text:
\section{Introduction}

The population of gravitational waves (GWs) events has grown considerably since the first ever detection in September of 2015 \cite{Abbott:2016blz}. The latest release, GWTC-2 \cite{Abbott:2020niy}, added 39 new events during the first half of the third observing run, O3a, to the 11 already detected in the previous runs, O1 and O2 \cite{LIGOScientific:2018mvr}. With new observatories coming online such as KAGRA \cite{Akutsu:2018axf} and IndIGO \cite{Unnikrishnan:2013qwa}, and the upgrade of the already existing ones, the number of detections is expected to grow faster and faster in the next years. With the growth of GWs events number, the first detection of gravitational lensing (GL) of GWs is getting ever closer.

Accounting for the possibility of lensing would be fundamental in future observing runs. If lensing is not taken into consideration, it might lead to biased results in determining the source properties \cite{Cao:2014}, since lensing magnification changes the inferred luminosity distance and source masses. Moreover, in the limit of strong lensing, there could be also distortions in the lensed signal when higher modes, precession or eccentricity are relevant \cite{Ezquiaga:2020gdt}. Future analyses would require including lensed waveforms in the template bank when performing parameter estimation \cite{Nakamura:1997sw,Hannuksela:2019kle,Lai:2018rto,Pang:2020qow,Dai:2020tpj}, or adopting different strategies, as deep learning \cite{Singh:2018csp,Kim:2020xkm}. Lensing would also be relevant for population analyses since it would affect the inferred redshift and mass distributions, as the amplification of the signals allow to hear events further away.

For present detectors, though, the probability of a lensing event is pretty low. Such a probability will depend on both the mass of the lens and on the mass and distance of the source, but one can approximately state that, for binary black hole (BBH) systems, the rate of detection of strong lensing is $\approx 0.06$ yr$^{-1}$ for sources at redshifts $z<1$ \cite{Oguri:2019}. More in detail, in \cite{Ken:2018} it is shown that strong lensing rate of GWs produced by elliptical galaxies is $\approx 0.2$ yr$^{-1}$ for LIGO and $\approx 1$ yr$^{-1}$ for aLIGO, in accordance with \cite{Li:2018}. Moreover,  studies made on the existing data reject the possibility that observed GWs from BBH might be lensed by (known) clusters at $\leq 4\sigma$ \cite{Smith:2018}, as well as no evidence for any lensing event has been found for GW$190521$ \cite{Abbott:2020mjq}, although ``intriguing'' candidates might be already at our disposal \cite{Dai:2020tpj}.

Things will change as long as the GW events catalog becomes more numerous and detector's horizons expand. For example, in \cite{Li:2018} it is additionally stated that for the nominal Einstein Telescope sensitivity, the rate of lensing events could rise to $\approx 80$ yr$^{-1}$, while in \cite{Christian:2018vsi} it is discussed how third generation detectors will be able to enlarge the possible range of lensing events to lens masses as low as $1$ M$_{\odot}$.

Once a GW event is recognised as lensed, another problem, intrinsic to gravitational lensing, must be taken into consideration: the mass-sheet degeneracy (MSD) \cite{Falco85_MSD,Gorenstein88_MSD,Schneider:2013sxa,Schneider:2013wga}. The MSD relies on the fact that the simultaneous scaling of the lens mass and of the source plane leave the geometrical lensing outputs unchanged. Thus, in practice, we have an intrinsic indeterminacy when we analyze observational data to reconstruct the mass (model) of the lens from the images structure. This is a well-known problem in electromagnetic (EM) GL. For example, such degeneracy can affect the mass modelling of the lens objects and can highly bias the estimation of the Hubble constant, $H_0$ \cite{Schneider:2013sxa,Kochanek:2019ruu}, contributing to the systematic error budget. This is the reason why MSD is carefully analyzed in projects like H0LiCOW \cite{Chen:2019ejq,Wong:2019kwg}, TDCOSMO \cite{Millon:2019slk} and in the Time Delay Lens modelling Challenge \cite{Ding:2018hai,Ding:2020jmg}.

In EM studies, the MSD is always defined in the \textit{geometrical optics} (GO) approximation, when the wavelength of the lensed signal is much smaller than the typical size of the lens. In this limit,
the MSD can be broken, or at least attenuated, when: $1)$ multiple images of the source are available \cite{Lubini:2013mta,Grillo:2020yvj}; $2)$ the same gravitational object act as lens for multiple sources at different redshifts \cite{Saha:2000kn}; $3)$ independent mass estimations of the lens are available, e.g. from spatially resolved kinematical observations of the lens \cite{Schneider:2013sxa,Birrer:2020tax}; $4)$ by a proper combination of the previous astrophysical data with cosmological geometrical data \cite{Chen:2020knz}.

In the context of GL of GWs, given the much lower frequency of the signal, the \textit{wave optics} (WO) regime can also be relevant as GW wavelengths can be comparable to the lens size \cite{Takahashi:2016jom,Cremonese:2018cyg,Cremonese:2019tgb} and its features have been analyzed recently \cite{Jung:2017flg,Dai:2018enj,Sun:2019ztn,Liao:2020hnx}. In this limit, the MSD can be broken, at least in case of multiple lenses. In fact, the diffraction pattern is strictly connected to the geometrical configuration of multiple lenses \cite{Feldbrugge:2020ycp,Feldbrugge:2020tti}. Note that even in the EM GO regime, substructures in the lens may contribute to the uncertainty in the $H_0$ estimation \cite{Gilman:2020fie}.

