diff --git a/README.md b/README.md index 743d8d5..57a2675 100644 --- a/README.md +++ b/README.md @@ -1,6 +1,6 @@ Paper repository for _Constraining gravitational wave amplitude birefringence with GWTC-3_ by Ng, Isi, Wong & Farr (2023). -The paper can be reproduced programmatically from this repo using [_showyourwork_](http://show-your.work). Scripts to produce figures and tables can be found in [src/scripts](src/scripts), and data in [zenodo](https://zenodo.org/record/7935107). +The paper can be reproduced programmatically from this repo using [_showyourwork_](http://show-your.work). Scripts to produce figures and tables can be found in [src/scripts](src/scripts), and data in [Zenodo](https://zenodo.org/record/7935107).

diff --git a/Snakefile b/Snakefile index 2872a8a..fb55ea2 100644 --- a/Snakefile +++ b/Snakefile @@ -27,6 +27,7 @@ rule compute_kappa_constraint_restricted: "src/tex/output/improvement_Okounkova.txt", "src/tex/output/restricted_absolute_kappa_90.txt", "src/tex/output/kappa_Wang.txt", + "src/tex/output/kappa_Zhu.txt", "src/tex/output/comparison_summary.txt" script: "src/scripts/kappa_constraint_restricted.py" diff --git a/showyourwork.yml b/showyourwork.yml index f8b0c00..349d768 100644 --- a/showyourwork.yml +++ b/showyourwork.yml @@ -53,11 +53,26 @@ dependencies: - src/data/GW191105_143521_GR.json.gz - src/data/samples_posterior_birefringence.feather + src/scripts/posterior_MPV.py: + - src/data/samples_posterior_birefringence.feather + src/tex/ms.tex: - src/tex/output/mu_median.txt - src/tex/output/sigma_median.txt - - src/tex/output/restricted_kappa_median.txt - src/tex/output/generic_kappa_median.txt + - src/tex/output/M_PV_constraint.txt + - src/tex/output/restricted_kappa_median.txt + - src/tex/output/CL_kappa_0.txt + - src/tex/output/restricted_absolute_kappa_68.txt + - src/tex/output/improvement_Okounkova.txt + - src/tex/output/restricted_absolute_kappa_90.txt + - src/tex/output/kappa_Wang.txt + - src/tex/output/kappa_Zhu.txt + - src/tex/output/comparison_summary.txt + - src/tex/output/best_events_kappa.txt + - src/tex/output/GW170818_constraint.txt + - src/tex/output/GW200129_constraint.txt + - src/tex/output/bimodal_events_mass.txt - src/tex/bib.bib datasets: diff --git a/src/scripts/birefringence.py b/src/scripts/birefringence.py index adc04dd..c360bc0 100644 --- a/src/scripts/birefringence.py +++ b/src/scripts/birefringence.py @@ -18,7 +18,7 @@ # frequency array parameters df = 0.125 f_min = 20 -f_max = 2048 +f_max = 2048*16 # using higher frequency to produce smoother TD waveform f_ref = 100 # source parameters @@ -81,7 +81,7 @@ axs[0].loglog(freq, np.abs(hr_fd), ls=':', c=c2, label=r"R (GR)") axs[0].loglog(freq, np.abs(hl_bi_fd), c=c1, label=r"L (BR)") axs[0].loglog(freq, np.abs(hr_bi_fd), c=c2, label=r"R (BR)") -axs[0].set_xlim(f_min+1,f_max-22) +axs[0].set_xlim(f_min+1,2048-22) # force x_max to the max value for IMRPhenomD axs[0].set_ylim(1e-30,1e-20) axs[0].legend() axs[0].set_ylabel(r"$|\tilde{h}|$") diff --git a/src/scripts/kappa_constraint_restricted.py b/src/scripts/kappa_constraint_restricted.py index 26c6ff4..a1d75d4 100644 --- a/src/scripts/kappa_constraint_restricted.py +++ b/src/scripts/kappa_constraint_restricted.py @@ -1,12 +1,13 @@ import numpy as np import pandas as pd +from bilby.gw.cosmology import get_cosmology import scipy import paths # constants h = 4.135667696e-15 # eV s c = 299792.458 # km/s -H_0 = 68.3 # km/s/Mpc +H_0 = get_cosmology().H0.value # km/s/Mpc result_dict = {} result_DataFrame = pd.read_feather(paths.data/"samples_posterior_birefringence.feather") @@ -49,13 +50,12 @@ with open(paths.output/"restricted_absolute_kappa_90.txt", "w") as f: f.write(f"$|\kappa| < {restricted_absolute_kappa_90:.2f} \\, \\mathrm{{Gpc}}^{{-1}}$") -# Wang's constraint in kappa +# Constraint comparison M_PV_Wang = 1e-22 # GeV kappa_Wang = (h*np.pi*H_0/c)*(1e3)*(100)*np.reciprocal(M_PV_Wang*1e9) with open(paths.output/"kappa_Wang.txt", "w") as f: f.write(f"$|\kappa| \lesssim {kappa_Wang:.2f} \, \mathrm{{Gpc}}^{{-1}}$") -# Constraint comparison kappa_Okounkova = 0.74 M_PV_Okounkova = (h*np.pi*H_0/c)*(1e3)*(100)*np.reciprocal(kappa_Okounkova)/1e9 @@ -64,6 +64,8 @@ M_PV_Zhu = 4.1e-22 # GeV kappa_Zhu = (h*np.pi*H_0/c)*(1e3)*(100)*np.reciprocal(M_PV_Zhu*1e9) +with open(paths.output/"kappa_Zhu.txt", "w") as