
    N_{\mathrm{long}} = \int_{V_\mathrm{c}} n(\vec{x})\,\mathrm{d}\vec{x}.
\end{equation}
This equation can be rewritten in terms of redshift,
\begin{equation} \label{eqn:convert_to_z}
    N_{\mathrm{long}} = \int_{z_1}^{z_2} n(z)\frac{\mathrm{d}V_\mathrm{c}}{\mathrm{d}z}\mathrm{d}z,
\end{equation}
where the redshifts $z_1, z_2$ are the bounds of the volume $V_\mathrm{c}$, $n(z)$ is the number density of sources in terms of redshift, and the factor $\mathrm{d}V_\mathrm{c}/\mathrm{d}z$ is the comoving volume element corresponding to redshift $z$, which is given by~\citep{ryden_introduction_2002},
\begin{equation}
    \frac{\mathrm{d}V_\mathrm{c}}{\mathrm{d}z} = \frac{4\pi c}{H_0} \frac{D_\mathrm{c}(z)^2}{E(z)},
\end{equation}
where $D_\mathrm{c}(z)$ is the comoving distance at redshift $z$ and $E(z)$ is defined in Eq.~\eqref{eqn:cosmo_factor}.
Changing variables from spatial coordinates to redshift significantly simplifies the evaluation of the integral for $N_{\mathrm{long}}$.

Because we use a discrete simulation for our calculations, Eq.~\eqref{eqn:convert_to_z} must be approximated as a discrete sum over individual sources in our \texttt{COSMIC} populations and a Riemann sum over redshift,
\begin{equation} \label{eqn:long_sum_final}
      N_{\mathrm{long}} = \sum_i \sum_j \frac{I_i^X(z_j)}{V_{\mathrm{box}}} \frac{\mathrm{d}V_\mathrm{c}}{\mathrm{d}z_j}\Delta z_j ,
\end{equation}
where we have re-written the spatial density of sources as $n(z_j)=\sum_iI_i^X(z_j)/V_{\mathrm{box}}$.
Here, $I_i^{X}(z_j)$ indicates if the $i$-th system in the sample is in state $X$ at redshift $z_j$, and $V_{\mathrm{box}}$ is the volume of the comoving box that we consider for our resampled population.
Eq.~\eqref{eqn:long_sum_final} allows for a quick, simplified computation of $N_{\mathrm{long}}$.

\subsubsection{Short-lived Sources}\label{sec:shortlived}

We can also apply Eq.~\eqref{eqn:definition} to calculate the number of sources $N_{\mathrm{short}}$ in the case that their lifetimes are significantly shorter than the length of the observing window.
This includes \ac{BBH} mergers in the frequency band of ground-based \ac{GW} detectors which only exist as detectable sources for at most a few seconds before coalescence.
In this case, the length of the source lifetime approaches zero, which causes the indicator function to act as a delta function,
\begin{equation}
     N_{\mathrm{short}} = \int_{V_\mathrm{c}} \int_{t_1(\vec{x})}^{t_2(\vec{x})} n(\vec{x},t) \delta(t-t'(\vec{x}))\,\mathrm{d}t \mathrm{d}\vec{x} ,
\end{equation}
where $t'(\vec{x})$ marks the time at which state $X$ occurs at spatial position $\vec{x}$.
In the continuum limit, this expression is an integral over the rate density of sources $r(\vec{x}, t)$,
\begin{equation} \label{eqn:rate_density_int}
    N_{\mathrm{short}} = \int_{V_\mathrm{c}} \int_{t_1(\vec{x})}^{t_2(\vec{x})}  r(\vec{x}, t)\,\mathrm{d}t \mathrm{d}\vec{x}.
\end{equation}
When we observe for some time window $T$, the time dependence of the rate density in Eq.~\eqref{eqn:rate_density_int} integrates to $T$ with a factor of $1/(1+z)$ to account for transitioning between the source frame and observer frame time.
As with Eq.~\eqref{eqn:convert_to_z}, we use a change of variables to write the volume integral as an integral over redshift,
\begin{equation}
    N_{\mathrm{short}} = T \int_{z_1}^{z_2} r(z) \frac{\mathrm{d}V_\mathrm{c}}{\mathrm{d}z} \frac{1}{1+z}\mathrm{d}z.
\end{equation}
Rewriting this equation as a sum over individual sources, we obtain
\begin{equation} \label{eqn:short_sum_final}
    N_{\mathrm{short}} = T \sum_i \sum_j \frac{I_i^X(t_j, t_j + \tau)}{\tau V_{\mathrm{box}}} \frac{\mathrm{d}V_\mathrm{c}}{\mathrm{d}z_j} \frac{1}{1+z_j}\Delta z_j,
\end{equation}
where we have re-written the rate density of sources as $r(t_j)=\sum_i I_i^X(t_j,t_j+\tau)/[\tau V_{\mathrm{box}}]$, with $t_j = t(z_j)$. $I_i^X(t_j, t_j + \tau)$ indicates if the $i$-th system in the sample undergoes an event $X$ at redshift $z_j$ between times $t_j$ and $t_j + \tau$, where $\tau$ is a chosen time interval significantly shorter than the timescale on which the rate density evolves.
For practical purposes, $\tau$ can be thought of as a chosen bin width in time.
Eq.~\eqref{eqn:short_sum_final} allows for an efficient computation of $N_{\mathrm{short}}$.

\subsubsection{Detection Weights} \label{sec:weight_calc}

We apply the formalism developed in the previous subsections to assign relative detection weights to all binaries in our sample.
Specifically, the sums over redshift in Eq.~\eqref{eqn:long_sum_final} and Eq.~\eqref{eqn:short_sum_final} are the expressions we use to define the weight of source $i$ in state $X$.
As we consider relative weights, we can ignore the constant volume factor $V_{\mathrm{box}}$ and the time factors $T$ and $\tau$.
We use the formalism for long-lived sources to calculate detectable \ac{XRB} weights, and that for short-lived sources to calculate detectable \ac{GW} weights.

For \ac{HMXB} sources that emit for long times, this weighting is the sum over redshift in Eq.~\eqref{eqn:long_sum_final}, which becomes
\begin{equation}\label{eqn:xrb_weights}
    \mathcal{W}_{\mathrm{xrb}, i} \propto \sum_j I_i^{\mathrm{HMXB}_{\mathrm{obs}}}(z_j) \frac{\mathrm{d}V_\mathrm{c}}{\mathrm{d}z_j}\Delta z_j ,
\end{equation}
where the indicator $I^{\mathrm{HMXB}_{\mathrm{obs}}}(z_j)$ has value equal to unity if the \ac{HMXB} exceeds a given X-ray detection threshold at redshift $z_j$, and has value zero otherwise.
Thus, sources that emit above the detection threshold for longer periods of time will contribute more in the summation and be given a higher relative detection weighting.

The weights for merging \ac{BBH} sources are similar, but now the \ac{GW} detection probability is included with the indicator function to account for selection effects,
\begin{equation}
    \mathcal{W}_{\mathrm{gw}, i} \propto \sum_{j} I_i^{\mathrm{BBH}_\mathrm{m}}(z_j) p_{\mathrm{det}, i}\frac{\mathrm{d}V_\mathrm{c}}{\mathrm{d}z_j} \frac{1}{1+z_j}\Delta z_j,
\end{equation}
where the indicator $I_i^{\mathrm{BBH}_\mathrm{m}}(z_j)$ has value unity if the \ac{BBH} coalesces at redshift $z_j$ and zero otherwise.
Thus, for \acp{BBH} merging at redshift $z_m$, each system has a relative weight of
\begin{equation} \label{eqn:gw_weights}
    \mathcal{W}_{\mathrm{gw}, i} \propto p_{\mathrm{det},i}\frac{\mathrm{d}V_\mathrm{c}}{\mathrm{d}z_m} \frac{1}{1+z_m}\Delta z_m .
\end{equation}
Building this formalism for relative detection weights allows us to easily compare observable systems and their progenitors in Section~\ref{sec:results}.