Previous works have considered the possibility to infer the lens' masses from wave optics GW lensing in some specific conditions, see for example \cite{Takahashi:2003ix}. However, no explicit study is present in literature assessing the role of the MSD in GL of GWs. Here, for the first time, we take into consideration this problem. On the one hand we show how, in the classical GO case, also for GWs, the MSD still stands. On the other hand, when we are in a WO or WO-to-GO scenario (which we generally define as ``interference regime'') the GW waveform is sensitive to the MSD, exhibiting some intrinsic and characteristic features which thus allow us to break the MSD. It is important to stress that we find that, differently from the EM case \cite{Lubini:2013mta,Grillo:2020yvj,Chen:2019ejq}, the MSD can be solved even for a single observation of a single waveform, i.e. when no multiple images of the source are identified, and with one single lens object.

This paper is organized as follows. In Sec.~\ref{ch:GL} we introduce the GL of GW, explaining the difference between GO and WO, and showing how the MSD affects the lensing of the waveforms. Sec.~\ref{ch:GL_regimes} will cover the calculations in GO, in the interference regime, and in WO. Here, we show when the MSD can be solved, with some examples. In Sec.~\ref{ch:SNR}, an additional quantitative analysis to assess this point and to define how well the MSD can be broken is presented. A discussion on the limits of the MSD breaking in WO is done and conclusions are drawn in Sec.~\ref{ch:conclusions}.

\section{Gravitational Lensing}\label{ch:GL}

In this work we study the possibility to detect a GL event by some foreground object (lens) which alters the GW signal. The typical depiction of a gravitational lensing system is shown in \figurename~\ref{fig:lens_scheme}. Here, we have a BBH system as GW source, at an angular-diameter distance $D_S$ from the observer, and a lens, at distance $D_L$ from the observer and $D_{LS}$ from the source, which bends the trajectory of the GW and alters its waveform. The redshift locations of the source and the lens are $z_S$ and $z_L$, respectively. The position of the source and its image are described by the angles $\vec{\theta}_{S}$ and $\vec{\theta}$, respectively, while $\vec{\alpha}$ is the deflection angle. In this work, we always describe the lens with a point mass (PM) model. This is because we consider lenses that have $M_{L}\leq 10^4$~M$_\odot$. Within this mass range, and given the lens redshift which we work with, the most logic and realistic model is exactly the PM one.

\begin{figure}[h]
\subimport{}{images/lens_plot.tex}
\caption{Geometry of a gravitational lens system.}
\label{fig:lens_scheme}
\end{figure}

Usually, the GL of light can be described and studied in the GO regime. For GW, though, we need a WO approach, because the corresponding conditions are more naturally satisfied. GO can be defined from the stationary phase approximation as the regime in which
\begin{equation} \label{eq:GO_cond}
    f\cdot\Delta t_d\gg1~,
\end{equation}
where $\Delta t_d$ is the time delay between the different stationary points (images) and $f$ is the observed frequency of the lensed radiation. From Eq.~\eqref{eq:GO_cond}, one can show that GO approximation breaks when \cite{Takahashi:2016jom}
\begin{equation} \label{eq:GO_cond2}
M_{L}\leq10^5 M_{\odot}\left[\frac{(1+z_L)f}{\text{Hz}}\right]^{-1},
\end{equation}
where $M_{L}$ is the (rest frame) mass of the lens. Below this threshold, WO is the right approach to described the lensing of the GW waveform.

In GO, the time delay is defined as
\begin{equation}\label{timedelGO}
    t(\vec{\theta},\vec{\theta}_{S})=\frac{1+z_L}{c}\frac{D_L D_S}{D_{LS}}\left[\frac{1}{2}(\vec{\theta}-\vec{\theta}_{S})^2-
    \hat{\Psi}(\vec{\theta})\right] ,
\end{equation}
with $\hat{\Psi}(\vec{\theta})$, the effective lens potential, being
\begin{equation}\label{lenspot}
\hat{\Psi}(\vec{\theta}) = \frac{D_{LS}}{D_L D_S}\frac{2}{c^2}\int \Phi(D_L\vec{\theta},z')dz'\, ,
\end{equation}
where $\Phi$ is the gravitational potential of the lens and $z'$ is the line-of-sight coordinate. An alternative way to write the previous expression in terms of dimensionless quantities is
\begin{equation}\label{timedelNODim}
t(\vec{x},\vec{y}) = \frac{1+z_L}{c} \frac{D_S \xi^{2}_0}{D_L D_{LS}} \left[ \frac{1}{2} \left( \vec{x}-\vec{y} \right)^2 - \Psi(\vec{x})\right]\; ,
\end{equation}
where: $\vec{\xi}_0$ is a reference scale length on the lens plane whose value depends on the mass model of the lens (for PM models, it is generally assumed to be the Einstein radius, $\xi_0 = D_L \theta_E$); $\vec{x}=D_L\vec{\theta}/\xi_0$ and $\vec{y}=D_L\vec{\eta}/D_S \vec{\xi}_0=D_L\vec{\theta}_{S}/\vec{\xi}_0$ are the dimensionless relative positions of the image and of the source (on the lens plane); $\vec{\xi}$ and $\vec{\eta}$ are the physical lengths on the lens and source plane, respectively; and $\Psi(\vec{x}) = D^{2}_L/\xi^{2}_{0}\hat{\Psi}(\vec{\theta})$ is the dimensionless lens potential. We can also define the dimensionless time delay as
\begin{equation}\label{eq:tGOdimless}
T(\vec{x}, \vec{y})=\frac{c}{1+z_L}\frac{D_L D_{LS}}{D_S \xi^{2}_0}\cdot t(\vec{x},\vec{y})\, .
\end{equation}
When we consider GL of GWs, if $h(t)$ is the GW strain (i.e. the waveform) in time domain, and $\tilde{h}(f)$ is the strain in the frequency domain, then the lensed waveform in the frequency domain will be given by
\begin{equation}
    \tilde{h}_{L}(f)=\tilde{h}(f)\cdot F(f,\vec{y}).
\end{equation}
Here, $F(f,\vec{y})$ is the amplification factor which comes from the diffraction integral.
In dimensionless units this is defined as \cite{gralen.boo}
\begin{equation}\label{eq:ampfactor}
F(w,\vec{y})=\frac{w}{2\pi i}\int \mathrm{d}^2x\exp[iwT(\vec{x},\vec{y})]\; ,
\end{equation}
where $w=\frac{1+z_L}{c}\frac{D_S \xi^{2}_{0}}{D_LD_{LS}} 2\pi f$ is the dimensionless frequency of the GW. Assuming spherical symmetry \cite{Nakamura7}, we can express it as
\begin{eqnarray} \label{eq:AF}
F(w,y) &=& -iw e^{iwy^2/2} \\
&\times& \int_0^\infty \mathrm{d}x\,x\,J_0(wxy)\exp\left\{iw\left[ \frac{1}{2}x^2-\Psi(x)\right]\right\}\,, \nonumber
\end{eqnarray}
where $J_0$ is the Bessel function of zeroth order, and, since we are assuming spherical symmetry, we can write $x=|\vec{x}|$ and $y=|\vec{y}|$.
In simple terms, the amplification factor accounts for all possible trajectories of the signal around the lens. Independent images are formed when the stationary points are enough separated in time (i.e. in GO regime), leading to an amplification factor of the form
\begin{equation}\label{Fgeom}
    F\approx \sum_j\vert\mu(\vec\theta_j)\vert^{1/2}\exp\left(i\omega t_d(\vec\theta_j)-i\, \text{sign}(\omega)\frac{n_j\pi}{2}\right),
\end{equation}
where $\mu(\vec\theta_j)$ is the magnification of the $j$-th image located at $\vec\theta_j$, and $n_j$ is a frequency-independent phase shift associated with the type of image or extrema of the time delay surface ($n_j=0$ for type I, $n_j=1$ for type II, and $n_j=2$ for type III images, see \cite{gralen.boo} for details).