f: + f.write(f"$|\kappa| \lesssim {kappa_Zhu:.2f} \, \mathrm{{Gpc}}^{{-1}}$") with open(paths.output/"M_PV_constraint.txt", "w") as f: f.write(rf"$M_{{\rm PV}} \gtrsim {M_PV_this_work/1e-21:.1f} \times 10^{{-21}}\, {{\rm GeV}}$") diff --git a/src/scripts/posterior_MPV.py b/src/scripts/posterior_MPV.py new file mode 100644 index 0000000..265cd51 --- /dev/null +++ b/src/scripts/posterior_MPV.py @@ -0,0 +1,41 @@ +import matplotlib.pyplot as plt +import matplotlib as mpl +import pandas as pd +import numpy as np +import seaborn as sns +from scipy.stats import gaussian_kde +from bilby.gw.cosmology import get_cosmology +import paths + +# constants +h = 4.135667696e-15 # eV s +c = 299792.458 # km/s +H_0 = get_cosmology().H0.value # km/s/Mpc + +sns.set_theme(palette='colorblind', font_scale=1.2) + +plt.rcParams.update({ + "text.usetex": True, + "font.family": "serif", + "font.serif": ["Computer Modern Roman"] +}) + +result_DataFrame = pd.read_feather(paths.data/"samples_posterior_birefringence.feather") +events = result_DataFrame['event'].unique() + +M_PV_inv = np.linspace(0, 1.5e21, 1000) +kappa = (h*np.pi*H_0/c)*(1e3)*(100)*(M_PV_inv)/1e9 + +chosen_events = ['GW190727_060333', 'GW200112_155838', 'GW190513_205428', 'GW190915_235702', 'GW190803_022701', 'GW200311_115853'] + +kernel_dict = {} +for event in chosen_events: + kernel_dict[event] = gaussian_kde(result_DataFrame[result_DataFrame.event == event]['kappa']) + plt.plot(M_PV_inv, kernel_dict[event](kappa)+kernel_dict[event](-kappa), label=event) + +plt.legend(fontsize=14) +plt.xlim(0, 1.5e21) +plt.ylabel(r'$p\left(M_\mathrm{PV}^{-1}\right)$') +plt.xlabel(r'$M_\mathrm{PV}^{-1}\, \left(\mathrm{GeV^{-1}}\right)$') + +plt.savefig(fname=paths.figures/"posterior_MPV.pdf", bbox_inches="tight", dpi=300) diff --git a/src/scripts/reweighed_kappa.py b/src/scripts/reweighed_kappa.py index 9b8d20b..3cc0582 100644 --- a/src/scripts/reweighed_kappa.py +++ b/src/scripts/reweighed_kappa.py @@ -33,8 +33,9 @@ norm_alpha = matplotlib.colors.Normalize(vmin=-1E-3, vmax=zscores.max()*1.1) for i, (z,line) in enumerate(zip(zscores, g.get_lines()[::-1])): line.set_alpha(norm_alpha(z)) - if i < 3: + if zscores.index[i]=='GW170818': line.set_label(zscores.index[i].replace('_', r'\_')) + line.set_linestyle('--') else: line.set_label(None) g.legend(fontsize=14) diff --git a/src/tex/bib.bib b/src/tex/bib.bib index 9d3b194..9fa6054 100644 --- a/src/tex/bib.bib +++ b/src/tex/bib.bib @@ -253,6 +253,22 @@ @dataset{GWTC-2.1_dataset url = {https://doi.org/10.5281/zenodo.6513631} } +@article{LIGOScientific:2020tif, + author = "Abbott, R. and others", + collaboration = "LIGO Scientific, Virgo", + title = "{Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog}", + eprint = "2010.14529", + archivePrefix = "arXiv", + primaryClass = "gr-qc", + reportNumber = "LIGO-P2000091", + doi = "10.1103/PhysRevD.103.122002", + journal = "Phys. Rev. D", + volume = "103", + number = "12", + pages = "122002", + year = "2021" +} + @dataset{GWTC-3_dataset, author = {{LIGO Scientific Collaboration} and {Virgo Collaboration} and {KAGRA Collaboration}}, title = {{GWTC-3: Compact Binary Coalescences Observed by @@ -809,3 +825,75 @@ @ARTICLE{Yunes2009 adsnote = {Provided by the SAO/NASA Astrophysics Data System} } +@article{Wang:2021gqm, + author = "Wang, Yi-Fan and Brown, Stephanie M. and Shao, Lijing and Zhao, Wen", + title = "{Tests of gravitational-wave birefringence with the open gravitational-wave catalog}", + eprint = "2109.09718", + archivePrefix = "arXiv", + primaryClass = "astro-ph.HE", + doi = "10.1103/PhysRevD.106.084005", + journal = "Phys. Rev. D", + volume = "106", + number = "8", + pages = "084005", + year = "2022" +} + +@article{Haegel:2022ymk, + author = {Haegel, Le\"\i{}la and O'Neal-Ault, Kellie and Bailey, Quentin G. and Tasson, Jay D. and Bloom, Malachy and Shao, Lijing}, + title = "{Search for anisotropic, birefringent spacetime-symmetry breaking in gravitational wave propagation from GWTC-3}", + eprint = "2210.04481", + archivePrefix = "arXiv", + primaryClass = "gr-qc", + doi = "10.1103/PhysRevD.107.064031", + journal = "Phys. Rev. D", + volume = "107", + number = "6", + pages = "064031", + year = "2023" +} + +@article{Jenks:2023pmk, + author = "Jenks, Leah and Choi, Lyla and Lagos, Macarena