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

The detection of gravitational waves (GW)
using the LIGO--Virgo--KAGRA (LVK) detectors \cite{LIGODet,VirgoDet,KAGRADet}
from mergers of compact objects~\cite{gw150914,gw170817,gwtc1,LIGOScientific:2020ibl, LIGOScientific:2021djp,Nitz:2019hdf, Nitz:2021zwj, Venumadhav:2019lyq} has enabled precision tests of general relativity (\text{GR}) in the strong field regime~\cite{LSC_2016grtests,GWTC1TestofGR,LIGOScientific:2020tif, LIGOScientific:2021sio}.
Far away from the source, $\text{GR}$ predicts that GWs are well described as linear perturbations of  the background Friedmann-Robertson-Walker (FRW) metric ~\cite{Will:1997bb}
Existing propagation tests hence typically consider modifications over the FRW background and its effect on the GW signals as measured at the detectors \cite{Ezquiaga:2018btd, PhysRevD.97.104038, Nishizawa_2018}.

GR also dictates that GWs have only two tensor  polarisations ($+,\times$) which propagate independently from each other at the speed of light.
However, in alternative theories of gravity, extra degrees of freedom (tensor, vector, scalar)~\cite{Will:2014kxa} can mix with GWs as they propagate, producing phenomena similar to neutrino oscillations (i.e. due to interactions between different neutrino flavors \cite{Workman:2022ynf}). In Lorentz invariant theories, the symmetries of the FRW restricts mixing effects to tensor degrees of freedom, either fundamental (e.g. in bigravity) or composite (e.g. multiple vector fields)~\cite{Max:2017flc,Max:2017kdc,Belgacem:2019pkk,Jimenez:2019lrk,Ezquiaga:2021ler}. However, inhomogeneities spontaneously break Lorentz symmetry, allowing interaction between GWs and scalar or vector degrees of freedom~\cite{Ezquiaga:2020dao,Dalang:2020eaj}.
This leads to new, testable predictions, and opens new opportunities to probe the gravitational sector beyond the FRW limit.

The evolution of GWs on an inhomogeneous background is described via propagation eigenstates: linear combinations of the interaction eigenstates ($h_+,h_\times$ and perturbations of additional polarisations) with a well-defined dispersion relation (analogous to massive neutrinos).
As the relation between interaction and propagation eigenstates and their speed depends on position and direction, an inhomogeneous region of space splits the original signal into several components, each arriving with a relative time delay~\cite{Ezquiaga:2020dao}. Moreover, if deviations from $\text{GR}$ are small, two eigenstates correspond to \textit{mostly-tensorial polarizations (linear combinations of $h_+,h_\times$ plus a negligible correction distinguishing both), with a very small speed difference.%
\footnote{We will ignore the remaining eigenstates (perturbations of beyond-$\text{GR}$ fields plus negligible corrections) because 1) their emission needs to be suppressed to avoid dipolar radiation and 2) their speed can be substantially different, making an association with the the mostly-tensorial part of the signal difficult \cite{Ezquiaga:2020dao}. \label{foot:ignore_extra_pol}}
}

We will refer to the difference in propagation speed between the $+,\times$ polarizations as \textit{lens-induced birefringence} ($\text{LIB}$).
$\text{LIB}$ is analogous to the way a non-isotropic crystal, such as calcite, splits light in two beams. This splitting is caused by a difference in the refractive index of the linear electromagnetic polarizations, which depends on the alignment of the polarization vector with the crystal structure. In our case birefringence is caused not by anisotropies in a crystal, but by the background configuration of additional, non-$\text{GR}$ fields which spontaneously break Lorentz symmetry. Moreover, $\text{LIB}$ splitting is independent of the frequency (in the high frequency approximation assumed), which would correspond to a perfectly isochromatic birefringent crystal. Because GW detectors have excellent time resolution and bad sky localization, our main observable will be the time delay between splitted signals and not their angular separation.

If the arrival time difference between the mostly-tensorial polarisations is larger than the duration of the binary merger signal then we would see only one polarisation at a time, mimicking a binary with either a zero inclination angle (face-on binary) or  with $90\degree$ inclination angle (face-off binary), appearing as GW echoes.
Since the detectors are more sensitive to face-on binaries, one can expect excess of near zero inclinations in case of birefringence for the population of binaries.
If the delay between the polarisations is larger than typical observing runs or the amplitude of one of the polarisations decays faster than the other, e.g. in Chern-Simons gravity \cite{Okounkova:2021xjv}, one would also expect an anisotropic inclination distribution.
Current observations though show that the orientation distribution is consistent with
being isotropic~\cite{Vitale:2022pmu}.

However, when the time delay is less than the duration the signal, the GW waveform would be distorted or ``scrambled'' due to the interference of both polarisations.
Note that this effect is frequency-independent, and hence distinguishable from a different dispersion relation for the $+$ and $\times$ modes or the circularly polarized combinations (L-R), as predicted in \text{GR}~\cite{deRham:2019ctd,Andersson:2020gsj,Oancea:2022szu} and alternative theories~\cite{Wang:2021gqm,Haegel:2022ymk,Bombacigno:2022naf}. As it is not suppressed by the frequency, $\text{LIB}$ is the dominant effect in the high-frequency limit for theories in which this effect is present.

Our study analyzes for the first time
arrival time difference ($\Delta t_{12}$) between the two polarisation states due to different propagation speeds (frequency-independent
dispersion relations) as a result of $\text{LIB}$. This is a new, model-independent test of a basic prediction of $\text{GR}$. We use these generic results to constrain GW lensing effects beyond \text{GR}, for example in scalar-tensor theories with derivative interactions ~\cite{Ezquiaga:2020dao}.

$\text{LIB}$ signatures are not linked to a specific regime of gravitational lensing in GR, such as strongly magnified or multiple images. The scale on which $\text{LIB}$ can be observed is very sensitive to the theory parameters and independent of the Einstein radius $R_E$, which characterizes the regimes of gravitational lensing. Hence, for sufficiently strong deviations from $\text{GR}$, $\text{LIB}$ can be detected for impact parameters much larger than $R_E$, typically associated to weak lensing. Therefore, $\text{LIB}$-tests can be applied to all the GW detections.
In addition, $\text{LIB}$ can be important for lenses very close to the source or the observer, for which $R_E$ vanishes. This is particularly interesting for sources merging near massive objects (e.g. a supermassive BH) since the background configuration of the additional fields enhances $\text{LIB}$.

The rest of the paper is organised as follows. In Sec.~\ref{sec:method} we describe our $\text{LIB}$ waveform model, methods for data analysis and introduce parameterized $\text{LIB}$ observation probabilities. In Sec.~\ref{sec:results}, we perform the birefringence test over a set of simulated GW events, and then to real events using the Bayesian model selection framework. In Sec.~\ref{sec: implications}, we study the implications of the results in constraining LIB probabilities and beyond-GR theories. Finally, in Sec.~\ref{sec:conclusion} we summarize the main results and discuss future prospects.

\begin{figure*}[tbh!]
    \centering
    \includegraphics[width=0.96\textwidth]{figs/GW_signal_delay_examples_ampl_strain.pdf}
    \caption{GW polarisations (left) and detector strain (right) for a CBC $(30+30)M_\odot$ with birefringent time delays $\Delta t_{12}=5, 10, 100$ ms (top to bottom). The sky localization and detector orientation correspond to $F^+= -0.38$, $F^\times= 0.71$ and $\text{LIB}$ strain is given by Eq. ~\ref{eq:hmodGRsimp} }
    \label{fig:mg_waveform}
\end{figure*}

\section{Method}\label{sec:method}
In \text{GR}, GWs have only two  polarisations $(+, \times)$ which propagate independently at the speed of light over the background FRW metric.
A given ground-based detector $I$ measures the GW signal, $h(t)$ as a linear combination of these polarisations~\cite{Veitch:2014wba},
\begin{equation}
    h_I(t) = F^{+}_I h_{+}(t) + F^{\times}_I h_{\times}(t)
    \label{eq:ht}
\end{equation}
where, $F^{+}_I, F^{\times}_I$ are the detector antenna pattern functions.
In the case of compact binary coalescence (CBC), the relative amplitude of the polarisation modes depends on the inclination and polarisation angles of the binary  w.r.t to the line of sight, and also depends on the sensitivity of the detector for the source location at the time of arrival. The overall amplitude of the signal is inversely proportional to the luminosity distance of the source. Masses and spins of the source dictate the frequency evolution of the signal and its amplitude.

\subsection{Parameterized Lens-induced Birefringence Waveforms}\label{sec:phenom_model}

When there is any inhomogeneity along the travel path of a GW, e.g. an intervening galaxy, the GW can be gravitationally lensed.
Gravitational lensing of a GW can produce multiple images of the original signal (strong lensing) or cause distortions (microlensing), but, in \text{GR}, both polarisations are affected in the same way, i.e. the polarisation rotation is negligible for any sensible astrophysical lens \cite{Ezquiaga:2020gdt,Cusin:2019rmt}.
However, in alternative theories of gravity the additional fields can couple with the tensor polarizations around  the lens and modify the GW propagation eigenstates. These eigenstates are a linear combination of original GW polarisations that evolve independently, each with a different speed, thus reaching the detectors at different times.
We will assume spherically symmetric lenses, focus on the limit of small deviations from $\text{GR}$, so the mostly-metric propagation eigenstates correspond to linear combinations of $h_+,h_\times$ (depending on the projected angle between the lens and the source), and neglect the additonal eigenstates (See Ref. \cite{Ezquiaga:2020dao} and footnote \ref{foot:ignore_extra_pol}).

This class of $\text{LIB}$ of GWs
can be captured in a phenomenological manner as proposed in Ref.~\cite{Ezquiaga:2020dao}.
After diagonalizing the propagation equations, the propagation eigenstates can be computed and one gets the transformation matrix
${\mathcal S}$ relating the polarisation amplitudes in $\text{GR}$ and after the $\text{LIB}$:
\begin{equation}
[h_{+},h_{\times}]^T_{\text{LIB}} = \mathcal{S} [h_{+},h_{\times}]^T_\mathrm{GR}
\label{eq:h}
\end{equation}
where
\begin{equation}
    {\mathcal S}= \hat{\mathcal M} \ \text{diag}(1,\Delta)\hat{\mathcal M}^{-1}\,,
    \label{eq:S}
\end{equation}
\begin{equation}
\hat{\mathcal M} = \left[
\begin{array}{cc}
-\sin(2\phi_\mathrm{lens}) & \cos(2\phi_\mathrm{lens}) \\
\cos(2\phi_\mathrm{lens}) & \sin(2\phi_\mathrm{lens})
\end{array}\right]\,,
\label{eq:mix}
\end{equation}
and $\Delta=e^{-i\omega \Delta t_{12}}$
with $\Delta t_{12}$ is the time delay between the polarisations and $\phi_\mathrm{lens}$ as the angle between the lens and the source, relative to the direction of GWs propagation that dictates the polarisation mixing. \\

It is easy to note that for $\phi_\mathrm{lens} = \pi/2 $,  $\mathcal{S} = \mathrm{diag}(1,\Delta)$, hence the signal observed by the detectors will just be superposition of $(+,\times)$ arriving at different times.
\begin{equation}
h_I^{\text{LIB}}(t) = F^{+}_I h_{+}(t) + F^{\times}_I h_{\times}(t-\Delta t_{12})
\label{eq:hmodGRsimp}
\end{equation}
whereas, if $\Delta t_{12} = 0$ the $\text{LIB}$  waveform morphology will be identical to the $\text{GR}$ one, independent of $\phi_\mathrm{lens}$.
Fig.~\ref{fig:mg_waveform} compares $\text{GR}$ v/s $\text{LIB}$ waveform polarisations and the detector strains for various values $\Delta t_{12}$.
Under $\text{LIB}$, the polarisations interfere leading to waveform distortions.