\subsection{Mass-sheet degeneracy}\label{ch:MSD}

The MSD is based on the following set of transformations which leave the lensed signal observables unchanged \cite{Falco85_MSD,Gorenstein88_MSD}:
\begin{enumerate}
 \item scaling by the same factor $\lambda$ both $\vec{\alpha}$, the reduced angle of deflection, and $\vec{\theta}_{S}$, the source position (angle):
\begin{equation}\label{eq:msd1}
\vec{\alpha} \rightarrow \vec{\alpha}_{\lambda} = \lambda \vec{\alpha} + (1-\lambda) \vec{\theta}\,,
\end{equation}
\begin{equation}\label{eq:msd2}
\vec{\theta}_{S} \rightarrow \vec{\theta}_{S,\lambda} = \lambda \vec{\theta}_{S}\,.
\end{equation}
\item scaling the lens mass (expressed in terms of the convergence $\kappa$ \cite{gralen.boo}) by adding or subtracting a constant mass layer (sheet) which acts as an additional thin lens:
 \begin{equation}\label{eq:msd3}
\kappa \rightarrow \kappa_{\lambda} = \lambda \kappa + (1-\lambda)\,.
\end{equation}
\end{enumerate}
The redshifts and the intrinsic source luminosity are not affected by MSD transformations.
Note that the parameter $\lambda$ is nowadays more commonly recognized to technically take into account two different contributions \cite{Birrer:2016xku}: not only mass from the lens itself (generally called ``internal'' MSD), but also environmental mass effects, from the matter distributed out of the lens and projected along the line of sight, thus called ``external'' MSD.

Without going into all the details and consequences of the above transformations, which have been thoroughly studied \cite{Falco85_MSD,Gorenstein88_MSD}, we focus here on the main features which are relevant for our study. In particular, the dimensionless source position, the lens potential and the time delay can be found to change under MSD transformations as
\begin{equation}
\vec{y} \rightarrow \vec{y}_{\lambda} = \lambda \vec{y}\, ; \label{eq:yT}
\end{equation}
\begin{equation}
\Psi(\vec{x})  \rightarrow \Psi_{\lambda}(\vec{x}) = \lambda \Psi(\vec{x}) + (1-\lambda) \frac{{|\vec{x}|^2}}{2}\, ; \label{eq:potT}\\
\end{equation}
\begin{equation}
t(\vec{x},\vec{y}) \rightarrow t_{\lambda} = \lambda\, t
- \frac{\lambda(1-\lambda)}{2} \left( \frac{1+z_L}{c}\frac{D_S \xi^{2}_0}{D_{L} D_{LS}}\right) \vec{|y|}^{2}\, . \label{eq:tdelT}
\end{equation}
Therefore, the amplification factor needs to be modified. Starting from Eq.~(\ref{eq:AF}), inserting the right transformed quantities Eqs.~(\ref{eq:yT})~-~(\ref{eq:potT})~-~(\ref{eq:tdelT}), and always relying on spherical symmetry, we get:
\begin{align} \label{eq:AFtransf}
F_{\lambda}(w,y) &= -iw e^{iw \lambda^2 y^2/2} \\
&\cdot \int_0^\infty \mathrm{d}x\,x\,J_0(\lambda wxy)\exp\left\{iw \lambda \left[ \frac{x^2}{2}-\Psi(x)\right]\right\}\,. \nonumber
\end{align}

Since, as we have stated above, we are working only with PM lenses, we can solve this integral analytically, getting:
\begin{align}\label{eq:AFtransf_A}
    F_{\lambda}(w,y)
    &= \frac{1}{\lambda} \left(-\frac{i \lambda  w}{2}\right)^{1+\frac{i \lambda  w}{2}} \exp\left\{\frac{1}{2} i w (\lambda y)^2\right\} \,\\
   &\cdot \Gamma
   \left(-\frac{i w \lambda}{2} \right)\ _1F_1\left(1-\frac{i w \lambda }{2};1;-\frac{i w \lambda}{2} y^2 \right)~, \nonumber
\end{align}
where $\Gamma$ is the gamma function and $_1F_1$ is the confluent hypergeometric function. Note that in the limit $\lambda \rightarrow 1$ we get the standard result shown in \cite{Peters:1974gj,Takahashi:2003ix}. It is also to be noted that for PM, there will be only two images in the strong lensing limit. We will denote them as + and - referring to their magnification and de-magnification respectively.