and Yunes, Nicol\'as", + title = "{Parameterized Parity Violation in Gravitational Wave Propagation}", + eprint = "2305.10478", + archivePrefix = "arXiv", + primaryClass = "gr-qc", + month = "5", + year = "2023" +} + +@article{Chia:2020psj, + author = "Chia, Horng Sheng and Edwards, Thomas D. P.", + title = "{Searching for General Binary Inspirals with Gravitational Waves}", + eprint = "2004.06729", + archivePrefix = "arXiv", + primaryClass = "astro-ph.HE", + doi = "10.1088/1475-7516/2020/11/033", + journal = "JCAP", + volume = "11", + pages = "033", + year = "2020" +} + +@unpublished{tgrsel}, + author = {Magee, R and Isi, Max and Chatziioannou, Katerina and Vitale, Salvatore and Farr, Will and Pratten, Geraint}, + title = {Selection biases in tests of general relativity with gravitational waves}, + year = {in prep.} +} + +@article{LIGOScientific:2019zcs, + author = "Abbott, B. P. and others", + collaboration = "LIGO Scientific, Virgo, VIRGO", + title = "{A Gravitational-wave Measurement of the Hubble Constant Following the Second Observing Run of Advanced LIGO and Virgo}", + eprint = "1908.06060", + archivePrefix = "arXiv", + primaryClass = "astro-ph.CO", + reportNumber = "LIGO-P1900015", + doi = "10.3847/1538-4357/abdcb7", + journal = "Astrophys. J.", + volume = "909", + number = "2", + pages = "218", + year = "2021" +} \ No newline at end of file diff --git a/src/tex/ms.tex b/src/tex/ms.tex index b360133..79a0c36 100644 --- a/src/tex/ms.tex +++ b/src/tex/ms.tex @@ -4,6 +4,7 @@ \usepackage{amsfonts,amssymb,amsmath} \usepackage[nolist,nohyperlinks]{acronym} \usepackage{bookmark} +\usepackage[final]{changes} %\usepackage[all]{hypcap} \newcommand{\infd}{\mathrm{d}} @@ -11,6 +12,8 @@ \newcommand*{\mi}[1]{\textsf{\color{magenta} [\textbf{MAX:} #1]}} \newcommand*{\wf}[1]{\textsf{\color{cyan} [\textbf{WILL:} #1]}} +\definechangesauthor[name=Will, color=cyan]{WF} + \begin{document} \title{Constraining gravitational wave amplitude birefringence with GWTC-3} @@ -73,7 +76,8 @@ \section{Introduction} % What's new? In this study, we use a frequency-dependent birefringence model to constrain the strength of \ac{GW} amplitude birefringence by performing \ac{PE} on \ac{LVK} binaries. This model is a better approximation of the birefringence effect expected from theory than the frequency-independent model used in \citet{Okounkova_2022}. -Compared to other studies, we perform \ac{PE} on more events, including events new to GWTC-3 \citep{GWTC-3}, and use a phenomenology-oriented parametrization. +\added{Including the frequency dependence allows us to break the degeneracy between birefringence and source inclination, which we discuss below in Sec.~\ref{sec:inclination}.} +Compared to \replaced{Refs.~\citep{Yamada_2020,Wang_2021,Okounkova_2022}}{other studies}, we perform \ac{PE} on more events, including events new to GWTC-3 \citep{GWTC-3}, and use a phenomenology-oriented parametrization. We consider 71 binary black hole merger events with a \ac{FAR} $\leq1/\mathrm{yr}$, as listed in Table I of \citet{GWTC-3_population}. We discuss single-event results in detail, and identify degeneracies between birefringence and spin effects, in addition to the already known correlations with source orientation and distance. We use the results from individual events to place a collective population constraint on the strength of \ac{GW} amplitude birefringence from GWTC-3. @@ -106,15 +110,17 @@ \subsection{Birefringence} The latter can manifest in changes to the relative amplitude and phase of the polarizations that accrue as the wave propagates, giving us hope of detecting initially small effects that compound over long propagation distances. In particular, \emph{amplitude} birefringence would enhance one polarization mode over the other. -To first order, in theories like Chern-Simons gravity, the Fourier-domain waveform observed a comoving distance $d_C$ away from the source can be written as +\added{Following \cite{Alexander:2009tp}, the effect of birefringence can be modeled as a frequency-dependent amplification or dampening; similar derivations can be found in \cite{Zhao:2019xmm, Ezquiaga:2021ler, Zhu:2023wci,Jenks:2023pmk}.