Since lensing is an environmental effect which can occur through any local inhomogeneity in the path of GWs, the parameters $\Delta t_{12}$ and $\phi_{\rm lens}$ are expected to vary between GW events. The time delay distribution depends on the theory and the (usually unknown) lens properties and the configuration relative to the source. In general, one can only predict the probability of the birefringence parameters given a gravitational theory and matter distribution (unless further information or assumptions are employed about the source's location or the signal's trajectory), see Sec.~\ref{sec:cross_section_pheno}. This is in stark contrast to other tests of GW propagation (that are done with individual GW events) in which deviations represent a fundamental property of gravity (eg. massive graviton dispersion relations) and are thus the same across all events and only depend on their distance \cite{Will:2014kxa}.

\subsection{Template Mismatch Studies}\label{sec:mismatch}

In order to quantify distortions due to GW birefringence, we calculate the mismatch between the $\text{GR}$ and $\text{LIB}$ waveforms as seen the LIGO-Virgo detectors. At each detector ($I$), the mismatch between the injected waveform ($h_I^\mathrm{inj}$) and the recovery waveform ($h_I^\mathrm{rec}$)  is given by:
    \begin{equation}
    M_I = 1- \frac{( h_I^\mathrm{inj} | h_I^\mathrm{rec} )}{||h_I^\mathrm{inj}||.||h_I^\mathrm{rec}||}
    \end{equation}
where, $(\cdot\mid\cdot)$ symbolises the noise-weighted inner product:
\begin{equation}
(a~|~b)  \equiv 2\int_{f_{\mathrm{min}}}^{f_{\mathrm{max}}}\frac{\tilde{a}(f)\tilde{b}^{*}(f)}{S_n(f)}df\,.
\end{equation}
Here, $\tilde{a}, \tilde{b}$ represent the Fourier transform of the time series $a(t), b(t)$; $[f_{\mathrm{min}}, f_{\mathrm{max}}]$ is the frequency range over which the inner product is evaluated;  $^*$ represents complex conjugation and $S_n(f)$ is the colored Gaussian noise power spectral density (PSD) at the detector. The norm $||h|| = \sqrt{(h|h)}$ is the optimal SNR of a waveform.
We define the total mismatch ($ \mathcal{M} $) for network of detectors as the signal-to-noise ratio (SNR, $\rho_I$) squared weighted average of individual detector matches,

    \begin{equation}
    \mathcal{M} = \frac{\sum_{I} \rho_{I}^2 M_I}{\sum_I \rho_{I}^2}
    \end{equation}

Note that the mismatch is a normalized quantity and is maximised over time and phase shifts. Thus, the mismatch quantifies differences in morphology between the signals.
Whereas, during the parameter estimation (PE) from GW signals both the mismatch and SNR play a role as the log-likelihood  $\propto \mathcal{M}{\sum_I \rho_{I}^2}$ around the maximum likelihood parameters. We first wish to quantify the overall detectability of the birefringence. Later, we will estimate parameters using Bayesian inference, accounting for correlations between all parameters.
\begin{figure}[tbh]
    \centering
    \includegraphics[width = 0.98 \linewidth]{figs/GW150914-like-waveform-fdomain.pdf}
    \caption{$\text{GR}$ and $\text{LIB}$ detector frame waveform  amplitudes in frequency domain of GW150914-like CBC. The birefringence leads to additional frequency modulations and distorts the $\text{GR}$ waveform. The magnitude of these distortions are however dependent on the two parameters: $\Delta t_{12} $ and $\phi_\mathrm{lens}$. }
    \label{fig:waveform}
\end{figure}

\begin{figure*}[tbh]
    \includegraphics[width = 0.49 \linewidth]{figs/re-GW150914-like-gr.pdf}
    \includegraphics[width = 0.49 \linewidth]{figs/re-GW150914-like-nonngr.pdf}
    \includegraphics[width = 0.49 \linewidth]{figs/re-GW190814-like-gr-hom.pdf}
    \includegraphics[width = 0.49 \linewidth]{figs/re-GW190814-like-nongr.pdf}\\
    \includegraphics[width = 0.49 \linewidth]{figs/just_colorbar.pdf}
    \caption{Mismatch between $\text{GR}$ and $\text{LIB}$ waveforms for GW150914-like CBC (top) and GW190814-like CBC (bottom). left pannel: $\text{GR}$ injection i.e. $\Delta t_{12}^\mathrm{inj}=0$ and $\phi_\mathrm{lens}^\mathrm{inj}=0$.  the mismatch is minimum for  $\Delta t_{12}^\mathrm{rec} \simeq 0 $. right pannel: a $\text{LIB}$ injection with $\Delta t_{12}^\mathrm{inj}=10 $ ms  and $\phi_\mathrm{lens}^\mathrm{inj}=\pi/5$.The mismatch is minimum at $\Delta t_{12}^\mathrm{rec} \simeq \pm 10 \mathrm{ms}$ and $\phi_\mathrm{lens}^\mathrm{rec} \simeq \pi/5, \pi/4 +\pi/5$.}
    \label{fig:mismatch}
    \end{figure*}

Fig.~\ref{fig:waveform} shows frequency domain $\text{LIB}$ and $\text{GR}$ waveforms for a GW150914-like CBC. The waveforms are generated using the approximant $\textsc{IMRPhenomXPHM}$ \cite{IMRPhenomXPHM}, as implemented in the $\textsc{LALSimulation}$ module of the $\textsc{LALSuite}$ software package~\cite{lalsuite}.
The waveforms are then projected onto the LIGO and Virgo detectors using their antenna pattern functions, as implemented in the $\textsc{bilby}$ \cite{bilby} software package. The $\text{LIB}$  waveforms have  additional frequency modulations which depend on the two parameters: $\Delta t_{12} $ and $\phi_\mathrm{lens}$ (see~\ref{sec:phenom_model}). Therefore we calculate the mismatch between the $\text{GR}$ and $\text{LIB}$ waveforms, keeping all the other parameters identical and fixed for the two waveforms.
In practice, we calculate ${M_I}$ using  $\textsc{pycbc.filter}$ \cite{pycbc} module.
The detector noise is generated using the zero-detuned high-power PSDs of Advanced LIGO and Advanced Virgo at their design sensitivities \cite{AVirgoSensitivity,aLIGOSensitivity}.

We consider two systems of binaries, one whose parameters resemble to that of the first CBC detection GW150914 and other of a higher mass-ratio CBC GW190814 where the presence of higher-order modes (HoMs) of GWs are significant. For both the systems, we inject a $\text{GR}$ and a $\text{LIB}$ waveform and recover with $\text{LIB}$ waveform to calculate the mismatch. The parameters for both the CBCs are mentioned in Appendix~\ref{sec:app_inj}

Fig.~\ref{fig:mismatch} shows mismatches for a GW150914-like CBC (top) and GW190814-like CBC  (bottom).
As expected for a $\text{GR}$ injection i.e. $\Delta t_{12}^\mathrm{inj}=0$ and $\phi_\mathrm{lens}^\mathrm{inj}=0$, the mismatch is minimum for  $\Delta t_{12}^\mathrm{rec} \simeq 0 $, for all $\phi_\mathrm{lens}$ as expected from Eq. ~\ref{eq:S}.
 Additionally, the local minimum of mismatch is at $\phi_\mathrm{lens}^\mathrm{rec} \simeq \pi/3 $, which could be because of vanishing polarisation ($+$ or $\times$) as seen at the detectors which further makes the mismatch independent of the time delay $\Delta t_{12}$.
The waveform plots in Fig.~\ref{fig:waveform} confirms this as the $\phi_\mathrm{lens}^\mathrm{rec} \simeq \pi/3 $ waveform resembles the $\text{GR}$ ones more as compared to the $\phi_\mathrm{lens}^\mathrm{rec} \simeq \pi/5 $, especially in the Livingston (L1) detector for  GW150914-like CBC.

We also checked the mismatch for the $\text{LIB}$ injections (right panel Fig.~\ref{fig:mismatch}) with $\Delta t_{12}^\mathrm{inj}=10$ ms and $\phi_\mathrm{lens}^\mathrm{inj}=\pi/5$ and the mismatch is minimum at,  $\Delta t_{12}^\mathrm{rec}\simeq \pm 10$ ms and $\phi_\mathrm{lens}^\mathrm{rec} \simeq \pi/5, \pi/4 +\pi/5$.
We can infer the degeneracy between $\Delta t_{12}$ and the coalescence time ($t_c$) as follows,
from Eq.~\eqref{eq:ht}-\eqref{eq:mix} if $\phi_\mathrm{lens} \rightarrow \phi_\mathrm{lens} + \pi/4$ and $\Delta \rightarrow 1/\Delta$ then, one finds $S \rightarrow  S/\Delta$, which implies that the modification at $(\Delta t_{12}, \phi_\mathrm{lens}, t_c)$ is the same as at $(-\Delta t_{12}, \phi_\mathrm{lens} +\pi/4,t_c + \Delta t_{12})$. Note that, this degeneracy stems from the fact that we do not know the composition of $h_{+,\times}$ before it encounters the lens. Additional information about the source could break this degeneracy, but we leave these investigations for the future.

\subsection{Bayesian Inference}\label{sec:Bayes}

 Bayesian model selection allows us to assign evidences for various hypotheses pertaining to the observed data, and also derive posterior probability distributions of the model parameters conditioned on individual hypotheses. Given the set of data $\{d\}$ from a network of detectors, the marginalized likelihood (or, Bayesian evidence) of the hypothesis $\mathcal{H_A}$ can be computed by
\begin{equation}
P(\{d\}|\mathcal{H_A}) = \int d{\bm \theta} P({\bm \theta}|\mathcal{H_A}) \, P(\{d\}|{\bm \theta}, \mathcal{H_A}),
\label{eq:marg_likelihood}
\end{equation}
where ${\bm \theta}$ is a set of parameters that describe the signal under hypothesis $\mathcal{H_A}$ (including the masses and spins of the compact objects in the binary, location and orientation of the binary and the arrival time and phase of the signal), $P({\bm \theta}|\mathcal{H_A})$ is the prior distribution of ${\bm \theta}$ under hypothesis $\mathcal{H_A}$, and $P(\{d\}|{\bm \theta}, \mathcal{H_A})$ is the likelihood of the data $\{d\}$, given the parameter vector ${\bm \theta}$ and hypothesis $\mathcal{H_A}$. Given the hypothesis $\mathcal{H_A}$ and data $\{d\}$, we can sample and marginalize the likelihood over the parameter space using an appropriate stochastic sampling technique such as nested sampling~\cite{skilling2006}.

Bayesian model selection allows us to compare multiple hypotheses. For e.g., the odds ratio $\mathcal{O}^{\text{LIB}}_\mathrm{GR}$ is the ratio of the posterior probabilities of the two hypotheses $\text{LIB}$ and $\text{GR}$. When $O^{\text{LIB}}_\mathrm{GR}$ is greater than one then hypothesis $\text{LIB}$ is preferred over $\text{GR}$ and vice versa. Using Bayes theorem, the odds ratio can also be written as the product of the ratio of the prior odds $P^{\text{LIB}}_\mathrm{GR}$ of the hypotheses and the likelihood ratio, or Bayes factor $\ensuremath{\mathcal{B}^{\MG}_{\GR}}$:

\begin{align}
\mathcal{O}^{\text{LIB}}_\mathrm{GR} : &= \frac{P(\mathcal{H}^{\text{LIB}}|\{d\})}{P(\mathcal{H}^{GR}|\{d\})} = \frac{P(\mathcal{H}^{\text{LIB}})}{P(\mathcal{H}^{GR})} \times \frac{P(\{d\}|\mathcal{H}^{\text{LIB}})}{P(\{d\}|\mathcal{H}^{GR})} \\
& =  \mathcal{P}^{\text{LIB}}_\mathrm{GR} \times \ensuremath{\mathcal{B}^{\MG}_{\GR}}
\end{align}

Since $\text{GR}$ has been tested well in a variety of settings, our prior odds are going to be highly biased towards it, i.e., $\mathcal{P}^{\text{LIB}}_\mathrm{GR} \ll 1$. Hence, in order to claim evidence of birefringence the corresponding Bayes factor supporting the $\text{LIB}$ hypothesis has to be very large. Since the Bayes factor is the only quantity that is derived from data, for the rest of the paper, we focus on the Bayes factor, i.e. the ratio of evidences under the two hypothesis.