In the next sections, we will use this modified amplification factor (\ref{eq:AFtransf_A}) to compute the lensing of GWs under MSD. Although in general it is difficult to guess analytically the effect of the MSD through $\lambda$, this becomes more apparent in the GO. There one can see that $\lambda$ will only appear in the exponential of (\ref{Fgeom}) attached to the time delay. Therefore, we can anticipate that $\lambda$ will not introduce any characteristic distortion of the independent images, apart for a magnification effect, that would allow us to break the MSD.

\section{Optical Lensing regimes}\label{ch:GL_regimes}

\begin{figure*}[htb!]
\centering
\includegraphics[width=0.9\textwidth]{images/final/s0702_TD_M10-4_y1_L075_zs05.png}
\caption{Impact of the MSD transformations on the time domain waveforms of lensed GWs in the geometrical optics limit. We choose a lens with (rest) mass $M_{L}=10^4 M_\odot$ and at redshift $z_{L}=0.1$. The source has a (rest) total mass of $M_{\rm tot} = 60 M_\odot$ at redshift $z_S=0.5$ and $y=5$ with $q=0.1$, and $\left\{s_{1,z},s_{2,z}\right\}=\left\{0.7,0.2\right\}$. In each panel we show, after arbitrary time alignment, the waveform after MSD with $\lambda=0.75$ in green and the original one in red. We focus on the peaks of the positively magnified (left) and negatively magnified (right) images. The difference in the y-axis of each image accounts for the effect of (de-)magnification.}
\label{fig:MSD_go_s0702}
\end{figure*}

Now that we have defined how the MSD affects the lensing of GWs, we will show how it can be broken in some cases. The chosen procedure will be to compute the lensed waveforms for different values of the MSD parameter, in particular $\lambda=\{0.5,0.75,0.95,1.1\}$, and to compare them to the \textit{original} signal, i.e. when $\lambda=1$, with respect to which we aim to solve the degeneracy. It is important to stress here that such \textit{original} signal is the waveform that would be detected by a possible instrument, and whose analysis, if the MSD is not properly taken into account, might lead to biased estimations of the lens and source systems. For the sake of clarity and compactness, we are going to show only some selected explanatory examples to clarify the content of our statements. For all cases, we fixed the values of the inclination of the source w.r.t. the line of sight at $\iota=\pi/3$, and the sky position and orientation w.r.t. the detector \cite{Sathyaprakash:2009xs} at $\theta_d=0.3$, $\phi_d=0.4$ and $\Psi_d=1.5$.

\subsection{Geometrical optics regime}\label{sch:GO}

To show how MSD acts in the GO case and to crosscheck our pipeline, we first consider, as source of the GW signal, a BBH merger with a (rest) total mass of $M_{\rm tot} = 60$ M$_\odot$ at redshift $z_S=0.5$, and three different scenarios: \textit{a)} same BH masses, $q=m_2/m_1=1$, with $m_2\leq m_1$; \textit{b)} different BH masses, $q = 0.1$; \textit{c)} different BH masses, $q = 0.1$, and (aligned) spins $\{s_{1,z},~s_{2,z}\}=\{0.7,~0.2\}$. The lens has a (rest) mass $M_{L}=10^4\,M_\odot$ and a redshift $z_{L}=0.1$. Since we focus on PM lenses, the time delay between the two images is uniquely determined by the source position $y$ (see e.g. Eq. (18) in \cite{Ezquiaga:2020spg}). For this concrete system, when the source position is $y=5$, we are safely in GO.\footnote{Note though that at large $y$ one approaches the regime of weak lensing where the first image is weakly magnified, $\mu_+\simeq 1 + 1/y^4$, and the second image highly de-magnified, $\mu_-\simeq 1/y^4$.} In fact, if we take into consideration the initial detected frequency, which in this case is $f_i = 30$ Hz, we can see how the conditions from Eqs.~\eqref{eq:GO_cond} and \eqref{eq:GO_cond2} become $f_i\cdot\Delta t_d \approx 10^2 \gg 1$ ($\Delta t_d\approx3.6$ s) and $M_L = 10^4M_\odot > 3\cdot10^3 M_\odot= 10^5 \left[(1+z_L)f_i\right]^{-1}M_\odot$. Thus, one can easily understand that, for greater frequencies, which are achieved while approaching the merging of the BHs, these conditions are even more strongly satisfied.

The unlensed waveform in time domain, $h(t)$, is obtained using the \texttt{PyCBC} \cite{alex_nitz_2020_4134752} software in \textsc{python}, with the \texttt{IMRPhenomHM} approximant \cite{London:2018} which includes higher-modes for circular, (anti)-aligned spinning sources. We evaluate the amplification factor, Eq.~\eqref{eq:AFtransf_A}, varying the transformation parameter $\lambda=\{0.5,0.75,0.95,1.1\}$ using \textsc{python}.  Then, we use the software \texttt{FFTW}\footnote{http://www.fftw.org/} to compute both the Fourier transform, i.e. to get the unlensed $h(f)$, and the inverse Fourier transform to get the lensed waveform $\tilde{h}_{L}(t)$.