\subsubsection{Probability Expressions}

We use the counting expressions in Section~\ref{sec:longlived} and Section~\ref{sec:shortlived} to calculate probabilities in the form of Eq.~\eqref{eqn:probs_general}.
For observable \acp{HMXB}, this equation becomes
\begin{equation}
    P(\mathrm{HMXB}_{\mathrm{obs}}) = \frac{N_{\mathrm{HMXB}_{\mathrm{obs}}}}{N_{\mathrm{total}}},
\end{equation}
which, written in summation form, is
\begin{equation} \label{eqn:pHMXB_obs}
   P(\mathrm{HMXB}_{\mathrm{obs}})   = \frac{\sum_i \sum_j I_i^{\mathrm{HMXB}_{\mathrm{obs}}}(z_j)[\mathrm{d}V_\mathrm{c}/\mathrm{d}z_j]\Delta z_j}{\sum_{i'} \sum_{j'} I_{i'}^\mathrm{ZAMS}(z_{j'})[\mathrm{d}V_\mathrm{c}/\mathrm{d}z_{j'}]\Delta z_{j'}},
\end{equation}
where $I_{i'}^{\mathrm{ZAMS}}(z_{j'})$ indicates if the $i$-th system in the sample has been initialized at \ac{ZAMS} and exists as any form of bound binary at redshift $z_{j'}$.
These are the systems we count as members of the $N_{\mathrm{total}}$ sample of binaries.

For merging \acp{BBH}, the probability of detecting a source is
\begin{multline} \label{eqn:pBBHm_obs}
    P(\mathrm{BBHm}_{\mathrm{obs}}) = \\ \frac{T}{\tau}\frac{\sum_i \sum_jI_i^\mathrm{BBHm}(t_j, t_j + \tau)p_{\mathrm{det},i}[{\mathrm{d}V_\mathrm{c}}/{\mathrm{d}z_j}][{1+z_j}]^{-1}\Delta z_j }{\sum_{i'} \sum_{j'}I_{i'}^\mathrm{ZAMS}(z_{j'})[{\mathrm{d}V_\mathrm{c}}/{\mathrm{d}z_{j'}}]\Delta z_{j'}}.
\end{multline}
Note that this probability heavily depends on choices for the observing window $T$ and time bin width $\tau$, and thus does not have a strong physical meaning.

We also consider \emph{conditional} probabilities for sources in our sample.
For example, the probability an observable \ac{HMXB} will become a \ac{BBH} that merges within a Hubble time can be written as
\begin{equation}
    P(\mathrm{BBHm}_\mathrm{H} | \mathrm{HMXB}_{\mathrm{obs}}) = \frac{P(\mathrm{BBHm}_\mathrm{H} \cap \mathrm{HMXB}_{\mathrm{obs}})}{P(\mathrm{HMXB}_{\mathrm{obs}})},
\end{equation}
which, in summation form, is
\begin{multline}
    P(\mathrm{BBHm}_\mathrm{H} | \mathrm{HMXB}_{\mathrm{obs}}) = \\ \frac{\sum_i \sum_j I_i^{\mathrm{HMXB}_{\mathrm{obs}} \cap \mathrm{BBHm}_\mathrm{H}}(z_j)[{\mathrm{d}V_\mathrm{c}}/{\mathrm{d}z_j}]\Delta z_j}{\sum_{i'} \sum_{j'}  I_{i'}^{\mathrm{HMXB}_{\mathrm{obs}}}(z_{j'})[{\mathrm{d}V_\mathrm{c}}/{\mathrm{d}z_{j'}}]\Delta z_{j'}},
\end{multline}
where $I_i^{\mathrm{HMXB}_{\mathrm{obs}} \cap \mathrm{BBHm}_\mathrm{H}}(z_j)$ indicates if the $i$-th system in the sample is an \ac{HMXB} emitting above a given observable flux threshold at redshift $z_j$ that will \emph{also} become a merging \ac{BBH} within a Hubble time.

These probability expressions, along with those for the relative detection weights in Eq.~\eqref{eqn:xrb_weights} and Eq.~\eqref{eqn:gw_weights}, provide all the mathematical tools necessary to quantify the relationships between observable sources and their progenitors.

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

To determine the impact of  X-ray and \ac{GW} selection effects on their observed populations, we apply detection weights as calculated in Section~\ref{sec:weight_calc} to our sampled binaries from \texttt{COSMIC}.
To fully understand these results, we must also understand the emission behavior of our X-ray and \ac{GW} populations.
This provides context for their evolutionary behavior and how it relates to their observability, especially in the \ac{HMXB} case where emission behavior has not been well-characterized for large theoretical populations.
We also examine the mass distributions of detected \ac{HMXB} and \ac{BBH} populations and calculate probabilities describing the relationships of sources to one another.
Through this analysis, we illuminate possible causes of the observational discrepancies in \ac{BH} masses and evolutionary predictions between X-ray and \ac{GW} sources.

\subsection{HMXB Emission Properties} \label{sec:xrb_results}

The emission properties of our \ac{HMXB} populations, as calculated in Section \ref{sec:xrb_calc}, provide insight into the nature of their detectability.
Figure~\ref{fig:max_lum_kde} shows the maximum X-ray luminosity $L_{\mathrm{max}}$ reached by all \acp{HMXB} in our $z_{<0.05}$ and $z_{<20}$ samples, as well as the time-averaged X-ray luminosity $L_{\mathrm{avg}}$ of \acp{HMXB} during the X-ray phase for the $z_{<0.05}$ sample.
We mark the Eddington luminosities for $10~M_{\odot}$ and $30~M_{\odot}$ \ac{BH} accretors with vertical dotted lines.
Almost $70\%$ of the $z_{<0.05}$ \acp{HMXB} achieve peak luminosities between 10$^{38}$ and 10$^{39}~\mathrm{erg\, s^{-1}}$ during their \ac{XRB} phase, which is roughly within one order of magnitude of their Eddington luminosity.
The overall maximum emission behavior of the $z_{<0.05}$ and $z_{<20}$ \acp{HMXB} is similar across all luminosities.
Both exhibit a steep drop in the distribution close to the Eddington limit along with a tail to lower X-ray luminosities.

The time-averaged luminosity during the \ac{XRB} phase follows approximately the same distribution as the maximum luminosity for values $< 10^{33}~\mathrm{erg\, s^{-1}}$.
At higher luminosities, however, these distributions diverge, with a larger discrepancy between the average and maximum luminosities reached during the \ac{XRB} phase.
This is because in our models, most systems emitting at higher luminosities only reach their peak emission for a short time and emit at luminosities a few orders of magnitude lower for most of the \ac{XRB} phase.

The detectability of \acp{HMXB} is influenced by the time for which they maintain high luminosities, which varies between binaries.
Figure~\ref{fig:emit_duration_kde} shows the duration of observable emission $T_{\mathrm{obs}}$ for \acp{HMXB} in our $z_{<20}$ and $z_{<0.05}$ samples that emit above the minimum X-ray detection threshold of $5\times10^{-15}~\mathrm{erg\, s^{-1}\, cm^{-2}}$ for a nonzero period of time.
We also show the total duration of the \ac{XRB} phase $T_{\mathrm{xrb}}$ for the same binaries in our $z_{<0.05}$ sample.
Once again, the emission behavior between the $z_{<20}$ and $z_{<0.05}$ samples is nearly identical.
More than $98\%$ of \acp{HMXB} emit observably for less than $1~\mathrm{Myr}$, with the $T_{\mathrm{obs}}$ distributions peaking near $0.3~\mathrm{Myr}$.
For the majority of binaries, this time is less than half of the duration of the X-ray binary phase, which is governed by the remaining lifetime of the donor star after the formation of the first \ac{CO}.

\begin{figure}
\includegraphics[width=0.5\textwidth]{figures/HMXB_max_lum.pdf}
\caption{The distribution of the maximum X-ray luminosity $L_{\mathrm{max}}$ reached by all \acp{HMXB} in the $z_{<20}$ (solid gold line) and $z_{<0.05}$ (dashed red line) populations, along with the time-averaged X-ray luminosity $L_{\mathrm{avg}}$ of binaries during the \ac{HMXB} phase for the $z_{<0.05}$ population (solid black line).
The left and right vertical dotted lines mark the Eddington luminosity limits for $10 M_{\odot}$ and $30 M_{\odot}$ \ac{BH} accretors, respectively.}
\label{fig:max_lum_kde}
\end{figure}

\begin{figure}
\includegraphics[width=0.5\textwidth]{figures/HMXB_emit_duration.pdf}
\caption{Distributions of the duration of observable X-ray emission $T_{\mathrm{obs}}$ for all \acp{HMXB} in the $z_{<20}$ (solid gold line) and $z_{<0.05}$ (dashed red line) populations that emit above the minimum X-ray detection threshold for a nonzero period of time.
For the $z_{<0.05}$ population, we also show the total duration of the X-ray binary phase $T_{\mathrm{xrb}}$ (solid black line).}
\label{fig:emit_duration_kde}
\end{figure}

\subsection{Comparing Selection Effects}

Here we compare the detectable \ac{HMXB} and detectable merging \ac{BBH} populations in order to understand the existing observational discrepancies between \acp{BH} detected in X-ray versus \ac{GW} sources.
Namely, we investigate the lack of observed \acp{HMXB} that are predicted to become \ac{BBH} mergers, as well as the lack of high-mass \acp{BH} found in \acp{XRB}.
We use Eq.~\eqref{eqn:xrb_weights} and Eq.~\eqref{eqn:gw_weights} to calculate detectability weights for \acp{HMXB} ($\mathcal{W}_{\mathrm{xrb}}$) and \acp{GW} ($\mathcal{W}_{\mathrm{gw}}$), respectively.