The waveforms under the $\text{GR}$ and $\text{LIB}$ hypotheses at each detector is the same way as described in Sec.~\ref{sec:mismatch}.
We use the standard Gaussian likelihood model for estimating the posteriors of the parameters under different hypotheses~(see, e.g.,~\cite{Veitch:2014wba}).
We use uniform priors in redshifted component masses of the binary, isotropic sky location (uniform in $\alpha, \sin \delta$) and orientation (uniform in $\cos \iota, \phi_0$), uniform in polarisation angle $\psi$, and a prior $\propto d_L^2$ on luminosity distance. Additionally for the $\text{LIB}$ hypothesis,  we choose the priors on $\Delta t_{12}$ as uniform $\in [-100  , 100 ]$ ms   and $\phi_\mathrm{lens}$ as uniform $\in [0,\pi/2] $. To estimate the posterior distribution and evidences for the $\text{GR}$ and $\text{LIB}$ hypotheses, we use the open-source parameter estimation package \textsc{bilby} package \cite{bilby} coupled with the dynamical nested sampler \textsc{dynesty}~\cite{Dynesty:2020}.

\subsection{Lensing Probabilities}\label{sec:cross_section_pheno}

The (non-)observation of birefringence can help us put constraints on theories beyond $\text{GR}$ that predict $\text{LIB}$. According to \text{GR}, the strong lensing of GWs caused by galaxies occurs when sources lie inside the Einstein radius of the lens,
which depends on lens mass and profile. This is the relevant scale determining the probability of lensing.
However, birefringence beyond $\text{GR}$ is in principle independent of the ratio between the impact parameter and the Einstein radius, changing the probability of observing $\text{LIB}$ compared to strong lensing.
It is thus possible to have $\text{LIB}$  time delays without multiple images, but birefringence could also occur for strongly lensed GWs, in which case it applies to each image separately, as typical time delays between images will be larger than $\Delta t_{12}$ \cite{Ezquiaga:2020dao}.

Assuming that the lenses are randomly distributed, birefringence detection is described by Poisson statistics.
A series of observations with $L$ lensed and $U$ unlensed GW events has an associated probability,
\begin{equation}
    P=\exp\Big({-\sum_i^U\lambda_{i}}\Big)\prod_j^L\left(1-e^{-\lambda_{j}}\right)\,.
\end{equation}
 The result depends on  $\text{LIB}$ rate for the $i^{th}$ event: $\lambda_i = \int dz_s d\vec p_L d\vec p_S \tau(z_s, \vec p_L) \mathcal{P}_i(z_s,\vec p_S)P(\vec p_S,\vec p_L)$.
Here $S,L$ denote parameters corresponding to the source and lens/theory (i.e. beyond GR), $\mathcal{P}_i$ is the posterior distribution of the source parameters and $P$ is the prior, which includes relations between parameters (i.e. the measured $\Delta t_{12}$ as a function of lens mass and beyond-$\text{GR}$ parameters).%
\footnote{A more complete treatment should account for the selection function \cite{Vitale:2020aaz}. In terms of gravitational lensing, we expect that $\Delta t_{12}$ correlates with magnification especially at sizeable impact parameters (e.g. single image regime, first magnified image if multiple images are formed), which dominate the lensing cross-section
Then events with larger $\Delta t_{12}$ are more likely to be observed and neglecting this correlation is conservative.}
The \textit{birefringence optical depth}, $\tau(z_s,\vec p_L)$ is the fraction of the sky for which $\text{LIB}$ is detectable for sources at a redshift $z_s$.
Hereafter we will assume the posterior to be sharply peaked at the mean source redshift $z_s$ and include the integration on the lens model parameters ($\vec p_L$) in the definition of the optical depth, so $\lambda_i \approx \tau(z_{s,i})$. If birefringence is excluded in all events, the probability only depends on the total optical depth $\tau_{tot} \approx \sum_{i=1..N} \tau_i \approx \sum_{i=1..N} \lambda_i$.
In Appendix~\ref{sec:beyond_poisson_stats} we will comment on opportunities to study GW birefringence beyond the Poisson statistics.

The lensing optical depth $\tau\left(z_{s}\right)$ depends on the angular cross section $\hat{\sigma} \left(z_{s}, \vec p_L \right)$ and the density of lenses $\hat n\left(\vec p_L \right)$  \cite{Xu_2022}.
In the following we will explicitly write the lens redshift $z_L$ and let $\vec p_{L^{\prime}}$ denote the remaining properties (i.e. lens mass \& theory parameters).
The total density of lenses at redshift $z_{L}$ is then $\int \hat n\left(z_{L}, \vec p_{L^{\prime}}\right) d \vec p_{L^{\prime}}$. The optical depth is computed directly by adding-up the cross-sections weighted by the density at different redshifts, i.e.
\begin{equation}
\tau\left(z_{s}\right)=\int_0^{z_{s}}d z_L \int d\vec p_{L^{\prime}}\frac{d V_c}{\delta \Omega d z_{L}} \hat n\left(z_{L},  \vec p_{L^{\prime}}\right) \hat{\sigma}\left(z_{s}, z_{L}, \vec p_{L^{\prime}}\right)
\label{eq:tau_z}
\end{equation}

where $d V_c =\delta \Omega D_{L}^2 \frac{dz} {(1+z) H(z)}$ is the physical volume given the solid angle $\delta \Omega$, angular diameter distance to the lens $D_{L}$  and  the Hubble parameter $H(z)$. For simplicity, we will assume point mass lenses of mass $M$ throughout. In GR, the lensing cross-section is  $\sigma = \pi \theta_E^2$,  where $\theta_E = R_E/D_L=(\frac{4 G M  D_{LS}}{ c^2 D_L D_S})^{1/2}$ is the Einstein angle, $D_S$ is the distance to the source from the earth and $D_{LS}$ is the distance between lens and source.

The relation between the $\text{LIB}$-time delays, the theory parameters and the configuration of the lensed system is complex (See Sec.~VI of Ref.~\cite{Ezquiaga:2020dao} for a worked-out example in a viable Horndeski theory). For this reason we will first consider two phenomenological models with generic
dependences of lensing cross section.
As a first example, we will assume that the relevant $\text{LIB}$ scale is proportional to the Einstein
angle, $\theta^E_X=\alpha_X\theta_E$, so that the cross section becomes
\begin{equation}\label{eq:cross_section_rE}
\sigma^E_{X} = \pi \alpha_X^2 \theta_E^2\,.
\end{equation}

Then the optical depth is given by Eq. (74) in Ref. \cite{Ezquiaga:2020dao} where the lenses have been assumed point-like.

Under these assumptions, lensing probabilities are independent of the mass function.%
\footnote{The mass independence also appears for the strong-lensing cross section for a distribution of point lenses \cite{Pei93,Zumalacarregui:2017qqd}. This differs from strong-lensing cross-section for extended lenses, where the lens mass affects the formation of multiple images \cite{Xu_2022}.}

In our second example we assume that the relevant $\text{LIB}$ scale is given by a constant physical scale associated to each halo. Moreover, we assume that this scale depends on the halo mass as a power law. Accordingly, the cross-section reads,
\begin{equation}\label{eq:cross_section_pheno}
    \sigma_{\rm ph}=\pi \frac{R_{12}^2}{D_L^2} \left(\frac{M}{10^{12}M_\odot}\right)^{2n}\,.
\end{equation}
The scale $R_{12}$ fixes the probability of lensing for halos with $M=10^{12}M_\odot$, while $n$ allows us to extrapolate to different halo masses. Below we will discuss some cases of interest.

We now generalize the expression for the optical depth presented in Ref. \cite{Ezquiaga:2020dao} (Eq. 76) to include a realistic halo mass function.
The optical depth from Eq.~\eqref{eq:cross_section_pheno} is given by
\begin{equation}\label{eq:tau_phys_halo}
    \tau^{\rm ph}(z_s,n) = \Omega_M h \left(\frac{R_{12}}{22\rm kpc}\right)^2 \hat \tau(z_s,n)\,,
\end{equation}
where

\begin{equation}\label{eq:cross_section_integrals}
\begin{multlined}
\hat \tau(z_s,n) = \int_0^{z_s}dz\frac{(1+z)^2}{H(z)/H_0} \int d\log(M) \\ \times \left(\frac{M}{10^{12}M_\odot}\right)^{2n-1}
{f(M,z)}.
\end{multlined}
\end{equation}

Here $f(M,z)=\frac{M^2}{\rho_0}\frac{d \hat n}{dM}$ is the scaled differential mass function (dimensionless) with $\rho_0$ as the matter density of universe at $z=0$. We will use the Tinker \textit{et al.} form~\cite{Tinker:2008ff} as implemented in the \textsc{Colossus} package~\cite{Diemer:2017bwl}.
As our approach is phenomenological, we assume a Planck $\Lambda$CDM cosmology~\cite{Planck:2018vyg}.
The true optical depth of a consistent $\text{LIB}$ model will typically depend more strongly on the theory parameters (e.g. entering Eq.~\eqref{eq:cross_section_pheno} via $R_{12}$) than on the precise values of $H(z), f(M,z)$ of the underlying $\text{LIB}$ cosmology, including the effects of deviations from $\text{GR}$ in cosmological expansion and structure formation. This is the case for the example theory discussed in Sec.~\ref{sec:theory_connection}: GW lensing effects are orders of magnitude more sensitive than solar-system tests (cf. Fig.~\ref{fig:horndeski_constrained}), in turn more stringent than current cosmological observations \cite{Zumalacarregui:2020cjh,Alonso:2016suf} (for theories without a screening mechanism).

In addition to the Einstein radius scaling, Eq. \eqref{eq:cross_section_rE}, we will consider three cases of interest:
\begin{itemize}
    \item $n=1$: the physical scale is proportional to the total halo mass, much like the Schwarzschild radius. The rates are dominated by large masses and saturate at $z_s\gtrsim 1$, as the more massive halos are exponentially suppressed at early times. This case captures the dependence of the time delay in a Horndeski model (Sec.~\ref{sec:theory_connection}).
    \item $n=1/2$: the scale has the same mass scaling as the Einstein radius and leads to rates independent of $M$. However, the overall redshift dependence is different, as $R_E$ depends also on $D_S,D_{LS}$.
    \item $n=1/3$: this mass scaling favors lighter halos and thus grows very rapidly with redshift. It is motivated by the mass-dependence of the Vainshtein radius $R_V$, i.e. the classical strong-coupling scale \cite{Vainshtein:1972sx}. For $n=1/3$ the contribution from lighter halos diverges and a low mass cutoff needs to be included (we will take $M>10^7 M_\odot$). We will see this mass dependence when considering a binary merging near an active galactic nuclei in a Horndeski theory (Sec.~\ref{sec:theory_190521}).
\end{itemize}
The optical depths for each of the cases as a function of the source redshift are plotted in Fig.~\ref{fig:optical_depths_constrained}.
Note that these phenomenological models assume that the cross section is independent of $\Delta t_{12}$, and thus common for all the analyzed events. Dependence in the time delay can be included, e.g. by multiplying Eqs.~\eqref{eq:cross_section_rE},~\eqref{eq:cross_section_pheno} by a factor $(\Delta t_{12}/10{\rm \mathrm{ms}})^{-k}$. For the sake of simplicity, we will not include this dependence and instead interpret the obtained values of $\alpha_X$, $R_{12}(n)$ at the median 95\% c.l. from all analyzed events.