In the simplest case, $q=1$, we verify how the observables are properly transformed under the MSD transformations. In particular, we define $p^{i}_{j}$ so that, for example, $p^{+}_{0.75}$ refers to the positively magnified maximum peak of the waveform with $\lambda= 0.75$. We identify the peak as the strain value of the positive maximum of the waveform; since we are in GO and no modulation is present (which is instead a typical feature of the WO regime), this is a safe definition. Then, we also calculate the time delay between the maximum peaks of the two image. We use the notation $\Delta t_{i}$ to indicate, for example, that $\Delta t_{0.75}$ is the ratio between the time delay difference with $\lambda=0.75$ with respect to the time delay difference with $\lambda = 1$.

In Table~\ref{tab:MSD_values} we present results of this first calculations. We can see how, as expected \cite{Falco85_MSD,Gorenstein88_MSD}: the ratio of the peaks/magnifications, for each separated image, with respect to the original waveform, scales as $\lambda$ (columns $2$ and $3$); the arrival time differences scale as $\lambda$ (column $4$).

\setlength\extrarowheight{5pt}
\begin{table}[htb!]
\centering
\begin{tabular}{c|c|c|c}
\toprule
\multirow{2}{*}{$\lambda$}	&	$p_1/p_\lambda$	& $p_1/p_\lambda$	&	\multirow{2}{*}{$\Delta t_\lambda$}	\\
 & $+$ im. & $-$ im. & \\
\hline
\hline
0.5  &	0.500	&	0.505	&	0.500 \\
0.75 &	0.750	&	0.754	&	0.750 \\
0.95 &	0.950	&	0.951	&	0.950 \\
1  	 &	1   	&	1   	&	1     \\
1.1	 &	1.101	&	1.097	&	1.100 \\
\hline
\end{tabular}
\caption{Scaling of observable under MSD transformations in geometrical optics. The source has a (rest) total mass of $M_{\rm tot} = 60 M_\odot$, at redshift $z_S=0.5$ and $y=5$ with $q=1$. The lens has a (rest) mass of $M_{L}=10^4 M_\odot$, and it is at redshift $z_{L}=0.1$. }
\label{tab:MSD_values}
\end{table}

Overall, the MSD is not solved in GO. The same conclusion can be visually derived also from Fig.~\ref{fig:MSD_go_s0702}, where we only show the case with mass asymmetry and spin for $\lambda = 0.75$, but results apply for all $\lambda$ values and in all three scenarios stated at the beginning of the section.

It is to be noted that, even in this scenario where the waveform has characteristic features due to the presence of higher order modes ($q=0.1$ and $\left\{s_{1,z},s_{2,z}\right\}=\left\{0.7,0.2\right\}$), which should ease the visualization of possible differences between the original and the transformed waveform, we cannot visually detect any sign of distortion induced by the MSD. The MSD only changes the magnification. If we look at the negatively magnified image in the right panel of Fig.~\ref{fig:MSD_go_s0702}, we also see no evidence at all for any kind of distortion.
Indeed, it is in type II images, where, if any, we would expect to find a characteristic feature dependent on the MSD due to the distortions with respect to general relativity (unlensed) waveforms due to the lensing phase shift when higher order modes are dominant \cite{Ezquiaga:2020gdt}.
As we can see, though, no such feature is present and the MSD cannot be broken. This visual analysis will be quantified precisely in Sec. \ref{ch:SNR} when considering the effect in the detector signal-to-noise.

\subsection{Interference regime}\label{ch:WO}

Since we did not found any distinctive signature of the MSD transformation in the GO limit, in this section, we analyze how the MSD affects the waveform when we are in the intermediate regime, transitioning from GO to WO, where most of the interference features should appear since the time delay between the images is of the order of the signal duration.

\begin{figure}[htb!]
\centering
\includegraphics[width=0.45\textwidth]{images/final/q1_TD_M500_y05_L075_zs01.png}\\
\includegraphics[width=0.45\textwidth]{images/final/q1_TD_M500_y1_L075_zs01.png}\\
\includegraphics[width=0.45\textwidth]{images/final/q1_TD_M500_y1_L075_zs05.png}\\
\caption{Impact of the MSD transformations on the time domain waveforms of lensed GWs in the interference regime, time domain. Lens with (rest) mass $M_{L}=500~M_\odot$ and at redshift $z_{L}=0.01$. Source with a (rest) total mass of $M_{\rm tot} = 100~M_\odot$ and: $z_S=0.1$, $y=0.5$ in the top panel; $z_S=0.1$, $y=1$ in the middle panel; $z_S=0.5$, $y=1$ in the bottom panel. All sources have $q=1$. After arbitrary time alignment, we show the waveform after MSD with $\lambda=0.75$ in green and the original one, i.e. $\lambda=1$, in red.}
\label{fig:MSD_INT_TD_1}
\end{figure}

As source of our GW signal, we consider a BBH merger with a (rest) total mass of $M_{tot} = 100$ M$_\odot$ at redshift $z_S=0.1$, and the three different scenarios for source masses and spin considered before. The lens now has a (rest) mass $M_L=500$ M$_\odot$ and a redshift $z_L=0.01$. We consider a PM model as mass distribution for the lens and we focus on two source positions on the lens plane, namely $y=1$ and $y=0.5$. A third case we consider consists of the same lens and the same source mass positioned at $y=1$, but $z_S=0.5$. The tools used here to compute the waveforms and the amplification factor are the same as in the previous section.