In all of these comparisons, we use the $z_{<0.05}$ \ac{HMXB} population and the $z_{<20}$ \ac{BBH} population.
This is because observable \acp{HMXB} in both populations have comparable properties (in addition to the emission properties in Figure~\ref{fig:max_lum_kde} and Figure~\ref{fig:emit_duration_kde}, their redshift, metallicity, and mass distributions are nearly identical), but the effective sample size of observable \acp{HMXB} in the $z_{<0.05}$ population is $\simeq50$ times larger, and thus allows for more robust statistics.
More than $99.9\%$ of \acp{HMXB} in the $z_{<20}$ sample are too distant to be observed, even if they achieve high luminosities.
In addition, the maximum \ac{ZAMS} formation redshift of observable \acp{HMXB} in our $z_{<0.05}$ sample is $\simeq8 \times 10^{-3}$, which is much smaller than the upper sampling bound of $z=0.05$.
Thus, even if we sampled a $z_{<20}$ population with one billion systems, we still would not find any observable \acp{HMXB} beyond $z \sim 0.01$.

\subsubsection{Redshifts and Metallicities}

In Figure~\ref{fig:zf_comp_kde}, we compare the redshift distributions of \acp{HMXB} formed by today (HMXB$_{z0}$) and \acp{BBH} merged by today (BBHm$_{z0}$), weighted by their respective detectability
In Figure~\ref{fig:met_comp_kde}, we do the same for the metallicity distributions.
As expected, the progenitors of observable \acp{HMXB} form at very low redshifts ($z_\mathrm{f} \lesssim 0.01$), whereas the progenitors of observable \acp{BBH} mergers form at much higher redshifts ($z_\mathrm{f} \gtrsim 1$), leading to detectable \ac{BBH} mergers in the redshift range $0.01 \lesssim z_\mathrm{m} \lesssim 1$.

As a consequence of the differing redshift distributions of observable \ac{HMXB} and \ac{BBH}-merger progenitors, there are also differences in the population metallicities, with the detection-weighted \acp{HMXB} having higher typical metallicities than the merging \acp{BBH}.
The metallicity distribution of detectable \acp{HMXB} peaks near $0.5 Z/Z_{\odot}$ while that for detectable merging \acp{BBH} peaks near $0.08 Z/Z_{\odot}$.
There is almost no support for detectable \ac{HMXB} formation below $0.1 Z/Z_{\odot}$, as fewer than $0.05\%$ of observable \acp{HMXB} have metallicities below this limit.
This is expected because the distance at which \ac{GW} detectors are sensitive to merging \acp{BBH} is much larger than the distance to which we can observe \acp{HMXB}, and thus \acp{GW} probe higher redshifts and lower metallicites.

\begin{figure}
\includegraphics[width=0.5\textwidth]{figures/hmxb_bbhm_obs_redshift_kde.pdf}
\caption{Distributions of the \ac{ZAMS} formation redshifts $z_\mathrm{f}$ for $z_{<0.05}$ \acp{HMXB} and  $z_{<20}$ merging \acp{BBH}, as well as the merger redshifts $z_\mathrm{m}$ for $z_{<20}$ merging \acp{BBH}, weighted by their X-ray and \ac{GW} detectability, respectively.}
\label{fig:zf_comp_kde}
\end{figure}

\begin{figure}
\includegraphics[width=0.5\textwidth]{figures/hmxb_bbhm_obs_metallicity_kde.pdf}
\caption{Distributions of progenitor metallicities for $z_{<0.05}$ \acp{HMXB} and $z_{<20}$ merging \acp{BBH}, weighted by their X-ray and \ac{GW} detectability, respectively.}
\label{fig:met_comp_kde}
\end{figure}

\subsubsection{Binary Component Masses}

Next, we examine the mass distributions of the $z_{<0.05}$ \ac{HMXB} and $z_{<20}$ merging \ac{BBH} populations.
In Figure~\ref{fig:hmxb_masses}, we show distributions of the binary component masses at the beginning of the \ac{HMXB} phase for all $z_{<0.05}$ \acp{HMXB} formed by today (HMXB$_{z0}$) and all $z_{<20}$ \acp{HMXB} that will merge as \acp{BBH} by today (BBHm$_{z0}$).
The horizontal axis shows the \ac{BH} accretor mass, $M_\mathrm{BH}/M_{\odot}$, and the vertical axis shows the donor star mass, $M_\mathrm{donor}/M_{\odot}$, at the start of the \ac{HMXB} phase.
In the top panel of Figure~\ref{fig:hmxb_masses}, we show the underlying mass distributions for the HMXB$_{z0}$ and BBHm$_{z0}$ populations.
While the distributions mostly overlap, the BBHm$_{z0}$ distribution is centered at lower \ac{BH} masses ($< 20 M_{\odot}$) because the majority ($\sim70$\%) of \acp{HMXB} with high-mass \ac{BH} accretors form \ac{BBH} systems with long delay times that do not merge by $z=0$.
Finding \acp{BH} with longer delay times at higher redshifts could be an artifact of various population synthesis prescriptions for binary evolution, such as those for \ac{BH} kicks, common-envelope evolution, mass transfer, etc.~\citep[e.g.,][]{van_son_redshift_2022}.
However, because we are only interested in obtaining a fiducial model for binary evolution in this paper, we do not investigate these effects in detail.

In the central panel of Figure~\ref{fig:hmxb_masses}, we show the same distributions weighted by their respective selection effects.
We weight the HMXB$_{z0}$ population by their X-ray detection weights $\mathcal{W}_{\mathrm{xrb}}$ and the BBHm$_{z0}$ population by their \ac{GW} detection weights $\mathcal{W}_{\mathrm{gw}}$, defined in Eq.~\eqref{eqn:xrb_weights} and Eq.~\eqref{eqn:gw_weights}, respectively.
While the distributions still overlap, the detectable BBHm$_{z0}$ distribution is pulled to higher masses ($>20 M_{\odot}$), which is the result of heavier \ac{BH} mergers having higher \ac{GW} detection probabilities.
The detectable HMXB$_{z0}$ distribution remains at masses similar to the underlying distribution, as we do not find a strong correlation of $\mathcal{W}_{\mathrm{xrb}}$ with \ac{BH} or donor mass.
Most notably, however, is that there are very few detectable \acp{HMXB} that have \ac{BH} accretor masses $>35 M_{\odot}$, while there is significant support for these massive systems in the \ac{GW} population: fewer than $3\%$ of detectable \acp{HMXB} host \acp{BH} $>35 M_{\odot}$, while $\simeq 20\%$ of detectable merging \ac{BBH} progenitors have primary \ac{BH} masses that exceed this limit.

In the bottom panel of Figure~\ref{fig:hmxb_masses}, we plot the same HMXB$_{z0}$ distribution along with the subpopulation of $z_{<0.05}$ \acp{HMXB} that will form merging \acp{BBH} within a Hubble time of \ac{ZAMS} (BBHm$_{\mathrm{H}}$).
Here, the distributions are \emph{both} weighted by their X-ray detectability.
This compares the full observable population of \acp{HMXB} to the subpopulation that will become merging \acp{BBH} within a Hubble time.
Changing the (black) distribution of the \ac{BBH} sample from $z_{<20}$ in the central panel to $z_{<0.05}$ in the bottom panel creates extra support for low-mass \ac{BBH} mergers that is independent of changing the weights from $\mathcal{W}_{\mathrm{gw}}$ to $\mathcal{W}_{\mathrm{xrb}}$.
This spread to lower masses is a result of the $z_{<0.05}$ population residing at higher metallicities.