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

In order to test our method and understand the observing capabilities, we first apply our pipeline to injections. We then proceed to analyse the latest GW catalog (GWTC-3).

\begin{figure*}[t!]
\centering

\subfigure[$SNR$ $10$]{\label{fig:a}
\includegraphics[width=0.32\linewidth]{figs/PE_150914like_snr10_violin.pdf}}
\subfigure[$SNR$ $15$]{\label{fig:b}
\includegraphics[width=0.32\linewidth]{figs/PE_150914like_snr15_violin.pdf}}
\subfigure[$SNR$ $20$]{\label{fig:c}
\includegraphics[width=0.32\linewidth]{figs/PE_150914like_snr20_violin.pdf}}
\subfigure[$SNR$ $30$]{\label{fig:d}
\includegraphics[width=0.33\linewidth]{figs/PE_150914like_snr30_violin.pdf}}
\subfigure[$SNR$ $40$]{\label{fig:e}
\includegraphics[width=0.33\linewidth]{figs/PE_150914like_snr40_violin.pdf}}
\caption{Signal-to-noise ratio (SNR) dependence of $\Delta t_{12}(\mathrm{ms})$ posteriors and the  $\log \ensuremath{\mathcal{B}^{\MG}_{\GR}}$ (upper-x axis) for the GW150914-like injections with different values of $\Delta t_{12}^\mathrm{inj}$ (lower-x axis) and $\phi_\mathrm{lens}^\mathrm{inj} = \pi/5$.  Time delays ($\Delta t_{12}$) as small as $1ms$ are recovered well with SNR 30 \& 40 signals, and for SNR 10 signals time delays $<30ms$ are not measurable. Both model selection and time delay measurements (without the symmetry around $\Delta t_{12}=0$) improve with increase in SNR.}
\label{fig:sim_events}
\end{figure*}

\subsection{Injections}
We inject GW150914-like signals in simulated Gaussian noise with $\Delta t_{12}^\mathrm{inj} \in \{0, 1, 3, 10, 30\}$ ms and  $\phi_\mathrm{lens}^\mathrm{inj}= \pi/5$ rad and recover them by running the parameter estimation routines under the $\text{GR}$ and $\text{LIB}$ hypothesis, as mentioned in Sec.~\ref{sec:Bayes}.  This allows us to compute the Bayes factors $\ensuremath{\mathcal{B}^{\MG}_{\GR}}$ to compare the two hypothesis for each injection.

The injections are set to have $\mathrm{SNR} \in \{10, 15, 20, 30, 40\}$. These SNRs are achieved by inversely scaling the luminosity distance ($d_L$) of the injections. Fig.~\ref{fig:sim_events} shows the violin plots and the $\log$ Bayes Factors for these injections.
The posteriors on  $\phi_\mathrm{lens}$ are uninformative in all the cases and hence not shown in the figure.
$\text{LIB}$ time delays ($\Delta t_{12}$) as small as 1 ms are recovered well with SNR 30 and 40 signals, whereas for SNR 10 signals time delays $<30$ ms are not measurable. As one would expect, only with $\Delta t_{12}^\mathrm{inj} = 0$, i.e. $\text{GR}$ injection the $\log \ensuremath{\mathcal{B}^{\MG}_{\GR}} < 0 $, i.e. consistent with the $\text{GR}$ hypothesis, except for the SNR 10 case where $\log \ensuremath{\mathcal{B}^{\MG}_{\GR}} = 0.1 $ is  within the intrinsic sampling error on the calculation of evidence. For $\Delta t_{12}^\mathrm{inj} \in \{1, 3, 10, 30\}$ ms we find that $\log \ensuremath{\mathcal{B}^{\MG}_{\GR}} > 0 $, i.e. consistent with the $\text{LIB}$ hypothesis for all the SNRs except 10. Hence, both model selection and sensitivity to measure the time delays improve with an increase in SNR.

We also note that the time delays are measurable up to a  symmetry around $\Delta t_{12}=0$. This is because $\text{LIB}$ waveform, Eq. \eqref{eq:S}, is identical at $(\Delta t_{12}, \phi_\mathrm{lens}, t_c)$  and $(-\Delta t_{12}, \phi_\mathrm{lens} +\pi/4,t_c + \Delta t_{12})$, which we also saw during mismatch studies with $\text{LIB}$ injections, see right panel of Fig.~~\ref{fig:mismatch}.
It is possible that for asymmetric and inclined binaries with significant HoMs a better measurement of $\phi_\mathrm{lens}$  could break the $\Delta t_{12}$ parity as well, however, this needs to be investigated further and is left for future studies.

Overall, as the sensitivity of the detectors improve we shall be able to measure the birefringence time delays as small as $1$ ms. On the other hand, in the absence of birefringence we expect to see $\Delta t_{12}$(s) posteriors which are consistent with the $\text{GR}$ value, i.e. zero and bayes factors that favour the $\text{GR}$ hypothesis. Most events in GWTC-3 have SNR $<30$. The time delay posteriors are hence expected to be broad, however the Bayes factors should already indicate whether $\text{LIB}$ is present or not.

\subsection{GWTC-3 Events}
We now analyse 43 CBC events from the GWTC-3, that have low detection false alarm rate, $\mathrm{FAR}\lesssim 10^{-3} \ \mathrm{yr}^{-1}$. These are also the events that are considered for other tests of $\text{GR}$ performed previously~\cite{LSC_2016grtests,GWTC1TestofGR,LIGOScientific:2020tif, LIGOScientific:2021sio}.

Fig.~\ref{fig:real_events} shows the $\Delta t_{12}$ posteriors and the $\log$ Bayes Factors for the real events. We find that for almost all the events the  $\Delta t_{12}$ posteriors are broad containing zero, i.e. consistent with $\text{GR}$. This is mostly due to the low SNRs of the events, as seen in our injection study. We also find the tightest 90\% credible bounds on $|\Delta t_{12}| \lesssim 0.51$ ms coming from the event
GW200311\_115853 which has reasonably high SNR ($\simeq 17.8$) and moderate redshift ($z\sim 0.23$) as compared to other events. As expected $\phi_\mathrm{lens}$ posteriors are uninformative for almost all the events.

\begin{figure*}[tbh]
    \centering
    \includegraphics[ width = \linewidth,clip ]{figs/alice_violin_t12_vertical_all_plt_hor.pdf}
\caption{lens-induced birefringence ($\text{LIB}$) test of GWTC-3 events \cite{LIGOScientific:2021djp}. We show the posteriors on $\Delta t_{12}(\mathrm{ms})$ and Bayes Factors $\log \ensuremath{\mathcal{B}^{\MG}_{\GR}}$ (upper $x$-axis). Events with positive Bayes factors are highlighted in red.}
    \label{fig:real_events}
\end{figure*}

38 out of 43 events resulted in $\log \ensuremath{\mathcal{B}^{\MG}_{\GR}} < 0 $, and hence consistent with the $\text{GR}$ hypothesis. Only a few events showed preference to $\text{LIB}$ hypothesis ($\log \ensuremath{\mathcal{B}^{\MG}_{\GR}} >0$), with highest one for GW190521 (3.21)  and then  GW190910\_112807 (0.8), GW170823 (0.8), GW191109\_010717 (0.7) \& GW191129\_134029 (0.1).

The Bayes factors are known to be prior dependent and its value does not signify the confidence in preferring one hypothesis over the other, but rather the preference of one hypothesis over the other given a set of prior assumptions.
The model with extra parameters ($\text{LIB}$) could be either fitting the noise or the signal, therefore we take a frequentist approach to determine the significance by considering different realisations of noise. We focus on the event with the highest Bayes factors (GW190521) and estimate its significance.
We generate the background distribution of Bayes factors by injecting $\text{GR}$ signals in Gaussian noise using the power spectral density around the trigger time. To calculate the false alarm probability corresponding to the observed Bayes Factor for the event GW190521, we simulate a hundred $\text{GR}$ injections, whose parameters are taken from the posteriors of GW190521 event for the $\text{GR}$ hypothesis. Fig.~\ref{fig:190521_bg} shows the background distribution of the  Bayes factors
and the corresponding false alarm probability (FAP). The FAP corresponding to each $\ensuremath{\mathcal{B}^{\MG}_{\GR}} = \kappa $ is calculated as the fraction of the background events having  $\ensuremath{\mathcal{B}^{\MG}_{\GR}} > \kappa$. We find that for the observed $\log \ensuremath{\mathcal{B}^{\MG}_{\GR}} = 3.2$ for GW190521  is $0.48$, i.e. its significance is less than $1 \sigma$.

It is to be noted that GW190521 is a remarkably loud but short ($ < 100 $ ms) signal,  being easily fit by widely different hypotheses such as head on-collision of a boson star \cite{CalderonBustillo:2020fyi} or left-right (L-R), frequency-dependent  birefringence \cite{Wang:2021gqm}. For the interested reader, in Appendix~\ref{sec:pe_190521} (Fig.~\ref{fig:GW190521}), we also plot posteriors and the waveforms corresponding to the maximum a posteriori  parameters for both the hypothesis, over the whitened signals observed at each detector.
The plots show that the $\text{LIB}$ hypothesis is also fitting the noise, and might therefore give a high Bayes factor, $\ensuremath{\mathcal{B}^{\MG}_{\GR}}$. The other events with $\ensuremath{\mathcal{B}^{\MG}_{\GR}} > 0$ are also show similar behavior and as their preference for LIB is marginal, we conclude that none of the events have any significant Bayes factor and find \textit{no strong evidence} for birefringence.

\begin{figure}[h]
    \centering
    \includegraphics[width = \linewidth]{figs/190521_background_BF_ngr_gr_v1.pdf}
    \caption{Bayes factors distributions for GW190521-like CBC, calculated by doing PE with both the hypothesis, for $\sim 100$ $\text{GR}$ injections from the GW190521 posteriors in different realisations of gaussian noise. The false alarm probability for the observed $\log \ensuremath{\mathcal{B}^{\MG}_{\GR}} = 3.2$ is found to be $0.48$. }
    \label{fig:190521_bg}
\end{figure}

\section{Implications}
\label{sec: implications}
In our analysis of the latest GW catalog, we have found that the majority of the events disfavor birefringence. For a subset of them (most notably GW190521) while the Bayesian inference prefers the $\text{LIB}$ hypothesis, a follow-up background study indicates that most simulated $\text{GR}$ signals give comparable Bayes factors.
In the following, we present the implications of these results.
First, we consider the implications for generic $\text{LIB}$. Then, we study the constraints on a specific scalar-tensor theory that predicts $\text{LIB}$.
Finally, we  entertain the possibility that GW190521 was emitted in an active galactic nucleus (AGN) and is displaying evidence of birefringence.
\subsection{Constraints on generic $\text{LIB}$}

From the non-observation of birefringence in the 43 events from GWTC-3 and using their median redshift values \cite{LIGOScientific:2021djp}, we estimate the total optical depth for the $\text{LIB}$ models discussed in Sec.~\ref{sec:cross_section_pheno}. The non-observation of birefringence translates to constraints on the phenomenological model parameters, as summarized in Table~\ref{tab:constraints}. For reference, we also show the constraints obtained from the full GWTC-3 (90 events).

\begin{table}[h]
    \centering
    \bgroup
\def1.5{1.5}
    \begin{tabular}{l l   c}
$\sigma_{\rm \text{LIB}}^{1/2}\hspace{.2cm} $ \hspace{0.5cm}& 95\% c.l. & comment\\ \hline
$ \propto M$ & $R_{12} < 4.4\,(2.9)$ kpc &  Sec.~\ref{sec:theory_connection} \\ \hline
$\propto R_E$ &  $\alpha_E < 3.0\,(1.6)$ &  \\
$\propto M^{1/2}$ & $R_{12} < 20\,(12)$ kpc & \\\hline
$\propto M^{1/3}$   & $R_{12} < 12\,(6.9)$ kpc &\quad  \\\hline
    \end{tabular}
    \egroup
    \caption{Constraints on the phenomenological models Eq.~\eqref{eq:cross_section_rE} and Eq.~\eqref{eq:cross_section_pheno}, assuming no birefringence detected for analysed (all) GWTC-3 events.
    \label{tab:constraints}}
\end{table}

The higher redshift events have higher optical depth. Non-observation of birefringence in distant sources leads to more stringent constraints, although the SNR scales with the inverse luminosity distance: hence some of the highest redshift events will not be considered because of our FAR threshold. The final results depend strongly on the model via source redshift and halo mass function. Figure~\ref{fig:optical_depths_constrained} shows the redshift dependence of the optical depth for the parameterizations discussed, adopting the 95\% c.l. values found by our analysis along with the observed GWTC-3 redshift distribution.