} +To first order \added{in the equations of motion}, in theories like Chern-Simons gravity, the Fourier-domain waveform observed a comoving distance $d_C$ away from the source can be written as \begin{equation} h_{L/R}^{\mathrm{br}}(f) = h_{L/R}^{\mathrm{GR}}(f) \times - \exp\left(\pm\kappa d_C \frac{f}{100\,\mathrm{Hz}}\right)\,, + \exp\left(\pm\kappa\, d_C \frac{f}{100\,\mathrm{Hz}}\right)\,, \label{eq:waveform_modification} \end{equation} where the emitted waveform $h_{L/R}^{\mathrm{GR}}$ is modified by an exponential birefringent factor to yield the observed waveform $h_{L/R}^{\mathrm{br}}$. -The overall magnitude of this effect for a given frequency $f$ is set by an attenuation coefficient, $\kappa$, which encodes the intrinsic strength of the birefringence. +The overall magnitude of this effect for a given frequency $f$ is set by an attenuation coefficient, $\kappa$, which encodes the intrinsic strength of the birefringence: +\added{$\kappa^{-1}$ represents an ``attenuation length'' encoding the typical distance that yields an $e$-folding in the amplification or dampening of the polarizations at a fiducial signal frequency of 100 Hz at the detector.} The emitted waveform for a given source (i.e., the waveform observed in the near zone, very close to the source) will generally differ from the analogous waveform predicted by \ac{GR} \cite{Alexander:2009tp,Okounkova:2019zjf}; however, since we expect most viable modifications to \ac{GR} to be intrinsically small (e.g., \cite{Okounkova:2022grv}), it is standard to approximate the emitted waveform by the prediction from \ac{GR} (hence the notation ``$h^{\rm GR}$'' above). Although the intrinsic modification is small, the effect targeted by Eq.~\eqref{eq:waveform_modification} accumulates as the \ac{GW} propagates. @@ -124,15 +130,15 @@ \subsection{Birefringence} Equation \eqref{eq:waveform_modification} can be derived as the first order effect in an expansion away from \ac{GR} under multiple frameworks. In general, $\kappa$ will be a function of the theory parameters and the cosmological history, e.g., the value of the pseudo-scalar field and its derivative in Chern-Simons gravity \cite{Alexander:2009tp}. -Since it originates from a truncated series expansion% -\footnote{Often the birefringent effect is written as an expansion in redshift, $z$ \cite[e.g.][]{Zhao:2019xmm}, rather than distance. Here we use $d_C$ to emphasize that the effect accumulates \emph{per cycle}, or with propagation distance. Thus, current ground-based experiments, which can detect gravitational waves to $z \simeq 1$, $d_C \simeq d_H$ (Hubble distance), are already observing propagation over a large fraction of the universe and constrain this effect meaningfully.}, % -Eq.~\eqref{eq:waveform_modification} is a good approximation only for small exponents, +Since it originates from a truncated series expansion,% +\footnote{Often the birefringent effect is written as an expansion in redshift, $z$ \cite[e.g.][]{Zhao:2019xmm}, rather than distance. Here we use $d_C$ to emphasize that the effect accumulates \emph{per cycle}, or with propagation distance. Thus, current ground-based experiments, which can detect gravitational waves to $z \simeq 1$, $d_C \simeq d_H$ (Hubble distance), are already observing propagation over a large fraction of the universe and constrain this effect meaningfully.} % +Eq.~\eqref{eq:waveform_modification} is a good approximation only for small exponents, namely \begin{equation} -\left|\kappa\right| d_C \left(f/100\, \mathrm{Hz}\right) < 1\,, + \left(\frac{\left|\kappa\right|}{\added{\mathrm{Gpc}^{-1}}}\right) \left(\frac{d_c}{\added{\mathrm{Gpc}}}\right) \left(\frac{f}{100\, \mathrm{Hz}}\right) < 1\,, + \label{eq:small_exponent} \end{equation} -note that $d_C$ is in units of Gpc. -Otherwise, more frequency-dependent terms would enter the exponent of Eq.~\eqref{eq:waveform_modification} in a theory-dependent way. - +recalling that $\kappa$ has dimensions of inverse length. +Otherwise, more frequency-dependent terms could enter the exponent of Eq.~\eqref{eq:waveform_modification} in a theory-dependent way. \begin{figure} \script{birefringence.py} \includegraphics[width=\columnwidth]{figures/birefringence.pdf} @@ -201,12 +207,14 @@ \section{Method} \subsection{Single-event parameter estimation} To constrain birefringence, we reanalyze events from GWTC-3 \citep{GWTC-2.1, GWTC-3} implementing Eq.