We find that the X-ray detection-weighted BBHm$_{\mathrm{H}}$ progenitors have higher donor masses relative to the full detection-weighted \ac{HMXB} population.
This is because observable \acp{HMXB} are unlikely to form \acp{BBH} in our sample; in addition to the progenitors of these stars having lower \ac{ZAMS} masses, we find that many lose additional mass during the \ac{HMXB} phase, making it more likely that they will ultimately form a \ac{NS} or \ac{WD}.
In fact, only $\simeq 20\%$ of \acp{HMXB} that emit observably for $1~\mathrm{Myr}$ or longer (the upper tail of the emission distribution in Figure~\ref{fig:emit_duration_kde}) contain donor stars that will form \acp{BH}.
Of these \acp{HMXB} that emit for longer times \textit{and} form \acp{BBH}, over $80\%$ are too wide to merge within a Hubble time.

\begin{figure}
\includegraphics[width=0.5\textwidth]{figures/bbhm_hmxb_2d_mass_kde_all.pdf}
\caption{Distributions of \ac{HMXB} component masses at the start of the \ac{XRB} phase, weighted by different detector selection effects.
We plot the $50\%$ and $90\%$ probability levels.
The \emph{top} panel shows the \ac{HMXB} component masses for the full underlying $z_{<0.05}$ \ac{HMXB} population (HMXB$_{z0}$) and the subpopulation of $z_{<20}$ \acp{HMXB} that will form merging \acp{BBH} by $z=0$ (BBHm$_{z0}$).
The \emph{central} panel shows these populations weighted by X-ray detector ($\mathcal{W}_{\mathrm{xrb}}$) and \ac{GW} detector ($\mathcal{W}_{\mathrm{gw}}$) selection effects.
The \emph{bottom} panel shows the same HMXB$_{z0}$ population compared with the supopulation of $z_{<0.05}$ \acp{HMXB} that will form merging \acp{BBH} within a Hubble time (BBHm$_\mathrm{H}$), both weighted by X-ray detector selection effects ($\mathcal{W}_{\mathrm{xrb}}$).
Changing the \ac{BBH} sample from $z_{<20}$ in the central panel to $z_{<0.05}$ in the bottom panel creates extra support for low-mass \ac{BBH} mergers due to the $z_{<0.05}$ population residing at higher metallicities.}
\label{fig:hmxb_masses}
\end{figure}

Finally, in Figure~\ref{fig:BH_weighted_masses}, we plot the component \ac{BH} masses for \acp{HMXB} in the $z_{<0.05}$ sample that become merging \acp{BBH} in a Hubble time (BBHm$_\mathrm{H}$) and the component \ac{BH} masses for the population of \acp{BBH} in the $z_{<20}$ sample that merge by today (BBHm$_{z0}$).
We weight the BBHm$_\mathrm{H}$ population by their X-ray detectability and the BBHm$_{z0}$ population by their \ac{GW} detectability.
This allows for comparison of the component \ac{BH} mass distributions for observable \acp{HMXB} that become \acp{BBH} and observable \ac{BBH} mergers.
The population of \acp{HMXB} that become merging \acp{BBH} within a Hubble time of formation accounts for $<1\%$ of the full \ac{HMXB}$_{z0}$ population, and the number of these systems that are detectable in X-ray is even smaller ($<0.05\%$ of the \ac{HMXB}$_{z0}$ population).
Consequently, we also plot the individual points for these systems, with the point size scaling with their X-ray detection weighting.

We find that the detection-weighted \ac{BH} mass distributions overlap significantly.
This is primarily because there are a handful of \acp{HMXB} with large X-ray detection weights that will form high-mass \ac{BBH} mergers, even though the majority of detectable \acp{HMXB} in this population form lower-mass \acp{BBH} that are unlikely to be detected using \acp{GW}.
Thus, though observable \acp{HMXB} that result in \ac{BBH} mergers have lower typical masses than \ac{BBH} mergers detected via \acp{GW}, it is plausible that X-ray detectors could find a high-mass \ac{BH} in an observable \ac{HMXB} system that is predicted to become a \ac{BBH} \ac{GW} source.

\begin{figure}
\includegraphics[width=0.5\textwidth]{figures/hmxb_bbhmH_weighted_BH_masses.pdf}
\caption{Distributions of the final \ac{BH} component masses for the subpopulation of $z_{<0.05}$ \acp{HMXB} that will become merging \acp{BBH} in a Hubble time (BBHm$_\mathrm{H}$) and the full population of $z_{<20}$ \acp{BBH} that merge by today (BBHm$_{z0}$).
We plot the $50\%$ and $90\%$ probability levels.
The BBHm$_\mathrm{H}$ distribution is weighted by X-ray detector selection effects ($\mathcal{W}_{\mathrm{xrb}}$) and the BBHm$_{z0}$ distribution is weighted by \ac{GW} detector selection effects ($\mathcal{W}_{\mathrm{gw}}$).
As the number of \acp{HMXB} with $\mathcal{W}_{\mathrm{xrb}} >0$ that become BBHm$_\mathrm{H}$ in our population is small, we plot the points for individual systems, with the point size scaling with $\mathcal{W}_{\mathrm{xrb}}$.}
\label{fig:BH_weighted_masses}
\end{figure}

\subsubsection{Population Probability Results}

\begin{deluxetable}{c c}
\tablecaption{Probabilities of obtaining various sources in our sampled populations as calculated using the methods described in Section~\ref{sec:probs}.}
\label{table:probs}
\tablehead{
\colhead{Probability} &  \colhead{Approximate Value}
}
\startdata
$P(\mathrm{HMXB}_\mathrm{obs} | \mathrm{HMXB}_{z0})$ & $2.6 \times 10^{-6}$ \\
$P(\mathrm{BBHm}_\mathrm{obs} | \mathrm{BBHm}_{z0})$ & $7.9 \times 10^{-7}$ \\
$P(\mathrm{BBHm}_\mathrm{H} | \mathrm{HMXB}_\mathrm{obs})$ & $6.2 \times 10^{-3}$ \\
$P(\mathrm{HMXB}_\mathrm{obs} | \mathrm{BBHm}_\mathrm{H})$ & $9.3 \times 10^{-7}$ \\
\enddata
\end{deluxetable}

To further quantify our results, we calculate the probabilities of obtaining sources in our sampled populations using the method described in Section~\ref{sec:probs}.
These quantities are defined in the context of our sampled populations, which consist of binaries that have survived the first \ac{SN} with primary \ac{NS} or \ac{BH} progenitor stars and companion stars that are $\geq 5 M_{\odot}$ at first \ac{CO} formation.
The probability results are summarized in Table~\ref{table:probs}.

We find that the probability of detecting an \ac{HMXB} in our $z_{<0.05}$ sample is $P(\mathrm{HMXB}_\mathrm{obs} | \mathrm{HMXB}_{z0}) \simeq 2.6 \times 10^{-6}$.
This means that most \acp{HMXB} in our sample are not probable to be detected via their X-ray emission.
We calculate the analogous probability for the merging \ac{BBH} population in our $z_{<20}$ sample, and find the probability of detecting a \ac{BBH} within an observing window of $T=100$ years is $P(\mathrm{BBHm}_\mathrm{obs} | \mathrm{BBHm}_{{z0}}) \simeq 7.9 \times 10^{-7}$.
This means that most \acp{BBH} in our sample are also not probable to be detected as \ac{GW} sources.
Both observable \acp{HMXB} and observable \acp{BBH} are rare outcomes from binaries across the Universe.

The probability that a detectable \ac{HMXB} becomes a merging \ac{BBH} in a Hubble time is $P(\mathrm{BBHm}_\mathrm{H} | \mathrm{HMXB}_\mathrm{obs}) \simeq 6.2 \times 10^{-3}$.
This implies that even if we detect an \ac{HMXB}, which itself is improbable, the detected binary will most probably not merge as a \ac{BBH} within a Hubble time.
This is consistent with predictions for the fates of Cyg X-1, LMC X-1, and Cyg X-3, as discussed in Section~\ref{sec:intro}.
Conversely, we calculate the probability that a \ac{BBH} that merges within a Hubble time of formation underwent an observable \ac{HMXB} phase in the past.
We find this to be $P(\mathrm{HMXB}_{\mathrm{obs}} | \mathrm{BBHm}_{\mathrm{H}})\simeq 9.3 \times 10^{-7}$ for the $z_{<0.05}$ population.
This probability is considerably smaller than $P(\mathrm{BBHm}_{\mathrm{H}} | \mathrm{HMXB}_{\mathrm{obs}})$, which indicates that for the population of $\mathrm{BBHm}_{\mathrm{H}}$, far fewer will have undergone an observable \ac{HMXB} phase compared to the amount of observable \acp{HMXB} that will become merging \acp{BBH} within a Hubble time.
This is expected, as only binaries formed at low redshifts can become observable \acp{HMXB}.
Thus, though $>97\%$ of \ac{BBH} mergers experience an \ac{XRB} phase during their evolution, it is unsurprising that the \emph{observed} \acp{HMXB} are not the progenitors of merging \acp{BBH}.