Future observations will increase in number of events and their SNRs, allowing better constrain the birefringence probabilities and ruling out more of the parameter space in the alternative theories of gravity. Higher-redshift observations above our FAR threshold will be especially valuable to constrain $\alpha_E$ and $R_{12}$ for $n=1/3,1/2$ (see Fig.~\ref{fig:optical_depths_constrained}).

\begin{figure}[tbh]
    \centering
    \includegraphics[width = \linewidth]{figs/cross_section_constrained.pdf}
    \caption{Birefringence optical depth for the phenomenological models considered here, using the parameters correspond to the 95\% c.l. limit compatible with the non-observation of $\text{LIB}$. The dark (light) gray shaded histograms show the binned redshift distribution of analysed (all) GWTC-3 events. See Sec.~\ref{sec:cross_section_pheno} for details.}
    \label{fig:optical_depths_constrained}
\end{figure}

\subsection{GW birefringence in Horndeski theories}\label{sec:theory_connection}

Let us now use our results to a specific theory that predicts $\text{LIB}$. We will present the theory and translate the constraints of the phenomenological model (Table~\ref{tab:constraints}) into fundamental theory parameters. In the next subsection we will interpret a tentative detection of $\text{LIB}$ in GW190521 as an AGN binary within the same theory.
We will focus on a particular scalar-tensor theory within the Horndeski class~\cite{Horndeski:1974wa}, whose $\text{LIB}$ predictions have been analyzed in detail, cf. Sec. 6 in Ref.~\cite{Ezquiaga:2020dao}.
The model is described by two parameters describing couplings between the Ricci scalar ($R$) and the new field $\phi$: a linear coupling $p_{4\phi}$ and a derivative coupling suppressed by an energy scale $\Lambda_4$.
The Lagrangian of this theory can be written as \cite{Ezquiaga:2016nqo,Ezquiaga:2017ner}
\begin{equation}\label{eq:lagrangian_example_horndeski}
    \mathcal{L}\sim -\frac{1}{2}(\nabla\phi)^2+ \frac{M_P^2}{2}\left(1+\frac{p_{4\phi}\phi}{M_P}\right)R + \frac{\phi}{\Lambda_4^2}\nabla_\mu\nabla_{\nu}\phi G^{\mu\nu}\,,
\end{equation}
where $R$ is the Ricci scalar, $G_{\mu\nu}$ is the Einstein tensor, $M_P$ is the Planck mass in units of $c = h = 1 $,  and $\nabla$ the covariant derivative.
The $\text{GR}$ limit corresponds to $p_{4\phi}\to 0, \Lambda_4\to \infty$ .
The parameters of this model are stringently constrained by the speed of GWs on the homogeneous FRW metric \cite{Ezquiaga:2017ekz,Creminelli:2017sry,Baker:2017hug,Sakstein:2017xjx} (see also \cite{Brax:2015dma,Lombriser:2015sxa,Bettoni:2016mij}), as observed by the near-coincident arrival of GW170817 and its associated counterpart~\cite{Monitor:2017mdv}: $|c_g/c-1|\lesssim 10^{-15}$. Compliance with this limit requires \cite{Ezquiaga:2020dao}
\begin{equation}\label{eq:horndeski_170817}
p_{4\phi} \lesssim 10^{-8}\Lambda_4/H_0 \quad \text{(GW170817)}\,.
\end{equation}
While this constraint is extremely stringent, $\text{LIB}$ allows comparable limits.

Specifying a model allows one to derive concrete predictions. The dependence of the time delay contributions (Shapiro, geometric) with the lens and theory parameters is complex. Nonetheless, we observed that the time delay decreases monotonically with the impact parameter. Moreover, its slope changes and becomes very sharp beyond the Vainshtein radius
\begin{equation}\label{eq:r_vainshtein}
    r_V
 = 1.2{\rm Mpc}\, p_{4\phi}^{1/3}\left(\frac{M}{10^{12}M_\odot}\right)^{1/3}\left(\frac{H_0}{\Lambda_4}\right)^{2/3}
    \,.
\end{equation}
$r_V$ represents the scale at which the scalar field has a strong self-coupling near a massive object \cite{Vainshtein:1972sx}\footnote{For extended lenses one needs to consider the effective Vainshtein radius, such that $r_V(M(r_V^{\rm eff})=r_V^{\rm eff}$, (see Eq. (186) and Fig. 14 in Ref. \cite{Ezquiaga:2020dao}).}.
In many scalar-tensor theories this leads to \emph{screening}: a suppression of scalar field fluctuations for $r<r_V$, allowing the theory to approximately recover $\text{GR}$ around massive bodies. However, screening is not necessary in this model given the stringent constraint from GW170817 (Eq.~\eqref{eq:horndeski_170817}). In this case, the strong-interaction within $r_V$ represents a a large coupling between the scalar field and the Riemann tensor, the kind of interaction producing $\text{LIB}$.

For simplicity, we will focus on the Shapiro time delay. The geometric time delay is usually dominant for massive halos at intermediate distances. (Fig 12 in Ref.~\cite{Ezquiaga:2020dao}). It is proportional to the Einstein radius, and it could thus be captured generalizing Eq.~\ref{eq:cross_section_rE} to extended lenses. Neglecting the geometric time delay is conservative but reasonable, since our constraints involve events at relatively low redshift ($z\lesssim 0.6$).

The $\text{LIB}$ predictions have a simple dependence on the lens mass and theory parameters. We verified that $\Delta t_{12}\propto M \Lambda_4^{-4/3}$. The proportionality to the mass stems from the scaling with the Vainstein radius, as well as $\Delta t_{12}$, the impact parameter and the time spent by the GW on the region of sizeable birefringence are all $\propto r_V$. It allows us to directly connect the theory parameters to $R_{12}$ with $n=1$, as constrained in the phenomenological model \eqref{eq:cross_section_pheno}. The scaling with $\Lambda_4$ allows us then to find $R_{12}$ by equating $\Delta t_{12}(R_{12})$ to the constrained value for different $p_{4\phi}$, but keeping $M=10^{12}M_\odot$, $\Lambda_4$ fixed.
For simplicity, we will take a sensitivity of $\Delta t_{12}\sim 10 $ ms to define $R_{12}$.
Using the actual posteriors on $\Delta t_{12}$ for each of the GWTC-3 events analyzed will not qualitatively affect these constraints in any significant manner.

\begin{figure}[tbh]
    \centering
    \includegraphics[width = \linewidth]{figs/horndseki_talk_2.pdf}
    \caption{95\% c.l. constraints on the parameters of a quartic Horndeski theory \cite{Ezquiaga:2020dao} using the lens-induced birrefringence ($\text{LIB}$) test. Shaded regions are excluded according to GWTC-3 (this work, blue solid), GW170817 \cite{Ezquiaga:2017ekz,Creminelli:2017sry,Baker:2017hug,Sakstein:2017xjx} (green dashed) and GW190521 assuming an AGN binary
    ~\cite{Graham:2020gwr} (red dotted, see Fig.~\ref{fig:dt_agn_horndeski}).
    The $\text{GR}$ limit corresponds to $p_{4\phi}\to0, \Lambda_4\to \infty$, when the scalar field is decoupled from gravity and its derivative interactions suppressed.
    See sections~\ref{sec:theory_connection},~\ref{sec:theory_190521} for details.
    If GW190521 is associated to an AGN, the upper shaded region improves the overall GWTC-3 constraints for $\Lambda_4\gtrsim 3 H_0$.
    If we further assume a detection of $\text{LIB}$, then the bottom red shaded region excludes GR.
    For reference, we also indicate Solar System constraints (gray horizontal) and the region there the GW frequencies at LIGO-Virgo detectors are larger than the (non-linear) energy scale of the effective field theory (magenta vertical).}
    \label{fig:horndeski_constrained}
\end{figure}

The excluded region is shown in Fig.~\ref{fig:horndeski_constrained}, along with constraints from the GW speed on FRW and lunar laser ranging (no screening, $p_{4\phi}\ll1$, see Sec. VBc in Ref. \cite{Zumalacarregui:2020cjh}). The change in slope at low $\Lambda$ corresponds to a transition in which $R_{12}$ surpasses the Vainsthein radius \eqref{eq:r_vainshtein}. For $\Lambda_4 \ll H_0$ the birefringence constraints approach those of the GW speed: this happens when $r_V$ is so large that most GWs are effectively behind a lens. Then the constraints are satisfied in the limit $c_{\rm GW}\to c$, equivalent to Eq. \eqref{eq:horndeski_170817}. For the sensitivity of GWTC-3 this happens for  $\Lambda_4\lesssim H_0$, where LVK frequencies lie beyond the validity of our framework as a classical effective field theory \cite{deRham:2018red}. At increasing $\Lambda_4$ the constraints degrade, since probability becomes very suppressed, Eq. \eqref{eq:r_vainshtein}.
For $\Lambda^4\gtrsim 10^2H_0$ solar system constraints become more efficient than birefringence.

\subsection{GW190521 as an AGN binary} \label{sec:theory_190521}

Let us now discuss the implications of a possible birefringence detection associated to GW190521. Given the constraints from the speed of GWs \eqref{eq:horndeski_170817} on our example Horndeski theory, the chances of birefringence being caused by a lens in the line of sight are very small. We will instead interpret our result, $\Delta t_{12}\gtrsim 9.5$ ms as due to an environmental effect near the source. We will follow the scenario outlined in Ref.~\cite{Graham:2020gwr}, where a candidate electromagnetic counterpart from an  AGN J124942.3 + 344929, observed 34 days after the GW signal, suggests that the binary merged in the environment of a supermassive black hole (SMBH).
Note that there are important uncertainties, both regarding the counterpart association (given large GW localization uncertainties \cite{Palmese:2021wcv}), and the significance of $\text{LIB}$ detection (given our analysis of random noise realizations, Fig.~\ref{fig:190521_bg}). This discussion is therefore not a statement on the status of $\text{GR}$. Instead, it proves the potential of identifying environments of GW sources to test gravity theories.