~\eqref{eq:waveform_modification} to directly obtain a posterior on $\kappa$ from the strain of each event. -We analyze the 71 \acp{BBH} that were detected with $\mathrm{FAR} < 1/\mathrm{yr}$; to avoid extended computations on longer signals and considering these are generally at closer distances, we do not analyze systems involving neutron stars in this work. +We analyze the 71 \acp{BBH} that were detected with $\mathrm{FAR} < 1/\mathrm{yr}$\added{, less stringent than the typical \ac{LVK} threshold of $1/1000\,\mathrm{yr}$ \cite{LIGOScientific:2020tif,LIGOScientific:2021sio}. +The \ac{FAR} values are typically determined by \ac{GR} pipelines, which could down-rank signals beyond \ac{GR} \cite{LIGOScientific:2020tif,Chia:2020psj,tgrsel}; however, since detectability through matched-filtering is most sensitive to the phase, not the amplitude, we expect only a minor decrease in sensitivity to the kind of birefringent signals explored here.} +To avoid extended computations on longer signals and considering these are generally at closer distances, we do not analyze systems involving neutron stars in this work. We procure strain data from \ac{GWOSC} \citep{GWOSC}. We estimate source parameters using a custom version of the \textsc{Bilby} software \citep{Bilby}, modified from the baseline version to apply Eq.~\eqref{eq:waveform_modification} for any \ac{GR} baseline waveform. We take the \ac{PE} configuration in \citep{GWTC-2.1, GWTC-3, GWTC-2.1_dataset, GWTC-3_dataset} as a starting point, with \textsc{IMRPhenomXPHM} \citep{Pratten:2020ceb} as the reference waveform. -We apply a distance prior corresponding to a uniform distribution over comoving volume and source-frame time, and set the prior on $\kappa$ to be uniform between $-1 \, \mathrm{Gpc}^{-1}$ and $1 \, \mathrm{Gpc}^{-1}$. +We apply a distance prior corresponding to a uniform distribution over comoving volume and source-frame time \added{(see, e.g., Eq.~10 in \cite{LIGOScientific:2019zcs})}, and set the prior on $\kappa$ to be uniform between $-1 \, \mathrm{Gpc}^{-1}$ and $1 \, \mathrm{Gpc}^{-1}$. For GW190521, we increase the maximum distance allowed by the prior to $1.5\times$ the original value in \cite{GWTC-2.1_dataset}, as the birefringence effect results in posterior support at larger distances. For GW190720, we decrease the analysis segment from 16 s to 8 s, in order to accommodate missing data near the edges of the 16 s segment in Virgo. @@ -273,7 +281,7 @@ \section{Results} \label{sec:Results} In this section, we present the results of our study. -We first show the $\kappa$ measurements of all events in our set, as well as the resulting global measurement of $\kappa$ that represents our primary constraint on birefringence (Sec.~\ref{sec:results:gwtc}). +We first show the $\kappa$ measurements from \deleted{of} all events in our set, as well as the resulting global measurement of $\kappa$ that represents our primary constraint on birefringence (Sec.~\ref{sec:results:gwtc}). We then assess the collection of measurements in more detail through a hierarchical analysis (Sec.~\ref{sec:results:hier}). Finally, we discuss some special events individually, and outline the degeneracies that arise between birefringence and orbital precession (Sec.~\ref{sec:results:notable}). @@ -297,6 +305,7 @@ \subsection{GWTC-3 result} Figure \ref{fig:violin_kappa} makes it clear that not all \acp{BBH} in GWTC-3 are equally informative about birefringence. When considered individually, the events that best constrain $\kappa$ are listed in Table~\ref{tab:best_events_kappa}, in order of increasing standard deviation $\sigma_i$. That table also shows the credible level (CL) at which the posterior supports $\kappa = 0$, whereby $\mathrm{CL} = 0$ ($\mathrm{CL} = 1$) means the posterior supports that value with high (low) probability. +\added{This quantity represents the relative height of the probability density function at $\kappa = 0$, and is not tied to a symmetric interval, making it particularly useful in assessing bimodal posteriors.} Judging by $\mu_i/\sigma_i$, the two events that show the largest tension with $\kappa = 0$ are GW170818, for which \variable{output/GW170818_constraint.txt}, and GW200129\_065458 (henceforth GW200129), for which \variable{output/GW200129_constraint.txt}. However, as we discuss in Sec.