The probability of a detectable \ac{HMXB} forming a detectable \ac{BBH} merger, $P(\mathrm{BBHm}_{\mathrm{obs}} | \mathrm{HMXB}_{\mathrm{obs}}$), is essentially zero.
This is because the delay time between the \ac{HMXB} phase and the \ac{GW} merger is much longer than the observing window $T$.
This probability is also highly sensitive to the choice of $T$, as one could theoretically choose a very long observing window and force this probability to be non-zero.
Thus, $P(\mathrm{BBHm}_{\mathrm{obs}} | \mathrm{HMXB}_{\mathrm{obs}})$, and similarly $P(\mathrm{HMXB}_{\mathrm{obs}} | \mathrm{BBHm}_{\mathrm{obs}})$, do not have strong physical meanings, as they are largely determined by the choice of observing period.

\section{Discussion} \label{sec:discussion}

We investigate the major current discrepancies in \ac{HMXB} and \ac{GW} \ac{BH} observations, namely the lack of observed \acp{HMXB} that are predicted to become \ac{BBH} mergers as well as the lack of high-mass \acp{BH} found in \acp{XRB}, and find that X-ray and \ac{GW} detector selection effects can explain them.
To arrive at this conclusion, we applied X-ray and \ac{GW} selection effects to simulated binaries from \texttt{COSMIC} and examined their impact on population parameters.
Using our new probability formalism, we quantified the probability of obtaining observable sources in our samples of binaries.
We discuss the main conclusions of our study in detail in Section~\ref{sec:conclusions}, and summarise caveats and areas for future work in Section~\ref{sec:future}.

\subsection{Main Conclusions}\label{sec:conclusions}

From our population synthesis analysis, we conclude that:
\begin{enumerate

    \item \textbf{Observable \acp{HMXB} are not likely to host \acp{BH} $\mathbf{>35 \emph{M}_{\odot}}$ relative to observable \ac{BBH} mergers.}
The mass distributions in the central panel of Figure~\ref{fig:hmxb_masses} show that \ac{GW} detection-weighted merging \ac{BBH} progenitors pull to higher donor and \ac{BH} masses compared to the full observable \ac{HMXB} population.
This is expected because the \ac{GW} detection probability increases with mass, while we find no strong correlation between mass and X-ray observability.
Fewer than $3\%$ of observable \acp{HMXB} host \ac{BH} accretors $>35 M_{\odot}$, while $\simeq 20\%$ of observable \ac{BBH} mergers have primary \ac{BH} masses that exceed this limit.
These results indicate that \ac{GW} detectors will preferentially see more binaries of higher mass than X-ray surveys.

    \item \textbf{It is highly unlikely that observable \acp{HMXB} will form merging \acp{BBH} within a Hubble time.}
We calculate that the probability a detectable \ac{HMXB} will merge as a \ac{BBH} within a Hubble time is $\simeq 0.6\%$.
This result is within one order of magnitude of predictions for the fates of Cyg X-1, Cyg X-3, and LMC X-1 (Section~\ref{sec:intro}), even though population synthesis prescriptions for binary evolution vary between all studies~\citep{belczynski_high_2012, belczynski_cyg_2013,neijssel_wind_2021}.
This rarity implies that the population of \acp{HMXB} that we do observe is unlikely to form merging \acp{BBH}.
Since the sample size of observed \ac{HMXB} sources with well-constrained binary properties is small, our results confirm that it is unsurprising none of these systems are likely to form merging \acp{BBH}.

    \item \textbf{X-ray and \ac{GW} selection effects probe different redshifts and metallicities.}
The \ac{ZAMS} formation redshift distributions for detectable \ac{HMXB} and detectable merging \ac{BBH} sources in Figure~\ref{fig:zf_comp_kde} show that observable \acp{HMXB} form locally (around $13~\mathrm{Mpc}$, if these systems were in the Hubble flow), while observable merging \ac{BBH} sources form and merge at much farther distances.
As a result, observable \acp{HMXB} also form at higher metallicities near $0.5 Z/Z_{\odot}$ while observable \ac{BBH} mergers form at lower metallicities below $0.1 Z/Z_{\odot}$.
This behavior is expected, as it mirrors what we see in observations: most \acp{HMXB} with well-constrained binary parameters are in the Milky Way or nearby Local Group galaxies that have typical metallicities of $0.1 \lesssim Z/Z_{\odot} \lesssim 1$, whereas \ac{GW} sources are detected at farther distances where there are more environments with low metallicity~\citep[e.g.,][]{evans_chandra_2010, rosen_xmm-newton_2016, krivonos_15_2021, abbott_gwtc-1_2019, abbott_gwtc-2_2021, abbott_gwtc-3_2021, abbott_gwtc-21_2022}.

\end{enumerate}
The differences between the observational samples of \acp{HMXB} and \acp{GW} sources means that these measurements are complementary, each providing different probes of the evolution of massive stars and the formation of \acp{CO}.

\subsection{Caveats \& Future Work}\label{sec:future}

In performing our calculations, we made several simplifying approximations.
First, we assume that the local universe is homogeneous, and we do not model individual galaxies or changes in metallicity within those galaxies.
Variations in metallicity within galaxies can be significant \citep[e.g.,][]{williams_two-dimensional_2021, taibi_stellar_2022}, having potential effects on binary stellar evolution.
Our results should therefore be taken as approximate estimates for the $z_{<0.05}$ \ac{HMXB} population.

We have also approximated the selection function of X-ray sources with a single flux threshold, but in reality these detected sources may not be resolvable as \ac{BH} X-ray binaries~\citep[e.g.,][]{lutovinov_population_2013, clavel_nustar_2019}.
This discrepancy can be seen through the difference in redshifts between our sampled \ac{HMXB} distribution in Figure~\ref{fig:zf_comp_kde} and those found in observations~\citep[e.g.,][]{evans_chandra_2010, arnason_distances_2021, krivonos_15_2021}.
Most observed \acp{HMXB} with well-constrained binary parameters are ${< 50~\mathrm{kpc}}$ away, which is significantly closer than the peak of our redshift distribution.
To accurately model these sources, separate simulations of the Milky Way and local galaxies are required.
However, as we do not attempt to reproduce existing observations of individual \acp{HMXB}, our results give a good sense of how selection effects play a role in detecting larger X-ray populations.

While we sample our populations with redshift--metallicity evolution from the \texttt{Illustris-TNG} simulations~\citep{nelson_illustristng_2021}, the true distribution of $P(Z|z)$ is uncertain~\citep[e.g.,][]{neijssel_effect_2019}.
For example, though sampling with a truncated log-normal distribution with a spread of $\sigma = 0.5~\mathrm{dex}$ for $P(Z|z)$ gives similar fundamental results to sampling with our method using \texttt{Illustris-TNG}, using a tighter log-normal distribution with $\sigma = 0.1~\mathrm{dex}$ significantly alters our results.
We choose to sample with \texttt{Illustris-TNG} because this redshift--metallicity evolution is more physically motivated than simple analytic alternatives.
Uncertainties in the redshift--metallicity evolution can affect \ac{GW} populations by changing the initial conditions and evolutionary properties of their progenitor populations~\citep[e.g.,][]{chruslinska_chemical_2022}.
However, features in the \ac{BBH} mass spectrum are relatively robust to variations in the joint redshift and metallicity evolution~\citep{van_son_locations_2022}.

Rapid population synthesis codes like \texttt{COSMIC} necessarily use approximate prescriptions for stellar and binary evolution.
Therefore, they are known to have systematic effects on their produced binary populations~\citep[e.g.,][]{shao_population_2019, gallegos-garcia_binary_2021, marchant_role_2021}.
Most of the shortcomings associated with these approximations can be well-addressed with state-of-the-art population synthesis codes like \texttt{POSYDON}~\citep{fragos_posydon_2022}, which use detailed \texttt{MESA} simulations~\citep{paxton_modules_2011} to model the full evolution of binary systems.
Currently, such codes only cover a limited range of metallicities, and so cannot yet be used for studies such as ours.
However, our analysis framework can be adapted to use updated population synthesis results as they become available.