\begin{figure}[htb!]
\centering
\includegraphics[width=0.45\textwidth]{images/final/s0702_TD_M500_y05_L095_zs01.png}\\
\includegraphics[width=0.45\textwidth]{images/final/s0702_TD_M500_y1_L095_zs01.png}\\
\includegraphics[width=0.45\textwidth]{images/final/s0702_TD_M500_y1_L095_zs05.png}\\
\caption{Impact of the MSD transformations on the time domain waveforms of lensed GWs in the interference regime, time domain. Lens with (rest) mass $M_{L}=500~M_\odot$ and at redshift $z_{L}=0.01$. Source with a (rest) total mass of $M_{\rm tot} = 100~M_\odot$ and: $z_S=0.1$, $y=0.5$ in the top panel; $z_S=0.1$, $y=1$ in the middle panel; $z_S=0.5$, $y=1$ in the bottom panel. All sources have $q=0.1$ and $\left\{s_{1,z},s_{2,z}\right\}=\left\{0.7,0.2\right\}$. After arbitrary time alignment, we show the waveform after MSD with $\lambda=0.95$ in green and the original one, i.e. $\lambda=1$, in red.}
\label{fig:MSD_INT_TD_2}
\end{figure}

\begin{figure*}[htb!]
\centering
\includegraphics[width=0.32\textwidth]{images/final/case_d_M500_y1_fd.png}~
\includegraphics[width=0.32\textwidth]{images/final/case_d_M500_y05_fd.png}~
\includegraphics[width=0.32\textwidth]{images/final/q1_FD_M500_y1_zs05.png}
\caption{Interference regime, frequency domain. Lens with (rest) mass $M_{L}=500 M_\odot$ and at redshift $z_{L}=0.01$. Source with a (rest) total mass of $M_{\rm tot} = 100 M_\odot$ at redshift $z_S=0.1$ and $y=1$ (left panel) and $y=0.5$ (central panel), and source at $z_S=0.5$ with $y=1$ (rights panel). Here, $q=1$ always. Comparing cases with different values for the MSD parameter $\lambda$.}
\label{fig:MSD_INT_FD}
\end{figure*}

We find that the lensed waveforms with different $\lambda$ are significantly and intrinsically different with respect to the case with $\lambda=1$, which is the original signal. As an example, in Figs.~\ref{fig:MSD_INT_TD_1} and \ref{fig:MSD_INT_TD_2} we show, respectively, how the waveform is changed when $\lambda=0.75$ and $0.95$, respectively. Additionally, in Fig.~\ref{fig:MSD_INT_FD} we show what happens in the frequency domain for all the MSD parameters $\lambda$ for the $q=1$ scenario.

In all three plots, we can see how the interference regime is different from the GO one. In fact, now, the transformation acts on the waveform not only on the overall amplitude of the strain, i.e. magnifying positively or negatively the image, but also on its shape. Let us consider Fig.~\ref{fig:MSD_INT_TD_1}, where we have $\lambda=0.75$ and we show the source (always with $q=1$) with $z_S=0.1$ and $y=0.5$ in the top panel, with $z_S=0.1$ and $y=1$ in the middle panel, and $z_S=0.5$ and $y=1$ in the bottom panel. In the top panel, we can see how for $t<0$ one has mostly a modulated magnification of the waveform (any temporal shift is basically undetectable) which might lead to think that there is not such a big difference w.r.t the GO regime. For $t>0$, though, we see how the pattern is strongly changing, with an out-of-phase magnification, and a distortion of the waveform for the case $\lambda=0.75$ which has no correspondence in the original signal. These characteristic features are even more evident in the central panel, where we can clearly distinguish a (positive) maximum when $\lambda=0.75$, not present in the original (red) waveform. Then, mostly at $t\sim 0.01-0.02$ we can also see that, when $\lambda=1$, a peculiar feature is present, caused by destructive interference, differently from $\lambda=0.75$, where it is much less visible. Finally, in the bottom panel, we can see how the two waveforms are totally different for $t>0$.

In general, from the plots we can deduce that, when $\lambda<0.95$, the distinction between the waveforms is more evident. But still, with $\lambda = 0.95$, we can detect some differences, as shown in Figs.~\ref{fig:MSD_INT_TD_2} and \ref{fig:MSD_INT_FD}. In Fig.~\ref{fig:MSD_INT_TD_2}, the panels are divided as in Fig.~\ref{fig:MSD_INT_TD_1}, but we are now considering $\lambda=0.95$ and the source has $q=0.1$ and $\left\{s_{1,z},s_{2,z}\right\}=\left\{0.7,0.2\right\}$. Here, the differences are more subtle. On the top panel, we can see how the modulation of the two waveforms starts to be different at $t\sim-0.01$, with some smaller further shape difference at $t\sim 0.03$. The same can be observed in the middle panel, where the two waveforms are intrinsically different at $t\sim0.01-0.02$, and in the bottom one.

If we consider the corresponding frequency domain, in all panels  of Fig.~\ref{fig:MSD_INT_FD}, the differences are straightforward to identify. Every waveform has a different shape, and we can not move from one to another with a simple rescaling, which would be characteristic of a simple magnification, which implies that they are intrinsically different. Note that in Fig.~\ref{fig:MSD_INT_FD} we only show the case $q=1$, but the same conclusions hold also for the other scenarios, that we are not showing here just for the sake of clarity, because when higher order modes are present, the waveforms in frequency domain become more messy.