Following Ref.~\cite{Graham:2020gwr}, we will assume an AGN binary scenario where the mass of the SMBH is $M_{\rm SMBH}\sim 10^8M_\odot$ and the source is located in a migration trap at $r\sim 700 GM_{\rm SMBH}$. Then, using the framework of Ref.~\cite{Ezquiaga:2020dao} allows us to compute the time delay as a function of the angle between the observer and the source, relative to the SMBH. The results are shown in Fig.~\ref{fig:dt_agn_horndeski} for $p_{4\phi}=10^{-8}, \Lambda_4 = 10 H_0$, compatible GW170817  \eqref{eq:horndeski_170817}, and different distances to the SMBH (the dependence on $M_{\rm SMBH}$ is less pronounced, see below). The birefringent time delay becomes very large as $\theta\to 0$\footnote{Our calculation relies on small deviations from a straight trajectory. This assumption breaks down for small angles, where one needs to consider the geodesics of the SMBH space-time instead. However, our results are conservative since actual trajectories will bend toward the SMBH, thus increasing $\Delta t_{12}$ relative to the straight propagation.}. Ultimately,  the maximum time delay is limited by the existence of the horizon, $\theta_s \approx 2GM/r$.
The birefringence also vanishes as $\theta\to\pi$ because of geometric cancellations in spherical symmetry.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figs/AGN_theta_dependence_R.pdf}
    \caption{Birefringent time delay for a source near a SMBH as a function of the angle of the observer, relative to the SMBH. Each line corresponds to a different source distance, for model parameters compatible with GW170817  (see. Eq. ~\ref{eq:horndeski_170817}).
    The horizontal line corresponds to the lower bound on $\Delta t_{12} = 9.5 $ ms  from the analysis of GW190521. The region between the shaded areas encompasses 95\% probability for a random observer. The lowest $\theta$ represents trajectories passing at 10 Schwarzschild radii of the SMBH.
    }
    \label{fig:dt_agn_horndeski}
\end{figure}

We will translate these predictions into theory parameters and include the comparison to GW190521. We will take the values of $M_{\rm SMBH}$ and the source radius fixed, and consider the credible intervals as being determined by the angle $\theta$, cf. Eq. \eqref{eq:known_lens} and Appendix~\ref{sec:beyond_poisson_stats}.
As we do not know the emission angle, we will assume a flat prior on the sphere $P(\theta)=\sin(\theta)$, and take the upper/lower 95\% c.l. values based on $P(\theta)$ (excluding the shaded regions in Fig.~\ref{fig:dt_agn_horndeski}).
Limits on the theory parameters can be derived by noting that $\Delta t_{12}(\theta)\propto p_{4\phi}^{4/3}  \Lambda_4^{-2/3}  M_{\rm SMBH}^{1/3}$ including different assumptions about the SMBH mass. Note that $M_{\rm SMBH}$ enters with a different scaling than the lens mass in Sec.~\ref{sec:theory_connection}, due to the source being at a fixed distance from the SMBH and within its Vainsthein radius, rather than randomly located.

The implications of GW190521 for the example theory \eqref{eq:lagrangian_example_horndeski} are shown in Fig.~\ref{fig:horndeski_constrained}. The orange regions are excluded if we assume the AGN scenario as discussed above. The lower region excludes the $\text{GR}$ limit $p_{4\phi}\to 0, \Lambda_4\to \infty$ and relies on trusting the measured birefringence $\Delta t_{12}\gtrsim 9.5$ ms to be due to new gravitational physics. Even if the result is interpreted as noise (e.g. Fig.~\ref{fig:190521_bg}), assuming the AGN scenario leads to exclusion of the upper orange region (assuming sensitivity to $\Delta t_{12}\lesssim 9.5$ ms). Because of the different scaling with the theory parameters, the detection of an AGN binary becomes even more constraining than GW170817 for high $\Lambda_4$.
The beyond $\text{GR}$ interpretation can be further probed not only by AGN events but by high-redshift multi-messenger observations. In this case, the time delay between GWs and EM counterparts scales as $\approx 1s \left(10^8 p_{4\phi}\frac{H_0}{\Lambda_4}\right)^2\frac{D}{40\rm Mpc}$ and can be probed by distant neutron-star mergers.

\section{Summary and Outlook}\label{sec:conclusion}
In this paper, we explored $\text{LIB}$  as a test of $\text{GR}$ using observations of GWs.
$\text{LIB}$ produces a difference in the arrival times of the GW polarisations in signals from the binary mergers, predicted by some alternatives to the $\text{GR}$. Using the Bayesian model selection framework, not only we can identify the signatures of birefringence, but also measure the time delay between the arrival of both polarisations ($\Delta t_{12}$). We show that this difference can be measured with high accuracy, of order few milliseconds with existing events and is likely to improve in the future following detector upgrades.

Using the latest GW catalog, GWTC-3, we find no strong evidence for the observation of the birefringence, with the highest $ \log \ensuremath{\mathcal{B}^{\MG}_{\GR}}=3.21$ for the heaviest binary black holes so far, GW190521.
However, after simulating similar events under different noise realizations,
we determine that there is a
false alarm probability of 48\%.
This event has been associated with an AGN flare, possibly indicating that the merger occurred near a SMBH.
This AGN scenario is especially favorable for the observation of $\text{LIB}$ since the SMBH would
act as a strong source of $\text{LIB}$. However, the AGN flare-GW association has been disputed, see e.g.
\cite{Palmese:2021wcv}.
Moreover, the loudness and shortness of this event makes it susceptible to different astrophysical and fundamental physics interpretations. It has also been found to
be violating many tests of $\text{GR}$  and mimicking many exotic scenarios of compact binary such as head-on-collision of a boson star \cite{CalderonBustillo:2020fyi} or left-right (L-R), frequency-dependent  birefringence \cite{Wang:2021gqm}. The latter effect is related to our flavour of $\text{LIB}$, with two important differences: first, L-R birefringence is defined in the basis of circularly polarized waves (left vs right, rather than $+$ vs $\times$), and second, it depends on the GW frequency. Both features also appear in the Gravitational spin Hall effect in \text{GR}, although the L-R time-delay is very suppressed \cite{Andersson:2020gsj,Oancea:2022szu}.

Of the 43 analyzed events, we find that the tightest bounds on the time delay between the two polarisations is $\Delta t_{12} \sim 0.51  $ ms at 90\% credible intervals coming from the
GW$200311\_115853$ merger event, while the median is $\Delta t_{12} \simeq 80 $ ms. From the non-observation of $\text{LIB}$, we constrained the lensing optical depths in a phenomenological parameterization in which the lensing cross-section is proportional to the Einstein radius or a fixed physical radius with a power law scaling in the halo mass.

Our constraints can be translated to gravitational theories that predict $\text{LIB}$. As an example, we presented novel constraints on a Horndeski scalar-tensor theory featuring a new dynamical field and two free parameters. The theory is stringently constrained by the speed of GWs on the homogeneous FRW background following GW170817. Nevertheless, the lack of observed $\text{LIB}$ places stringent bounds, which can be orders of magnitude better than Solar System tests and in some limits as tight as the GW speed bound.
As a proof of principle of $\text{LIB}$ due to a known inhomogeneity, we interpret GW190521 as an AGN binary (assuming that the signal originated in close proximity to a SMBH \cite{Graham:2020gwr}) in terms of our example theory. Then, the large curvature is able to generate detectable $\text{LIB}$ even when deviations from $\text{GR}$ are minute.
Our $|\Delta t_{12}|\gtrsim 9.5$ ms results would then exclude \text{GR}, placing a minimum value of the theory parameters. When interpreting this result as a fluctuation and $\text{GR}$ to be correct, the AGN hypothesis is still able to produce very stringent bounds, that can even overcome those of the GW speed on FRW.

In future, the methods we developed here can be useful for studying new classes of events. Of particular interest will be signals where the merger is either near an SMBH or is known to have a lensed counterpart due to strong lensing. In such cases, the information about the lens may improve the constraints substantially, along the lines of the AGN-scenario we discussed. The increase in detection rate and a growing chance of strongly lensed GW identification makes $\text{LIB}$ test also relevant for future runs of LVK detectors and upcoming GW detectors such as  Einstein Telescope, Cosmic Explorer and LISA \cite{Kalogera:2021bya, Sathyaprakash:2012jk, Ding:2015uha, mesut_LISA}. Lastly, the addition of ground-based detectors such as LIGO-India and KAGRA can allow us to measure extra linear combinations of the GW polarisations and construct a null-stream~\cite{Chatziioannou:2012rf} to extract each of the polarisations individually. The extracted polarisations can then be used to test their consistency with $\text{GR}$ or other theories of gravity directly.

Strongly lensed GW signals may allow us to measure additional linear combinations of the same GW polarisations and hence improve various tests of $\text{GR}$ ~\cite{Goyal_2021}, including the one proposed here. Ultimately, developing $\text{LIB}$ predictions for other alternative theories and generalizing the model-independent parameterizations presented here will allow our results to further test the landscape of theories beyond $\text{GR}$.

\section{Acknowledgements}

We are grateful to A. K. Mehta, P. Ajith, G. Brando, S. Savastano, G. Tambalo, Y. Wang, H. Villarrubia-Rojo and J. Tasson for fruitful discussions. SG and AV are supported by the Department of Atomic Energy, Government of India, under Project No. RTI4001. AV is also supported by a Fulbright Program grant under the Fulbright-Nehru Doctoral Research Fellowship, sponsored by the Bureau of Educational and Cultural Affairs of the United States Department of State and administered by the Institute of International Education and the United States-India Educational Foundation.
J.M.E. is supported by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 847523 INTERACTIONS, by VILLUM FONDEN (grant no. 37766), by the Danish Research Foundation, and under the European
Union's H2020 ERC Advanced Grant "Black holes: gravitational engines of discovery" grant agreement no. Gravitas–101052587.
The numerical calculations reported in the paper are performed on the Alice computing cluster at ICTS-TIFR, with the aid of \textsc{LALSuite}~\cite{lalsuite}, \textsc{Bilby}~\cite{bilby}, \textsc{PyCBC}~\cite{pycbc} and \textsc{Colossus} \cite{Diemer:2017bwl} software packages. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation.

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« The media and American fears of ISIS | Main | Arab Uprisings Explained Tour hits DC »
What's up with those Houthis in Yemen?
The Yemen-based photojournalist Alex Potter posted this striking picture of Houthi fighters in Sanaa's Tahrir Square last week (follow all of her work here).
The Houthi advance into Yemen was as shocking in its own way as was the Iraqi army's collapse against ISIS in Mosul. But just as in Iraq, the long-developing problems which might explain the surprising turn of events were well-known to those who had been paying attention. The shortcomings of the National Dialogue process, the fracturing of Yemeni state institutions, and the rising power of both the Houthi and the southern insurrections were all amply detailed in POMEPS Briefing #19 Yemen's National Dialogue, with contributions by Danya Greenfield, Holger Albrecht, April Alley, Stephen Day, Peter Salisbury, Stacey Philbrick Yadav, Silvana Toska, Adam Baron, Lisa Wedeen and more. In June, April Alley of the Crisis Group published an extremely prescient report, "The Huthis: From Sadaa to Sanaa."
I was a bit perturbed by the initial lack of coverage and analysis of these events. A lot of what was published fell into a simplistic (or politically expedient) sectarian framing around "Iranian-backed Yemeni Shi'ites" or tedious U.S. political framing around "the failure of Obama's success story." A much more useful round of analysis has now started to emerge to help make sense of these new developments.
Here are some of the recent pieces which I've found helpful for understanding the Houthi movement and Yemeni politics, including two which I was delighted to be able to publish on the Washington Post's Monkey Cage:
Stacey Philbrick Yadav, "The limits of the sectarian framing in Yemen." The Monkey Cage.
Silvana Toska, "Shifting balances of power in Yemen's crisis." The Monkey Cage.
Charles Shmitz, "The Huthi Ascent to Power." Middle East Institute.
Iona Craig, "What the Huthi takeover of Sanaa reveals about Yemen's politics" Al-Jazeera America.
Peter Salisbury, "Yemen's Huthis redraw political map, upend transition." World Politics Review.
Gregory Johnsen, "Yemen's bloody weekend." Buzzfeed.
+ Danya Greenfield, "President Hadi's Gamble." Atlantic Council.
I'll update this list as appropriate, with a + to indicate the new ones!
Note: photo comes from Alex Potter's excellent and highly recommended Instagram page.