Finally, we only examine isolated binary evolution scenarios in this project.
In reality, we expect \ac{GW} sources to form through a number of channels~\citep[e.g.,][]{zevin_one_2021}, and these must be considered as well in order to fully understand \ac{GW} sources.
Considering additional formation channels may further differentiate the evolution of \ac{HMXB} sources from \ac{BBH} sources, which exemplifies the need to understand the selection effects (both in terms of source formation and detectability) unique to each of these populations.

As the LIGO--Virgo--KAGRA detector network prepares for its fourth observing run~\citep{abbott_prospects_2020}, new \ac{GW} data will better resolve the mass distribution of \acp{BBH}.
In addition, as \ac{HMXB} measurements continue to improve with upcoming missions like the \textit{eROSITA} survey set to complete in 2023~\citep{basu-zych_next_2020}, more binaries will be detected with resolved component masses and more accurate \ac{BH} mass measurements will be attained for previously observed systems.
Combining \ac{GW} and X-ray observations using studies like ours can help us build a more complete concordance model of binary evolution.

\section*{Acknowledgements}

The authors thank Scott Coughlin and Katie Breivik for their assistance with \texttt{COSMIC}.
We thank Jeff Andrews and Pablo Marchant for insightful conversations.
C.L.\ acknowledges support from CIERA and Northwestern University.
Support for M.Z.\ was provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51474.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.
C.P.L.B.\ acknowledges support from the CIERA Board of Visitors Research Professorship and from STFC grant ST/V005634/1.
Z.D.\ also acknowledges support from the CIERA Board of Visitors Research Professorship.
V.K.\ was partially supported through a CIFAR Senior Fellowship, a Guggenheim Fellowship, and the Gordon and Betty Moore Foundation (grant award GBMF8477).
This work utilized the computing resources at CIERA provided by the Quest high performance computing facility at Northwestern University, which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology, and used computing resources at CIERA funded by NSF PHY-1726951.

\textit{Software:} \texttt{COSMIC}~\citep{breivik_cosmic_2020}; \texttt{Matplotlib}~\citep{hunter_matplotlib_2007}; \texttt{NumPy}~\citep{van_der_walt_numpy_2011}; \texttt{Pandas}~\citep{mckinney_data_2010}; \texttt{seaborn}~\citep{waskom_seaborn_2021}.

redpajama_set_name: RedPajamaArXiv


text:
Home » Travel » France » Basilique du Sacré-Cœur; Montmartre
Basilique du Sacré-Cœur; Montmartre
Standing on top of the highest point in Paris, the hill of Montmartre, the basilica looks out over the beautiful city and has become one of Paris's most distinctive landmarks. It is one of the best places to visit in paris.
This Roman-Catholic basilica is by many perceived to be an old basilica, much in the same way as the Notre Dame de Paris, but this is not the case. The basilica was rather young, considering it built in the late 19th-and early 20th century. Basilique du Sacré-Cœur
It, constructed to honor the 58.000 who lost their lives in the Franco-Prussian War, a war that France lost disastrously. Basilique du Sacré-Cœur
It, built to inspire faith in the French people who struggled during what was a very trying era, not only with the defeat in the war but also with the turbulent time in Paris after the defeat with the rule of the Paris Commune.
The Paris Commune was a workers party who ruled Paris for two months in 1871. It didn't take too long before conflicts and fighting began with ultra-Catholics and royalists, backed up by the French army.
The Commune had lost the bloody battles, the archbishop of Paris ordered to build the basilica as a way to "expiate the crimes of the communards". Basilique du Sacré-Cœur
It caused of course much controversy, as both sides in the fighting had committed horrible crimes. One Parisian called it "a lunatic's confectionery dream" and an offended Zola declared it "the basilica of the ridiculous.
Some interesting facts about Montmartre
"Montmartre had been the site of the Commune's first rebellion and the hill a religious site for a long time, so the place for the new basilica was prominent.
Even though the basilica has a somewhat troublesome history, it is today loved and adored by almost all Parisians.
The architect of this beautiful piece was Paul Abadie, who won the architect competition after beating the other 77 architects.
You can see by looking at the basilica, and its architectural style is very different as compared to other churches in Paris.Montmartre
Instead of being a Neo-classical or a Gothic church like Notre Dame de Paris, the Sacré-Cœur inspired by eastern Byzantine architecture. The most obvious sign of this is the fact three big domes dominate the basilica.
Under the watchful eye of development began, an open law utility passed to hold onto land at the highest point of Montmartre. The establishment stone laid In 1875.
The passing of a law to seize the ground further caused controversy to the construction project, and the matter finally reached the Chamber of Deputies where the law was rescinded.
By this stage construction of basilica had come such a long way that it was allowed to be finished. The architect Abadie never got to see the basilica finished, as he died not long after the foundation stone had been laid. Montmartre
Five other architects carried on with his plans. The basilica stood finished 1914, 39 years after the start of the construction. However, due to the interference of the First World War, the basilica formally dedicated later in 1919.
The total construction cost, estimated at 7 million French francs and raised to a large degree by private donations, both from rich and poor.
A large part of the money came from visitors and pilgrims, who visited the provisional chapel and the construction site.
The basilica itself is white. The Sacré-Cœur, made of travertine stone quarried in Château-Landon. This stone-safe stone continually radiates calcite, which guarantees that the basilica stays white even with enduring and contamination.
Even though the building style was not regularly French at that point. The Sacré-Cœurhas many structure components which represent patriot subjects.
If you stand in front of the church and look towards the entrance, you will have the French national saint Joan de Arc to your right, and King Saint-Louis IX to you left.
Between and above the two bronze statues, you will see a statue of Jesus, welcoming you into the basilica. Below Jesus, you can read the Latin phrase; "Cor Jesu Sacratissimum" which translates to "The Sacred Heart of Jesus".
In French, Sacré-Cœur means just that, "Sacred Heart''. The Sacred Heart is a symbol of the love Jesus Has for all of humanity.
While the outside of the basilica is beautiful, the same is true inside. The inside is open to the public, just make sure you have dressed somewhat appropriately.
The most prominent part of the inside is the mosaic in the apse, entitled "Christ in Majesty". The mosaic is genuinely majestic, being the largest one in the world.
It pictures The Sacred Heart of Jesus, worshipped by the Virgin Mary, Joan de Arcand St. Michael the Archangel.
The basilica can also boast about having one of the most massive free-swinging bells in the world, as its bell "Savoyarde", weighs an impressive 19 tons.
Many people do not know that it is also possible to climb the main dome of the basilica. On the same side as the King Saint Louis IX statue, the left side of the basilica, you will find the entrance to the crypt and a staircase to the top.
While the 300 steps are tough to make, it will give you a magnificent view over Paris. Many say it's the second-best view possible in Paris. On a clear day, you can see as far as 48km. I highly recommend that you make an effort and visit the top.
basilica Basilique du Sacré-Cœur best place to visit in paris Byzantine architecture montmartre paris best monument paris famous roman church paris place to visit paris tourist attractions Sacré-Cœur things to do in paris
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redpajama_set_name: RedPajamaCommonCrawl


Recorded at Studio Mécanique, La Chaux-de-fonds by Boris Gerber, mixed by Hannes Bieger, Berlin, mastered by Calyx Mastering, Berlin.
Special Thanks to Boris Gerber for helping us produce the album.
Artwork by Allan Schaller and Gregory Terlikowski, photograph by Gregory Terlikowski.


The phoenix has a number of important associations for me and I feel that this print captures it with spectacular spirit and intensity.
I was pleased to find it and am very happy to own a copy. As a note - I feel that there is a better sense of depth in the print than is apparent in the image here on the site. The colors are very vibrant with good contrast and the attention to detail is really excellent.