~\ref{sec:GW200129}, we have reason to think that the preference for $\kappa < 0$ in GW200129 might be driven by noise anomalies in the Virgo detector; with that in mind, in the next section we consider the effect of excluding this event from the joint result (we find its impact to be minimal). @@ -352,7 +361,7 @@ \subsection{Hierarchical modeling} \includegraphics[width=\columnwidth]{figures/reweighed_kappa.pdf} \caption{ Individual-event $\kappa_i$ distributions of Fig.~\ref{fig:violin_kappa} reweighted in light of the population-level inference of Fig.~\ref{fig:corner_Gaussian}, marginalizing over $\mu$ and $\sigma$. - Curves are colored by the magnitude of the supported deviation, as represented by the respective posterior $|\mu_i / \sigma_i|$; we label the events most in tension with $\kappa_i = 0$ by that same measure. + Curves are colored by the magnitude of the supported deviation, as represented by the respective posterior $|\mu_i / \sigma_i|$. } \label{fig:reweighted_kappa} \end{figure} @@ -381,7 +390,9 @@ \subsection{Hierarchical modeling} \includegraphics[width=\columnwidth]{figures/posterior_kappa.pdf} \caption{ Restricted and generic posteriors on $\kappa$, which respectively do and do not assume that $\kappa$ is shared by all events (color). - Solid and dashed traces indicate whether the analysis included GW200129 or not. + \replaced{% + Solid (dashed) traces indicate the analysis included (excluded) GW200129.} + {Solid and dashed traces indicate whether the analysis included GW200129 or not.} The black dashed line marks the absence of birefringence ($\kappa=0$). The shaded distribution is the primary result in this work (blue band in Fig.~\ref{fig:violin_kappa}). } @@ -405,8 +416,7 @@ \subsubsection{GW170818} \includegraphics[width=\columnwidth]{figures/corner_GW170818.pdf} \caption{ GW170818 posterior on $\kappa$, luminosity distance $d_L$ and inclination $\cos\iota$ from our birefringence analysis (blue), compared to the GR result (orange). - The top right panel shows the marginalized posterior on $\chi_p$: allowing for birefringence reduces the preference for precession. - (See Fig.~\ref{fig:corner_GW170818_appendix} for a full corner plot.) + The top right panel shows the marginalized posterior on $\chi_p$: allowing for birefringence reduces the preference for precession. (See Fig.~\ref{fig:corner_GW170818_appendix} for a full corner plot.) } \label{fig:corner_GW170818} \end{figure} @@ -460,7 +470,8 @@ \subsubsection{GW190521} \includegraphics[width=\columnwidth]{figures/corner_GW190521.pdf} \caption{ GW190521 posterior on $\kappa$, luminosity distance $d_L$ and inclination $\cos\iota$ from our birefringence analysis (blue), compared to the GR result (orange). - The marginalized posterior on $\chi_p$ is shown in the top right panel. + The marginalized posterior on $\chi_p$ is shown in the top right panel. (See Fig.~\ref{fig:corner_GW190521_appendix} for a full corner plot.) + \added{Note that, although we show $d_L$ in this plot, our definition of $\kappa$ is associated with $d_C$, meaning that the condition in Eq.~\eqref{eq:small_exponent} is satisfied even for the largest values of $d_L$ supported in this posterior.} } \label{fig:corner_GW190521} \end{figure} @@ -570,7 +581,7 @@ \subsubsection{Okounkova et al.} On the other hand, our constraint from Sec.~\ref{sec:results:gwtc} is \variable{output/restricted_absolute_kappa_68.txt} at $100 \, \mathrm{Hz}$ at 68\% credibility. This is a factor of \variable{output/improvement_Okounkova.txt}more stringent than Ref.~\cite{Okounkova_2022}, besides arising from a less simplified model. -The result of \citet{Okounkova_2022} is also phrased in terms of a canonical Chern-Simons length scale,\footnote{This results from setting the so-called ``canonical'' Chern-Simons embedding into a FRW cosmology, where the Chern-Simons field evolves as $\theta \propto t$ \cite{Alexander:2009tp,Jackiw:2003pm,Yunes2009}.} +The result of \citet{Okounkova_2022} is also phrased in terms of a canonical Chern-Simons length scale\footnote{This results from setting the so-called ``canonical'' Chern-Simons embedding into a FRW cosmology, where the Chern-Simons field evolves as $\theta \propto t$ \cite{Alexander:2009tp,Jackiw:2003pm,Yunes2009}.