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A Review of Mein Kampf
Mein Kampf, Adolf Hitler's book describing his early life, wartime experiences and the early development of the National Socialist Movement, continues to be one of the best-selling political books in the world. In fact, the more it is condemned, the more it seems to fascinate. Just google 'sales of Mein Kampf'. Rational analysis is in far shorter supply.
In that vein, I ran across an interesting review of Mein Kampf the other day. The reviewer pointed out that prefaces in the English editions of Mein Kampf, published in the 1930s, were quite sympathetic to Hitler, no doubt due to the fact that the Depression-level unemployment in Germany had been greatly reduced, and the nation seemed to be getting on its feet again after its calamitous defeat in the First World War. Also, the 1936 Olympics in Berlin were a huge propaganda victory for the Hitler regime.
It is no use saying that hate, bigotry and madness and whatever else strikes your fancy are the reasons for the success of the fascist movements, and the Hitler movement in particular, in the 20s and 30s, as millions of ordinary, decent people supported them, not only in Germany but throughout Europe and as far away as India. And it is easy to say that Hitler was supported by big business to crush the socialist and communist parties of the time, but that would not have happened had he not "talked a great movement into existence already."
In the 1920s, there was a multitude of left- and right-wing parties, movements and aspiring demagogues vying for attention and power. Why did Hitler succeed when so many others failed? Our reviewer continues….
"But Hitler could not have succeeded against his many rivals if it had not been for the attraction of his own personality, which one can feel even in the clumsy writing of Mein Kampf, and which is no doubt overwhelming when one hears his speeches…The fact is there is something deeply appealing about him."
Obviously, some deep psychological need was being addressed, something way beyond mere opinions about political parties or national policies.
"One feels, as with Napoleon, that he is fighting against destiny…"
Indeed, there are numerous references and appeals to Destiny in Mein Kampf and no doubt Hitler felt that was his purpose, either to fulfill it or thwart it.
Prior to the First World War, or the Great War as it was then known, Europe possessed many powerful socialist movements. At times, it seemed that the old imperial orders across Europe were approaching their end. But when war broke out in August 1914, the international socialist parties across Europe folded like straws in the wind before the onrush of Nationalist Awakening. Nation and race have a far deeper psychological appeal than watery internationalism (something our reviewer noted elsewhere) and Hitler and Mussolini rose to power largely because they could recognize this fact and their opponents couldn't.
But there were nationalist leaders before that never evoked anything like the adulation that was awarded to Hitler and Mussolini. There were also reasons for this….
"…Also he has grasped the falsity of the hedonistic attitude to life. Nearly all western thought since the last war, certainly all "progressive" thought, has assumed tacitly that human beings desire nothing beyond ease, security and the avoidance of pain. In such a view of life there is no room, for instance, for patriotism and the military virtues."
At least intermittently, people yearn for struggle, self-sacrifice and devotion to a higher cause. Whether one considers these passions desirable is beside the point, they exist and are powerful. Hence…
"…however they are as economic theories, Fascism and Nazism are psychologically far sounder than any hedonistic conception of life. The same is probably true of Stalin's militarized version of Socialism….whereas Socialism, and even capitalism in a grudging way, have said to people 'I offer you a good time', Hitler has said to them 'I offer you struggle, danger and death,' and as a result a whole nation flings itself at his feet."
As if you hadn't already guessed, "our reviewer" is, of course, George Orwell, probably the most prescient writer on politics in the twentieth century. Orwell's essay (here) is not really a review of Mein Kampf, but an essay on the psychological basis of Hitler's appeal to millions of ordinary people. It is an important essay because it is written honestly and without fear, not of Hitler, but without fear of bullying and censorship in his homeland. Orwell was a man possessed of a terrible clarity of vision and a crystalline honesty, qualities entirely lacking in our current world of conformity, cowardice and mediocrity.
It is important to note that his essay was published in England, in March 1940, during a period of wartime censorship, when England was at war with Germany. It is highly unlikely that such an essay, analyzing Hitler's appeal to the good, as well as the bad, in people, would be able to be published in this country in peacetime, when, supposedly, we are guaranteed freedom of speech.
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November 24, 2017 Rebel Yell
Don Laird
Enjoyed the essay, both of them thought provoking.
This said however, if this apathy and self-loathing madness in the West does end and end soon, this will be the "horror without end".
One of the echoes of yesteryear I find most grievous is the apathy of our youth, the indifference to the sweat and blood poured by our ancestors, the same securing the future of our children, how remarkable now, our children turn their backs on their betters……
Here's a quote from something something I penned a few years ago, as I watched the Muslim boot descend on the necks of the English…..
…….."So England my friend…..what is your convenience?…..what is the depth of your character?…..what is the breadth of your conviction?….. what does Britain and your ancestors mean to you?….What will be your legacy?…..What will your grandchildren say when they speak of you?…….During November's winter whisper, on parade squares when silvered haired patriots and regiments assemble, when politicians and priests with hand on heart eulogize……will they speak of those who held the line or will they mumble, eyes cast aside, of those for whom courage failed, those who, when their country needed them, turned and ran? Will you honour the dying words of Lt Col. John McRae as he urged; "Take up our quarrel with the foe, to you from failing hands we throw the torch; be yours to hold it high. If ye break faith with us who die, we shall not sleep, though poppies grow in Flanders Fields"……
Excerpt, "An Open Letter To England By Don Laird
Ahhhh, you would think I would have learned all I needed to learn about treachery, ingratitude and betrayal when I read of the day they nailed the Christ to the cross.
But here below, is the last installment to Alfred Lord Tennyson's "Charge of The Light Brigade", its a poem by Rudyard Kipling, an ode to the brave men left to "the streets and the workhouse".
The Last of The Light Brigade.
There were thirty million English who talked of England's might,
There were twenty broken troopers who lacked a bed for the night.
They had neither food nor money, they had neither service nor trade;
They were only shiftless soldiers, the last of the Light Brigade.
They felt that life was fleeting; they knew not that art was long,
That though they were dying of famine, they lived in deathless song.
They asked for a little money to keep the wolf from the door;
And the thirty million English sent twenty pounds and four !
They laid their heads together that were scarred and lined and grey;
Keen were the Russian sabres, but want was keener than they;
And an old Troop-Sergeant muttered, "Let us go to the man who writes
The things on Balaclava the kiddies at school recites."
They went without bands or colours, a regiment ten-file strong,
To look for the Master-singer who had crowned them all in his song;
And, waiting his servant's order, by the garden gate they stayed,
A desolate little cluster, the last of the Light Brigade.
They strove to stand to attention, to straighten the toil-bowed back;
They drilled on an empty stomach, the loose-knit files fell slack;
With stooping of weary shoulders, in garments tattered and frayed,
They shambled into his presence, the last of the Light Brigade.
The old Troop-Sergeant was spokesman, and "Beggin' your pardon," he said,
"You wrote o' the Light Brigade, sir. Here's all that isn't dead.
An' it's all come true what you wrote, sir, regardin' the mouth of hell;
For we're all of us nigh to the workhouse, an' we thought we'd call an' tell.
"No, thank you, we don't want food, sir; but couldn't you take an' write
A sort of 'to be continued' and 'see next page' o' the fight?
We think that someone has blundered, an' couldn't you tell 'em how?
You wrote we were heroes once, sir. Please, write we are starving now."
The poor little army departed, limping and lean and forlorn.
And the heart of the Master-singer grew hot with "the scorn of scorn."
And he wrote for them wonderful verses that swept the land like flame,
Till the fatted souls of the English were scourged with the thing called Shame.
O thirty million English that babble of England's might,
Behold there are twenty heroes who lack their food to-night;
Our children's children are lisping to "honour the charge they made – "
And we leave to the streets and the workhouse the charge of the Light Brigade!
-The Last of the Light Brigade
-Rudyard Kipling
Like I said, treachery, ingratitude and betrayal are timeless.
Regards, Don Laird
PS, its Friday evening
Here's a clip that oughta wake ya up….
https://www.youtube.com/watch?v=-QNlebUDb8U
November 25, 2017 at 12:05 AM	Reply
old white guy
60 years ago as a teenager I read mein kampf, didn't think much of it then and even less now.
November 25, 2017 at 7:01 AM	Reply
Barry Thompson
If you looked at how World War I ended from Germany's point of view, and then projected that through the revenge that Versailles brought, followed by a lack-luster new democracy that steered them into a classic hyperflation, bankrupting the middle class … do you know why Nov 11 is called "Armistice Day"? It's because it was supposed to be an armstice where the terms of peace were to be negotiated, and it ended up that Germany had conditions imposed on it, including removing the monarchy. I could go on. Hitler probably gave legitimate voice to the darker feelings of Germans.
Even so, his plans to "purify" the population were never something that was used to gain votes There was serious harassment of Jews before the war, but it seems to have been aimed at driving them out of Germany. They didn't turn to camps and gas chambers until after the war started. The real abatoirs were not in Germany, the gruesome stuff was done in Poland.
Don't misunderstand me — I don't like Hitler, and am not even close to being a white supremacist. My focus is learning from the past, probably futile, but it's my hobby. It's a mistake to note the resentment in Hitler's appeal without putting it in the social context. It amounts to building a new stereotype.
We are in a kind of "Weimar" period in our own cultures at the moment. Huge corruption in the control centre. It's probably wise to soak up a little about how economic and political strains can drive a whole population to extremes.
November 25, 2017 at 12:52 PM	Reply
Barry Thompson has a point.
Back an entire nation into a corner, strip it of its pride, vilify its culture and heritage at every turn, loot its economy, grind its citizens into the ground with crushing taxation, lie to its citizens at every turn with a centrally controlled, like-minded media, promote sexual immorality, attack its foundational underpinnings, humiliate the entire nation on the world stage, and sooner or later there will rise voices that object, sooner or later those voices will give rise to a renewed sense of nationalism, sooner or later those voices will set in motion the mechanisms for a renewed strength and pride, sooner or later there will rise those who refuse to live on their knees with the boots of their oppressors on their necks……..and who among us could blame those voices?
Starting to see the similarities between post WW1 Germany and many of the civilized Western countries of today?
Starting to see the similarities illustrated in the vilification of the civilized West today, at every turn, at every opportunity, by a centrally controlled Marxist mass media?
Starting to see the similarities between Angela Merkel, Barack Obama and Justin Trudeau, devout Marxists all, treasonous and seditious all, who take great delight in the grinding humiliation and ravaging of their cultures and nations under the guise of White Guilt, multiculturalism, mass uncontrolled immigration and the sacred global village, and all fuelled with their seething sense of self-loathing?
Starting to see the similarities illustrated in the attacks, at every turn, at every opportunity, on every single brilliant accomplishment of civilized Western societies and their citizens as reflected in the actions of groups like La Raza, Black Lives Matter, Antifa, the mass media, social media groups like FaceBook, Twitter and YouTube and virtually the entirety of the Democratic party?
Starting to see the similarities as our judiciary engages in virtually criminal levels of activism by blocking, openly defying, our elected officials attempts to close the third-world floodgates to the West, essentially, and forgive me, but what can best be termed as "shitting in the punchbowl", wide open, unchecked, uncontrolled mass immigration, floodgates that pour in what amounts to human refuse, what amounts to parasitic economic opportunists, the criminal, the mentally ill, the chronically ill, the uneducated, the illiterate, the lazy, the fraudulent, the liars, the grifters and those, in their tens of thousands, who despise the West and the culture and the freedoms that we enjoy, the same tens of thousands sworn to our complete subjugation or annihilation, and not being picky as to which mechanism they choose to achieve their ends?
Yes, Barry Thompson has a point.
Here are a few quotes from previous posts and an essay pasted together, somewhat fractured but you get the idea…..…..
………."As Britons, as men, you stand idle and indifferent as your heritage, your culture, your history and your country are destroyed. You turn away as social engineers and sneering academia fill your children's heads with poisonous propaganda, turning them into haters of their own people and lovers, nay, worshippers of the black hearted, murderous cancer of Islam. The hands of welcome and generosity you extended to Muslims have been spat on and bitten. Your sacred traditions, customs and observances including your places of worship have been attacked, mocked and defiled by your Muslim guests and you turn away, indifferent. Even your very monarchy suffers threat and derision and you hang your heads and look away"………
……"A letter authored by a collection of self-loathing Liberals who grovel at the feet of the Muslim student activists who stalk the halls of our universities smearing the excrement of Islamic fundamentalism on everything within their reach, demanding immediate deference and capitulation to their every Islamic demand and whim.
A letter authored by those academic social engineers and university administrators who force and harass normal students and faculty into validating the mental illness and sexual depravity of a few shrieking, primping, preening "trans" activists who scamper about the university campuses extolling the joys and benefits of their pending publicly funded genital renovations.
A letter authored by nothing more than a collection of academic parasites who, for the past 3 decades, have squandered literally billions of dollars in taxpayer funding in their destruction of the very foundations of sound academic institutions and in so doing, ravaged and corrupted the minds and characters of our children and placed the stewardship of our nations in the hands of the weak and the mentally ill.
A letter authored by "women and men of colour" whose hyphenated terms of their cultural origin and racial identification betray them for that collection of whining, publicly funded, grifters they are, grifters who vomit their derision on the brilliant success of Western societies all the while ignoring the fact that their very own racial and cultural origins hail from continents, cultures and countries that have been spectacular and amusing failures for millennia.
A letter authored by malevolent, murderous Leftists who celebrate the virtual toilets that are the majority of African and Middle Eastern countries, all the while attacking, at every opportunity, the greatest success story in the history of Mankind: civilized, secular, capitalist, democratic, enlightened Western societies that rest on a foundation of Anglo Saxon Protestant Judaeo Christianity"……
As for me, I'm sick and tired of apologizing for the colour of my skin and the fact that I am part of the greatest success story in the history of Mankind.
I am sick and tired of apologizing for the magnificent accomplishments of the West, from our advancements in medicine, to engineering, to science, to technology, to our love of the rule of law, to our capitalism and our love of entrepreneurship, and this, all of this, which has brought us to the point where we now stand on our tiptoes and reach into the blackness of space in an effort to answer those questions that have burdened us since the darkness was upon the waters, who are we?, where do we come from? where are we going? and quite simply, why? and to, as the poet said…."slip the surly bonds of earth, to dance the skies on laughter silvered wings, to reach out our hands and touch the face of God"
And if that, if all of that, makes me a "racist" or an "Islamophobe" or "intolerant" or "homophobic" or an "extremist", or a "nationalist", or "xenophobic", or "transphobic" , or a "Nazi", or a "hater", well then so be it, if those are the labels for my life, my character, my love of my country and my heritage, labels that fall from the mouths of the treasonous, the seditious and the self-loathing, then they are labels I wear with pride.
Sincerely, Don Laird
PS, by the way, pour a cup of coffee and spend 45 minutes with Jared Taylor. And of course, with all the truth Mr. Taylor speaks and the fact that I have posted this link, will make all of us Nazi's.
https://www.youtube.com/watch?v=Ol6GA4dMZw4
November 25, 2017 at 2:27 PM	Reply
Pat Saxton
I would agree with Don Laird on many points except his inclusion of capitalism as a virtue. If he had said free enterprise instead I could understand his argument. Capitalism is happier in totalitarian countries like the fascists of the second world war, or the many dictators of African and South America and now China. Capitalism leads to monopoly in every historical situation. We in the western democracies decided to limit the ability of monopolies to overwhelm our democracies, while allowing free enterprise to flourish. That is a balancing act that we are may be losing. It is the middle class that pays for and protects our western institutions. They are under attack from the far left and far right, similar to what happened in Germany.
I also read mein kampf recently because my granddaughter wanted to know why the Nazis murdered the Jews. You have to understand something before you can critic it. That motto is now lost in our universities. Like my beloved CBC they are unwilling to understand anything or see another view point without labeling it. Don't offend anyone who is not in the majority since everything is the majority's fault. I don't consider them leftists. Tommy Douglas was NDP and he would not have tolerated the censorship that is going on in the media and our universities.
Most of the immigrants entering Canada are not supporters of the far right or left. I have worked with enough to know that most of them are here because of the middle class values of peace, order and relatively good government that we created in this imperfect but great country. We should be able to speak out against anyone who brings values contrary to our hard won secular society. That would mean excluding people who have values contrary to secularism and want to push our values aside. Keep your religion in your church or house. Don't bring it to the public sphere. Quebec has made that a law and had more balls then the rest of Canada.
The anger that many of us feel is because too many hateful and mindless voices are given free reign because they are so called minorities. Though most of them are more privileged then we are. We have to fight back each time we are attacked, similar to what was done in the Halifax paper two weeks ago, where a well reasoned argument was written to point out that racism can be both ways. The article addressed an ugly rant by a privileged so called minority student who hated white people.
Therefore I would agree that our voices are being silent by part of the media, and other valid points of view are suppressed because they come from a 'majority' person. Let the others rant while we support those that speak for a realistic view of what is going on in the world.
I need to make clear to the Thought Police and one or two concerned readers that the authors of Barrelstrength do not agree to or will countenance totalitarian doctrines or measures of any kind: whether predicated on racial, economic, social class, national, tribal or any other form of (pseudo) justification. To cite the motto of a newspaper I revile: the Chief Magistrate will neither advise nor consent to arbitrary measures.
Dear Msr. Dalwinnie
Do go on……….seriously…….expand and expound……..a few more paragraphs will not hurt…….
Sincerely Don Laird
About racism… it's a scary topic to discuss, and it's what is used to vilify people who think government budgets should balance — at least most of the time. We call them the 'right wing', which is what we also call the leader of the National Socialist German Workers' Party.
The problem is that racism was taught in the universities of the world as the next great idea for improving humanity. Evolution, as a theory, had itself evolved into the eugenics movement. Planned Parenthood, in the USA, was started by a first wave feminist, and was aimed at reducing the number of black children being produced. The sainted Tommy Douglas wrote a sociology MA thesis in which he located two and three generation welfare families, and traced their family trees to document the social pathology of the whole group. Eugenic thinking became mainstream after 1925, and was considered to be "progressive" thinking. It may surprise people, but the US federal civil service wasn't racially segregated until Woodrow Wilson became President — the first Democrat in a long time.
The take-away point is that what would today be called extreme racism that nobody would find tolerable was a major part of "progressive" thinking — until Hitler. When Hitler picked up on those ideas and got involved in a lot of racial purity stuff, those ideas were academically certified!
And academics have been running away from that ever since. Now we have a anti-racism hysteria centred on campuses … as well as a 'gender' hysteria, modelled on the treatment the ex-slaves got. It has become a redemptive racism, aimed at white males. And men in Canada are too gutless to face the world they are leaving their sons.
It's gone too far. Consider this: the modern mood is negative towards our own culture. Life is felt to be purposeless and everything is merely arbitrary and oppressive. It is felt that what gives people meaningful lives has something to do with their narratives, which suggests all we need to do is create a new cultural mythology. But this is the contradiction: the real narratives of our lives are history, and it involves racial groups, right down to families. We have lost this sense of family history, or 'belonging' to something bigger, because our kinship system has been obliterated by the state. But any revival of narratives will almost certainly involve racial and national narratives. Think on that!
That's just my point of view.
I agree with most of your statements since the belief in Eugenics was wide spread. However you are making the same argument that those people attacking our society are making. We will ever be responsible for the dark side of our history which exempts the others from taking responsibility for theirs. Whatever good was done by people like Tommy Douglas is ignored since we find something bad and peripheral to his achievements. History used properly is not a myth building tool, it is a tool to understand how we got here and what are the consequences of our historically created beliefs. My point was that the left in the past was in the vanguard to protect freedom of speech. It was not to justify everything they believed in the past.
What is going on in universities and the media today is a consequences of dumbing down of critical thought by accepting a minority's assessment without critical review. In universities it started in the 1970s (with the acceptance that all students can't fall because it is hard on them) to where we are now: students will be traumatized if they hear contrary arguments. So not only do they not have to study, they also don't have to think.
The media in its push to be multicultural has given up its role of watchdog against hypocrisy and has accepted that it is racist to challenge minority beliefs or statements. That leaves a lot of Canadians who think, with a skeptical view of both institutions.
December 6, 2017 at 10:49 AM	Reply