I usually work on my Mac Mini that runs with 8 GB RAM. Along with that I use an Apple Thunderbolt Display and Apple Wireless keyboard. I also have an old Sony Vaio E-Series laptop that tri-boots into Ubuntu 12.10, Fedora 19 and Windows 7. As crazy as it may sound, I really love to work on different operating systems throughout the day.
For my designing work, I use a Wacom Intuos tablet along with a Moleskine for quickly sketching out wireframes. I have Moleskines of every size and shape and they accompany me wherever I go. I also have a second generation iPad that I mostly use for consuming news and watching movies. I heavily rely on my iPhone 4S for messaging, Skype calls, meeting reminders and for checking the vital stats of my startup on the go. I read on my Kindle Paperwhite, and it's my best companion while I'm travelling.
I'm also an amateur astronomer and use a Celestron Astromaster 76 EQ for stargazing that is supported by an eyepiece and filter kit. Believe me, there is nothing as satisfying as watching the stars after a stressful day! Whenever I get time, I also play with my Arduino board which is usually connected to a WiFi or a motor shield.
For development, I usually work with Vim with a highly customized .vimrc file. I have also tried a range of other editors like Emacs, Sublime Text and Brackets. I use Git as a version control system and either GitHub or BitBucket for my online repositories. For performance monitoring, I use New Relic and Pingdom and collaborate with my team using HipChat and Basecamp.
For designing, I use Moqups for wireframing. I prefer designing in the browser as it gives you a better idea of site responsiveness and performance. However, I am looking forward to using Macaw, which seems to be a friendly tool for today's web designers. I use CloudApp for sharing design work with my coworkers and peers. For deciding on color schemes, I use ColorSnapper and Adobe Kuler. I prefer Type Cast for deciding on web typography and TypeKit for rendering beautiful fonts on the web.
On the iPhone, I use iMessage, Fantastical, Prismatic, Alien Blue and native apps for Pingdom and New Relic. I use Sky Guide as my stargazing assistant, Duolingo for learning new languages and ARGUS to stay fit! I also use Gmail, Dropbox, Skype, MindNode and iA Writer on a daily basis.
I would love to have a multi-monitor setup running on high-end GPUs and GBs of RAM. For portability, I would love to have a MacBook Air with more memory and a Retina screen. To be honest, I would be more than eager to get my hands on anything that provides better computational power with an equally impressive user experience.


text:
Q: curve_fit impossible if the function calls minimize Say I have a function f1, of an independent variable x and a parameter a.
The function f1, in turn, needs a parameter m that can be obtained minimizing a second function f2, and this function needs the variable x as a parameter.
I have no problem calling f1(x,a) with any value, but if I try to make a fit with curve_fit of it, it throws a ValueError: setting an array element with a sequence. I guess it does it since it assigns not a single x value, but the whole array.
There is no way to make curve_fit work ?
Toy code, to check the problem:
import matplotlib.pyplot as plt
from scipy.optimize import minimize
from scipy.optimize import curve_fit
from random import uniform

def f1(x,a):
    m =  minimize(f2,0.,args=(x)).x[0]
    return (x + a + m)**2
def f2(theta,a):
    return (theta-a)**2

xs = np.linspace(-3.,3.,10)
ys = [round(f1(x,0.5)-uniform(-1,1),2) for x in xs]
plt.scatter(xs,ys)
popt,pcov = curve_fit(f1, xs, ys, [0.3])
plt.plot(xs,[f1(x,popt[0]) for x in xs])

Full traceback:
ValueErrorTraceback (most recent call last)
<ipython-input-87-5db669cee585> in <module>()
     10 plt.scatter(xs,ys)
     11 plt.plot(xs,[f1(x,0.5) for x in xs])
---> 12 popt,pcov = curve_fit(f1, xs, ys, [0.3])
     13 plt.plot(xs,[f1(x,popt[0]) for x in xs])

/usr/lib64/python2.7/site-packages/scipy/optimize/minpack.pyc in curve_fit(f, xdata, ydata, p0, sigma, absolute_sigma, check_finite, bounds, method, jac, **kwargs)
    749         # Remove full_output from kwargs, otherwise we're passing it in twice.
    750         return_full = kwargs.pop('full_output', False)
--> 751         res = leastsq(func, p0, Dfun=jac, full_output=1, **kwargs)
    752         popt, pcov, infodict, errmsg, ier = res
    753         cost = np.sum(infodict['fvec'] ** 2)

/usr/lib64/python2.7/site-packages/scipy/optimize/minpack.pyc in leastsq(func, x0, args, Dfun, full_output, col_deriv, ftol, xtol, gtol, maxfev, epsfcn, factor, diag)
    381     if not isinstance(args, tuple):
    382         args = (args,)
--> 383     shape, dtype = _check_func('leastsq', 'func', func, x0, args, n)
    384     m = shape[0]
    385     if n > m:

/usr/lib64/python2.7/site-packages/scipy/optimize/minpack.pyc in _check_func(checker, argname, thefunc, x0, args, numinputs, output_shape)
     25 def _check_func(checker, argname, thefunc, x0, args, numinputs,
     26                 output_shape=None):
---> 27     res = atleast_1d(thefunc(*((x0[:numinputs],) + args)))
     28     if (output_shape is not None) and (shape(res) != output_shape):
     29         if (output_shape[0] != 1):

/usr/lib64/python2.7/site-packages/scipy/optimize/minpack.pyc in func_wrapped(params)
    461     if transform is None:
    462         def func_wrapped(params):
--> 463             return func(xdata, *params) - ydata
    464     elif transform.ndim == 1:
    465         def func_wrapped(params):

<ipython-input-87-5db669cee585> in f1(x, a)
      1 from scipy.optimize import minimize
      2 def f1(x,a):
----> 3     m =  minimize(f2,0.,args=(x)).x[0]
      4     return (x + a + m)**2
      5 def f2(theta,a):

/usr/lib64/python2.7/site-packages/scipy/optimize/_minimize.pyc in minimize(fun, x0, args, method, jac, hess, hessp, bounds, constraints, tol, callback, options)
    595         return _minimize_cg(fun, x0, args, jac, callback, **options)
    596     elif meth == 'bfgs':
--> 597         return _minimize_bfgs(fun, x0, args, jac, callback, **options)
    598     elif meth == 'newton-cg':
    599         return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback,

/usr/lib64/python2.7/site-packages/scipy/optimize/optimize.pyc in _minimize_bfgs(fun, x0, args, jac, callback, gtol, norm, eps, maxiter, disp, return_all, **unknown_options)
    961     else:
    962         grad_calls, myfprime = wrap_function(fprime, args)
--> 963     gfk = myfprime(x0)
    964     k = 0
    965     N = len(x0)

/usr/lib64/python2.7/site-packages/scipy/optimize/optimize.pyc in function_wrapper(*wrapper_args)
    291     def function_wrapper(*wrapper_args):
    292         ncalls[0] += 1
--> 293         return function(*(wrapper_args + args))
    294
    295     return ncalls, function_wrapper

/usr/lib64/python2.7/site-packages/scipy/optimize/optimize.pyc in approx_fprime(xk, f, epsilon, *args)
    721
    722     """
--> 723     return _approx_fprime_helper(xk, f, epsilon, args=args)
    724
    725

/usr/lib64/python2.7/site-packages/scipy/optimize/optimize.pyc in _approx_fprime_helper(xk, f, epsilon, args, f0)
    661         ei[k] = 1.0
    662         d = epsilon * ei
--> 663         grad[k] = (f(*((xk + d,) + args)) - f0) / d[k]
    664         ei[k] = 0.0
    665     return grad

ValueError: setting an array element with a sequence.

redpajama_set_name: RedPajamaStackExchange


Original forms inspired by floral motifs characterize Tuliss, which enhances the soft sofa in the middle of the living room with its sinuous lines. The fine structure in foam, lying on the floor, back and seat cushions contain soft and thick. The curved shape and the distinctive design suggest an immediate feeling and a relaxation assured. A sofa in a modern style in various colors for outlining distinctively their own luxurious lounge.


A Quick Aerial of Cypress Village Irvine, CA Jasmine Marigold Magnolia, Laurel, Mulberry Grand Canal . Just surveying (bird's eye view) the new Cypress Village build progress for a friend who bought a house there. I used 2.7k, 30fps, w/out ND filter this time and wow look at the jellos!?!