Of course, one would expect that the smaller the changes (with $\lambda$), the more difficult it is to break the degeneracy. In Sec.~\ref{ch:SNR}, we will quantify more precisely such differences with a match-filter analysis.
However, it seems clear that, in principle, one single lensed waveform in the interference regime can break the MSD. The reason for that is given by the fact that the MSD transformation changes the time delay between the images and we are here very close to the boundary between WO and GO. In fact, since the two images, well separated in the GO case, here are interfering, a change in the time difference is translated into a different interference pattern.

\begin{figure*}[htb!]
\centering
\begin{minipage}[c]{.48\linewidth}
{\includegraphics[width=\textwidth]{images/final/q1_TD_M100_y1_L075_zs01.png}}\\
{\includegraphics[width=\textwidth]{images/final/q01_TD_M100_y1_L075_zs01.png}}\\
{\includegraphics[width=\textwidth]{images/final/s0702_TD_M100_y1_L075_zs01.png}}
\end{minipage}
\hfill
\centering
\begin{minipage}[c]{.48\linewidth}
{\includegraphics[width=\textwidth]{images/final/case_d_M100_y1_fd_sp.png}}
\end{minipage}
\caption{Wave optics regime, time domain (left panel) and frequency domain (right panel). Lens with (rest) mass $M_{L}=100 M_\odot$ and at redshift $z_{L}=0.01$. Source with a (rest) total mass of $M_{\rm tot} = 100 M_\odot$ at redshift $z_S=0.1$ and $y=1$. In the time domain, we show the cases for $\lambda=0.75$ and: $q=1$ (top panel); $q=0.1$ (middle panel); $\left\{s_{1,z},s_{2,z}\right\}=\left\{0.7,0.2\right\}$ (bottom panel). In the frequency domain, we compare cases with different values for the MSD parameter $\lambda$ for the $q=1$ scenario only. In the inset, we show a comparison between $\lambda=0.5$, $\lambda=1$ and a rescaled version of $\lambda=1$. More details in the text.}
\label{fig:MSD_WO}
\end{figure*}

\subsection{Wave optics regime}

Finally, we want to understand what happens when we get further into WO. Here, the possible breaking of the MSD is not always so clear. Actually, one should note that in the case of pure WO, i.e. when $w\rightarrow 0$, from Eq.~\eqref{eq:AFtransf_A} we get $F\to1$\footnote{Note that $F_{\lambda}$ is totally equivalent to Eq.~(\ref{eq:AF}) when expressed in terms of $y_{\lambda}$ and $\Psi_{\lambda}$. In that case, $F_{\lambda}\rightarrow 1$ for $w \rightarrow 0$. But our Eq.~\eqref{eq:AFtransf_A} is in terms of $y$ and $\Psi$ which correspond to $\lambda=1$. Thus, in Eq.~\eqref{eq:AFtransf_A}, we must perform both the limits $w \rightarrow 0$ and $\lambda \rightarrow 1$ in order to recover $F_{\lambda} \rightarrow 1$.} . Since no lensing is present, the MSD cannot be assessed at all in this limit.

Apart from this extreme case, as an example, we take the same source as in Sec.~\ref{ch:WO}, with same mass and redshift, but a lens with smaller mass, $M_{L}=100$ M$_\odot$. Even at $y=1$, with the lens being less massive, we are further into WO regime w.r.t. the previous cases. If we look at the right panel of Fig.~\ref{fig:MSD_WO}, which shows the frequency domain waveform in this regime for the $q=1$ scenario, we can see that no characteristic features are present in a very striking way, at least, not as much as in the interference regime. Nonetheless, we can see how the transformed waveforms are not simply a rescaling of the original one, with $\lambda=1$. This can be seen in the inset of the right panel of Fig.~\ref{fig:MSD_WO}. There, we plot again the lensed waveform when $\lambda=1$ and $0.5$, together with a rescaled version of the original signal, in black. The rescaling is computed in order to have the same strain at the beginning of the observation, i.e. at $f=10$ Hz. We can clearly see that, differently from the GO case, the transformation here does not introduce only a magnification. In fact, the shape of the waveform changes. This is valid also for the waveforms with other $\lambda$, and even for $\lambda=0.95$ and $1.1$.

The same, even though with more difficulties, can be observed when we move to the time domain (left panels of Fig.~\ref{fig:MSD_WO}). We can see how the most notable differences result in a rescaling of the strain magnitude. Still, some less evident features are present. For example, see the negative peak at $t\approx0.025\,s$ of red waveform on the bottom panel of Fig.~\ref{fig:MSD_WO}, which is not present when $\lambda=0.75$. This means that, even in this case, the MSD might be broken, even though it might be more difficult to check this breaking.

\section{Template matching}\label{ch:SNR}

How to quantify the precision with which the MSD could be broken in the interference regime? This is an important question to answer, because setting the range where the MSD is clearly broken means that we are able to determine where the source distance and mass lens estimation are not biased. The most direct way to assess this point would be by adopting an inference approach, for example using the inference algorithm present in \texttt{PyCBC} \cite{alex_nitz_2020_4134752}, including specific lensing routines in it. Although this is not feasible because of our hardware capabilities, we can still get important insights by calculating the matched-filtered signal-to-noise ratio (SNR) of all the cases studied above.