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Exelon Recognized for Industry-Leading Sustainability Performance for Eleventh Year in a Row
Company added 558 megawatts of clean energy last year, including new wind and solar
CHICAGO — In recognition of its industry-leading corporate sustainability performance, Exelon Corporation (NYSE: EXC) has been named to the 2016 Dow Jones Sustainability Index (DJSI) for North America, marking the eleventh consecutive year Exelon has appeared on the index.
Exelon was one of seven electric and gas companies to be named to the DJSI for North America out of 600 total companies considered. Exelon ranked particularly high among peers in the areas of corporate governance, market opportunities and environmental policy and management systems. The independent DJSI assessment is conducted each year by sustainability investment specialist RobecoSAM. It is based on a comprehensive review of environmental performance, health and safety, corporate citizenship, corporate governance, risk management and human resources development.
"We're proud to have been named to the Dow Jones Sustainability Index for the eleventh year, demonstrating our long-term commitment to the ideals it represents," said Chris Crane, president and CEO of Exelon "This recognition validates our continuing efforts to incorporate clean energy, efficiency and smart grid technology as part of our strategy to build a sustainable and reliable next-generation energy grid."
Continuing to make progress toward its sustainability goals, Exelon in 2015:
Invested $3.7 billion in electric and gas utility infrastructure to make the grid smarter, more reliable and more resilient;
Invested $1.1 billion to maintain the safety and reliability of one of the cleanest, lowest cost generation fleets in the nation;
Increased current and future investment in renewable generation, with our wind portfolio now totaling 1,500 megawatts across 10 states;
Helped utility customers save more than 10.5 million megawatt-hours of electricity through energy efficiency programs;
Joined with CB&I and 8 Rivers Capital to build NET Power's first-of-its-kind demonstration power plant that produces zero atmospheric emissions;
Grew Constellation Technology Venture investments in electric vehicles, local generation, energy storage, renewable generation and intelligent buildings;
Enhanced corporate citizenship, with employees increasing charitable giving by 12 percent over the prior year and volunteering nearly 130,000 hours in community service;
Gave more than $36 million in corporate contributions to community nonprofits
Spent more than $1.3 billion with diversity-certified suppliers, far surpassing 2014 spend;
Achieved first-quartile performance among peers for the length and duration of power outages at our utilities;
Increased companywide recycling rate to 74 percent, the highest ever.
Exelon uses rigorous performance and risk management programs to track its sustainability performance against specific, quantifiable goals while accounting for a wide variety of factors, from changes in the prices of electricity and natural gas to the risks associated with climate change and drought.
Exelon provides a detailed review of its sustainability strategy and performance in its annual sustainability report. The 2015 report is available at: http://www.exeloncorp.com/sustainability/interactive-csr
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