As an activist for atheism, Madalyn Murray O'Hair was such a controversial figure that in 1964 Life magazine called her "the most hated woman in America." O'Hair was one of the litigants in the case of Murray vs. Curlett, which led the U.S. Supreme Court, in a 1963 decision, to ban organized prayer in public schools. The decision made O'Hair the country's most famous atheist. She founded the group American Atheists in 1963 and remained its leading spokesperson until 1995, when she and two of her adult children vanished after leaving a note saying they would be away temporarily. The trio appeared to have taken with them at least $500,000 in American Atheist funds; one private investigator concluded that they had fled to New Zealand. Eventually suspicion turned to David Roland Waters, an ex‐convict who had worked at the American Atheist offices. Police concluded that he and accomplices had kidnapped the O'Hairs, forced them to withdraw the missing funds, and then murdered them. Waters eventually pled guilty to reduced charges and in January 2001 he led police to three bodies buried on a remote Texas ranch, which proved to be Madalyn Murray O'Hair and her children.
The children who disappeared with Madalyn Murray O'Hair were Jon Garth Murray, her son, and Robin Murray O'Hair, her granddaughter by another son, William. O'Hair also had adopted Robin Murray, making her both her daughter and her grandchild… O'Hair's son William announced his conversion to Christianity on Mother's Day in 1980 and became an outspoken evangelist for his new faith.


The Michelin Pilot Alpin PA3 N-Spec is a winter tire that offers excellent grip and control in winter driving environments. They meet the industry's severe snow service requirements. The Pilot Alpin PA3 N-Spec is designed with a tread compound that decreases hydroplaning and increases handling ability. It also has a 30,000 mile tread life warranty for all sizes.


text:
Kerry Coombs set to reunite with Buckeyes, again lead secondary
Buckeyes to host Luke Wypler, CJ Hicks for first ever college games
Ohio State could have record group of early-enrollees in 2020 class
Enokk Vimahi eases recruiting burden, receiver commits put on show
Should Buckeyes use one of final 2020 spots on offensive line?
Luke Wypler ready for Ohio State, eager to compete, learn
Luke Wypler is making an unofficial visit to Ohio State for his first ever college football game. (Luke Wypler/Twitter)
Stay in touch or be left out because Ohio State recruiting is a 24-hour, seven-day-per-week obsession. What's the latest news as the Buckeyes build for the future? Get that and more Sunday through Thursday on Lettermen Row. The midweek notebook talks to Luke Wyper and C.J. Hicks about their upcoming visits to Ohio State, a visit that will be the first-ever college football game experience for each of both.
Luke Wypler excited for return to Ohio State
He's been committed to Ohio State for more than a year, and he's a top-100 prospect nationally regarded as the country's top-overall center prospect in the Class of 2020. On the field, there's not much Luke Wypler isn't capable of doing, and off the field he's certainly seen plenty as well. But this weekend, the 6-foot-4, 290-pound St. Joseph's Regional (Montvale, N.J.) star will see something new — a college football game.
"I'm going back to Ohio State this weekend," Wypler told Lettermen Row on Tuesday night. "It's my first time to a game. Any college game. I may be the only national recruit that can say that."
Wypler is making the visit with his father, who has never been to a college football game either. Wypler pointed out that it's not as though he's made a decision to miss games in the past, but that circumstances — namely his high school football games — haven't afforded him the opportunity. He knows that makes him a bit unique, and he's perfectly fine with that.
"I love being a little different," New Jersey's top-ranked 2020 prospect said. "It's not like I waited, though, it's just that my high school plays maybe one Friday night game a year and the rest are on Saturday afternoons. I just haven't been able to visit."
Wypler will arrive in Columbus on Friday morning. He was previously on campus over the June 14 weekend when he was in town for his official visit. That trip gave him a chance to get a bit closer to a pair of offensive linemen who also pride themselves on being a bit different: freshman Harry Miller and sophomore Nicholas Petit-Frere. He's been close to Petit-Frere for a while now, and having those types of guys on the Ohio State roster are part of the reason he's never looked back since making his college decision in the summer of 2018.
"Nick is one of my closest friends on the team," Wypler said. "The day before I committed, me and him were on the phone for three hours. That's when I decided to pull the trigger. [The people are] one of the things I immediately fell in love with at Ohio State."
Ohio State to host top in-state 2022 prospect
While it's kind of crazy that Wypler hasn't ever been to a college football game before, it's sort of understandable considering his location in New Jersey. There's no real immediate draw for him or any kid growing up there to visit a place like Rutgers unless there are family ties. That's not the case in Ohio where schools like Ohio State, Cincinnati, Miami and others are all within a short drive of Dayton. But C.J. Hicks, a top 25-player in the Class of 2022, will also see his first college game this weekend at Ohio Stadium.
Lettermen Row first introduced Hicks to readers in June, and now he's getting his first chance to see the Buckeyes since catching their attention at a pair of one-day recruiting camps.
"I'm going to be watch and see how well the defense on both teams play," the 6-foot-4, 190-pound Hicks told Lettermen Row. "They each have really good playmakers on offense, so it'll be worth watching. I'm excited also just to see what it's like at a college football game. I have never been to one ever.
"This is my first visit, so I'm excited."
Hicks will travel to Columbus with a pair of Dayton-area stars. Kale Steneman, a 2020 prospect who plays quarterback at Alter, and Northmont (Clayton) 2021 standout receiver prospect Markus Allen will also be at the Horseshoe.
Buckeyes offer Illinois two-way lineman
Ohio State added another name to its list of offered offensive lineman in the 2021 class earlier this week when David Davidkov announced he'd picked up a scholarship opportunity from the Buckeyes.
Blessed to receive an offer from The Ohio State University! #GoBucks pic.twitter.com/V4tqrrjnB3
— David Davidkov (@davidkov_d) September 3, 2019
The second-ranked junior prospect in Illinois, Davidkov is a 6-foot-6, 265-pound athlete who plays both sides of the ball for New Trier in Winnetka, a suburb north of Chicago. The Ohio State offer was the 18th scholarship for Davidkov, who also has LSU, Wisconsin and Penn State on his early list.
Davidkov is a talented and versatile athlete who has also wrestled and played rugby for the Trevians. This read by Mike Clark of the Chicago Sun-Times gives a bit more background on him and is worth checking out.
Need more Ohio State recruiting #Stuff? Join the conversation below in the comments and sign up for Scarlet Sunrise, Lettermen Row's morning newsletter that will get your day started with all the important Ohio State news you need to know.
Related Topics:Featured, luke wypler, cj hicks, david davidkov
Senior Tackle: Jaxon Smith-Njigba has huge first week in tough loss
Are Ryan Day, Buckeyes adjusting running back recruiting?


text:
> Main > Affirming Gender Mainstreaming at a National Level
Affirming Gender Mainstreaming at a National Level
Objectives:
This initiative sought to affirm the concept of gender mainstreaming at a national level in all spheres of activity. A gender mainstreaming strategy is not aimed at changing men or women, but it rather accepts their different situations, priorities and needs as being of the same importance. This is the starting point in (re)designing organisations and institutions to be gender equal.
This initiative aimed at providing directors, policy makers, human resource managers and equality bodies with the impetus to update policies and practices to meet the obligations of EU and Maltese commitments towards the implementation of gender mainstreaming.
The principle of gender mainstreaming was promoted through the following training sessions:
Training seminars for directors, managers and policy makers coming from both the Private and Public Sector.
Training for various Equality Committees within every Government Ministry.
Awareness Raising:
The following awareness raising activities were held as part of this initiative:
Adverts were issued on all local Sunday newspapers to disseminate information about the training seminars, the aims and components of this initiative and the benefits of introducing gender mainstreaming in the business environment.
An information pack on the implementation of gender mainstreaming and on NCPE's role in this regard was produced specifically for directors, policy makers and H.R. managers, and distributed among all business organisations in Malta and Gozo.
A brochure targeting the general public was produced, printed and distributed to all households in Malta and Gozo, disseminating information on gender equality; gender equality in employment; discriminatory advertising; sexual harassment and where to seek advice and assistance.
Information Pack for Business Organisations / Ġabra ta' Informazzjoni għall-Kumpaniji: Part 1 / Part 2
Information Pack for Households / Ġabra ta' Informazzjoni għad-Djar: Part 1 / Part 2
Meetings with Equality Committees in various ministries were held to further promote the principle of gender mainstreaming, explaining the role of equality committees and suggesting ideas and concepts that help to develop or enhance the equality policy and equality principle.
A radio advertisement campaign, consisting of four different radio public service announcements (PSAs) in Maltese and English, was carried out on the main national radio stations. These PSAs raised awareness with the general public on what constitutes gender discrimination by using various examples. Simultaneously, a television advertisement campaign, consisting of two PSAs, was also carried out on the main national television stations. These addressed sexual harassment and gender discrimination at the workplace.
Project Ref. ESF 48
Project Name Affirming Gender Mainstreaming at a National Level
Fund EU Structural Funds Programme for Malta 2004-2006
Start Date May 2007
End Date June 2009


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