}, \begin{equation} l_0 = \frac{c d_H \kappa}{3 \pi f} = 1400 \, \mathrm{km} \left( \frac{\kappa}{1 \, \mathrm{Gpc}^{-1}} \right) \left( \frac{100 \, \mathrm{Hz}}{f} \right); \end{equation} @@ -578,6 +589,7 @@ \subsubsection{Okounkova et al.} \subsubsection{Wang et al.} +\label{sec:comparison_Wang} \citet{Wang_2021} performed \ac{PE} on GWTC-1 events with a frequency-dependent model of birefringence resembling ours. Following \cite{Zhao:2019xmm}, that reference parameterized the birefringent waveform in terms of some amplitude and phase modifications to the linear polarizations ($\delta h$ and $\delta \Psi$ respectively), such that @@ -591,18 +603,50 @@ \subsubsection{Wang et al.} &\approx h^{\rm GR}_{L/R}(f) \exp(\mp \delta h \mp i \delta \Psi) \, , \end{align} assuming a small $\delta h$ and $\delta \Psi$ in the last line. -To consider only amplitude birefringence, as in this work, we must compare to the result in Ref.~\citep{Wang_2021} that set $\delta \Psi = 0$ and parameterized $\delta h = \pi f z / M_{\rm PV}$, where $M_{\rm PV}$ is the energy scale of the birefringent (parity-violating) correction.% -\footnote{Concretely, \citet{Wang_2021} write $\delta h = - A_\nu \pi f$ with $A_\nu = M_{\rm PV}^{-1} \left[\alpha_\nu (z=0) - \alpha_\nu(z) \left(1+z\right)\right]$, for $\alpha_\nu$ some function of redshift encoding the evolution of a birefringence-mediating field; to produce their constraint, however, they further set $\alpha_\nu(z) = 1$, yielding $A_\nu = -z / M_{\rm PV}$ and hence $\delta h = \pi f z / M_{\rm PV}$.} -Comparing to our Eq.~\eqref{eq:waveform_modification} and approximating $z \approx d_C H_0/c$ via the Hubble constant, $H_0$, this means +To consider only amplitude birefringence, as in this work, we must compare to the result in Ref.~\citep{Wang_2021} that set $\delta \Psi = 0$ and parameterized $\delta h = \pi f z h_P / M_{\rm PV}$, where $h_P$ is Planck's constant and $M_{\rm PV}$ is the energy scale of the birefringent (parity-violating) correction.% +\footnote{Concretely, \citet{Wang_2021} write $\delta h = - A_\nu \pi f$ with $A_\nu = M_{\rm PV}^{-1} \left[\alpha_\nu (z=0) - \alpha_\nu(z) \left(1+z\right)\right]$, for $\alpha_\nu$ some function of redshift encoding the evolution of a birefringence-mediating field; to produce their constraint, however, they further set $\alpha_\nu(z) = 1$, yielding $A_\nu = -z / M_{\rm PV}$ and hence $\delta h = \pi f z h_P / M_{\rm PV}$, multiplying by $h_P$ to obtain the right dimensions.} +Comparing to our Eq.~\eqref{eq:waveform_modification} and approximating $z \approx d_C H_0/c$ via the Hubble constant, $H_0$, this means that \begin{equation} - |\kappa| \simeq \frac{\pi H_0}{c} \frac{h \left( 100 \, \mathrm{Hz} \right)}{M_{\rm PV}} \simeq \frac{3.0 \times 10^{-22} \, \mathrm{GeV}}{M_\mathrm{PV}} \, \mathrm{Gpc}^{-1}, + \label{eq:kappa-to-Mpv} + |\kappa| \simeq \frac{\pi H_0}{c} \frac{h_P \left( 100 \, \mathrm{Hz} \right)}{M_{\rm PV}} \simeq \frac{3.0 \times 10^{-22} \, \mathrm{GeV}}{M_\mathrm{PV}} \, \mathrm{Gpc}^{-1} . \end{equation} -where $h$ is Planck's constant. \citet{Wang_2021} quote a constraint of $M_{\rm PV} > 10^{-22}\, {\rm GeV}$ at 90\% credibility, which translates into \variable{output/kappa_Wang.txt}. We obtained a tighter 90\% upper limit of \variable{output/restricted_absolute_kappa_90.txt}, or \variable{output/M_PV_constraint.txt}. -The parameterization of \citet{Wang_2021} is the same as that in the more recent work by \citet{Zhu:2023wci}. -That work, which appeared while this manuscript was finalized, reported a constraint of $M_{\rm PV} > 4.1 \times 10^{-22}$ based on GWTC-3. +\deleted{The parameterization of \cite{Wang_2021} is the same as that in the more recent work by \cite{Zhu:2023wci}. +That work, which appeared while this manuscript was finalized, reported a constraint of $M_{\rm PV} > 4.1 \times 10^{-22}$ based on GWTC-3.} + +\subsubsection{\added{Zhu et al.}} +\label{sec:comparison_Zhu} + +A more recent work by \citet{Zhu:2023wci}, which appeared while this manuscript was finalized, performed \ac{PE} on GWTC-3 events with a frequency-dependent model of amplitude birefringence. +Their model is the same as the one used in \citet{Wang_2021}, which allows us to compare to our results as we did in Sec.~\ref{sec:comparison_Wang}. +\citet{Zhu:2023wci} reported a constraint of $M_{\rm PV} > 4.1 \times 10^{-22}\, {\rm GeV}$ at 90\% credibility, which translates into \variable{output/kappa_Zhu.txt} per Eq.~\eqref{eq:kappa-to-Mpv} above. + +This constraint is weaker than ours in spite of the similar number of events considered, a discrepancy that we trace back to inference at the individual-event level. +Figure \ref{fig:posterior_MPV} shows the posteriors on $M_\mathrm{PV}^{-1}$ derived from our analysis via Eq.~\eqref{eq:kappa-to-Mpv} for each event highlighted in Fig.~1 of \citet{Zhu:2023wci}. +Our posteriors differ from those in \citet{Zhu:2023wci} in both overall scale and specific shape. +For example, the posterior for GW190727\_060333 peaks at zero in our analysis, but peaks at a nonzero value in \citet{Zhu:2023wci}. +All distributions are narrower in our analysis by a factor of ${\sim}10\times$. +% However, we could not compare the analysis in \citet{Zhu:2023wci} with ours in detail, as their analysis result is not publicly available. +This might be explained by a number of analysis differences. +The fact that posteriors disagree at the individual-event level suggests that the discrepancy originates in the different choice of parameterization and corresponding priors, in addition to potentially unstated differences in implementation. + +% First, there are differences in priors, since \citet{Zhu:2023wci} assume a distribution uniform in $z/M_{\rm PV}$ rather than $\kappa$ or $M_{\rm PV}$ itself. +% Second, as reported in \citep{Zhu:2023wci}, their combined result was computed by multiplying $M_{\rm PV}^{-1}$ posteriors from individual events; however, this is not the correct way of combining $M_{\rm PV}$ measurements from independent observations: instead of multiplying posteriors, one should multiply the \emph{likelihoods} to avoid double counting the prior every time a new event is added to the combined constraint. +% Finally, \citet{Zhu:2023wci} report \emph{ad hoc} exclusion of certain events from the combined constraint, without explicitly providing a list of events considered. +% All these factors combined preclude a clear interpretation of the results in \citep{Zhu:2023wci}. +% In \ref{fig:posterior_MPV}, we show the posterior on $M_{\rm PV}^{-1}$ for each event in our analysis. + +\begin{figure} + \script{posterior_MPV.py} + \includegraphics[width=\columnwidth]{figures/posterior_MPV.pdf} + \caption{ + Individual-event $M_\mathrm{PV}^{-1}$ posteriors from our analysis, converted from the $\kappa$ posteriors in Fig.~\ref{fig:violin_kappa} via Eq.~\eqref{eq:kappa-to-Mpv}. + Chosen events are those highlighted in Fig.~1 of \citet{Zhu:2023wci}. + } + \label{fig:posterior_MPV} +\end{figure} \subsection{Parameter degeneracies} \label{sec:degeneracies} @@ -656,11 +700,16 @@ \section{Conclusion} In particular, we identified the relevance of spins and their (partial) degeneracy with $\kappa$, in addition to expected correlations with source inclination and distance. The mass ratio can also play a role, by coupling with the spins or $\kappa$ directly (Sec.~\ref{sec:degeneracies}). +\added{% +Motivated by Chern-Simons gravity, this study only focused on amplitude birefringence. +Other parity-violating gravity theories also predict velocity birefringence, which can dominate over amplitude birefringence when both are present \cite{Zhao:2019xmm}. +Velocity birefringence was tested with GWTC-3 in \citet{Wang:2021gqm} and \citet{Haegel:2022ymk}. +} + This study was restricted to \acp{BBH} because of the expectation that they should dominate the birefringence constraint thanks to their larger redshifts. However, lower mass systems involving neutron stars may also be informative thanks to the wide band of frequencies spanned by their signals, in spite of their closer distances. Analysis of future catalogs will benefit from the inclusion of those events, as well as a much larger number of \ac{BBH} sources at increasingly greater distances. Future measurements will also be enriched by the addition of KAGRA \cite{KAGRA} and LIGO India to the \ac{GW} detector network, which will allow us to better disambiguate between polarization states and hence between models of birefringent propagation. - All our data are available in \citet{dataset}. \begin{acknowledgments}