In a match filtering analysis, such as the one used for the detection of GWs, the SNR is calculated comparing a signal $s(t)=h(t)+n(t)$, where $h$ is the GW signal and $n$ the noise, with a template $h_{T}(t)$ \cite{maggiore2008gravitational,Ezquiaga:2020gdt}
\begin{equation}
\rho = \frac{\left(s|h_{T}\right)}{\sqrt{(h_{T}|h_{T})}} \approx \frac{\left(h|h_{T}\right)}{\sqrt{(h_{T}|h_{T})}}\, ,
\end{equation}
where the right-hand side is given if we neglect the correlation of the noise and the template. The inner product $(a|b)$ in the Fourier space is defined as
\begin{equation}
    (a|b) = 4\,{\rm Re}\left[\int_0^\infty\frac{\tilde{a}(f)\cdot\tilde{b}^*(f)}{S_n(f)}df\right]\, ,
\end{equation}
where $\tilde{a}(f)$ is the waveform (signal) in frequency domain and $S_n(f)$ is the single-sided power spectral density \cite{LIGOScientific:2019hgc,maggiore2008gravitational}.
The optimal SNR is given when the signal matches the template, $h(t)\propto h_{T}(t)$, i.e. $\rho_{\rm opt}=\sqrt{(h|h)}$.
We compute here  $\rho/\rho_{opt}$, i.e. the ratio between the SNR obtained by a given template w.r.t. the optimal SNR. To compute these calculations we use the publicly available Livingston detector sensitivity during the first months of the observing run O3\footnote{``o3\_l1" in \url{https://dcc.ligo.org/LIGO-T1500293/public}.}.

It is important to note that in order to study the distinctive features introduced by the MSD transformations we are most interested in differences in the template matching and not in the absolute SNR values. Absolute SNR values are relevant, though, to answer the question of how well the possible differences between different MSD transformations could be distinguished for a given detector network.
Since current ground-based detectors cover a similar frequency range, the ratio $\rho/\rho_{opt}$ will not change significantly among them.

In order to quantify the region of parameter space that could break the MSD, we compute the change in the $\chi^2$ when using different templates. We follow a similar approach to that of \cite{Wang:2021kzt}.
In particular, the likelihood of a given GW event can be determined assuming that after the subtraction of the waveform from the signal, the noise is Gaussian \cite{maggiore2008gravitational}, i.e.
\begin{align}
\mathcal{L} &\propto \exp \left[-\frac{1}{2}(s|s)+(h|s)-\frac{1}{2}(h|h)\right] \nonumber \\
&\propto \exp \left[(h|s)-\frac{1}{2}(h|h)\right]\, ,
\end{align}
from which the $\chi^2$ is derived to be
\begin{equation}
\chi^2 = (h|h)-2(h|s) \approx \rho^{2}_{opt} \left[1-\frac{2\rho}{\rho_{opt}}\right]\,,
\end{equation}
where we do not include the common, constant term $(s|s)$, and neglect correlations between the noise and template.
Searching for a given confidence level w.r.t. to the best model corresponding to $\rho_{opt}$ we get
\begin{equation}
\Delta \chi^2 \approx 2 \rho^{2}_{opt} \left[1-\frac{\rho}{\rho_{opt}}\right] \,.
\end{equation}
For a given threshold in $\Delta \chi^2$, this relation allows us to determine the level of mismatch, $\rho/\rho_{opt}$, that can be distinguished as a function of the optimal SNR.
When two free parameters are involved in the analysis (as in our case, as we explain in the following), the $3\sigma$ confidence level roughly corresponds to $\Delta \chi^2\sim 11.8$.
In our examples we will consider a very loud event with $\rho_{opt} \sim 55$ and a typical near threshold event with $\rho_{opt} \sim 11.5$. This will convert into $\rho/\rho_\text{opt}\approx 0.998$ and $\rho/\rho_\text{opt}\approx 0.955$ respectively. These numbers set the benchmarks to determine the region of parameter space where the MSD can be broken.
\begin{figure}[h!]
    \centering
    \includegraphics[width=0.48\textwidth]{images/final/SNR_regimes.png}
    \caption{Effect of the MSD transformation parameter $\lambda$ on the detection of GWs. We compute the signal-to-noise ratio (SNR) using templates that vary $\lambda$, $\rho(\lambda)$, for an original signal with $\lambda=1$. We compare this SNR to the optimal SNR $\rho_{opt}=\rho(\lambda=1)$ and plot their ratio. We consider three different regimes: geometric optics (GO, red) with $M_L=500~M_\odot$ and $y=10$; interference regime (blue) with $M_L=500~M_\odot$ and $y=1$; and wave optics (WO, green) with $M_L=100~M_\odot$ and $y=1$. Lens at $z_L=0.01$, always. The source has a rest mass of $M=100~M_\odot$ at $z_S=0.1$; and $\rho_{opt} \approx 55$, which set the $3\sigma$ threshold at $0.998$.}
    \label{fig:SNR_regimes}
\end{figure}

In Fig.~\ref{fig:SNR_regimes}, we present a first insight on the behaviour of $\rho/\rho_{opt}$ depending on $\lambda$, for the three optical regimes we considered so far. The GO regimes case (in red in the figure) is given by a mass lens of $M_L=500~M_\odot$ and a source position $y=10$; the interference regime case (blue) has $M_L=500~M_\odot$ and $y=1$; and the WO case (green) has $M_L=100$ and $y=1$. The lens is always at $z_L=0.01$. The source is the same for the three cases and it is a BBH with total rest mass of $M=100~M_\odot$. The threshold is set at $0.998$ (gray region), i.e. a $3\sigma$ confidence given by $\rho_{opt}=55$.
Although this figure does not offer the full picture of the degeneracy breaking since it does not consider how $y$ and $M_L$ change with $\lambda$, it still provides valuable information of the MSD effect. In particular:
\begin{enumerate}
\item in the GO case, the MSD is not broken at all since the corresponding curve never leaves the confidence region (in fact the curve is, to our degree of numerical accuracy, flat);
\item in the interference regime, $\rho/\rho_{opt}$ decreases quickly when $\lambda\neq1$ and leaves the confidence region for $\lambda$ close to $1$, meaning that the MSD is broken for almost any $\lambda\neq1$;
\item in the WO case, $\rho/\rho_{opt}$ decrease outside $\lambda=1$, but not as quickly as in the interference case. This means that, here, the MSD is still valid for a greater interval in$~\lambda$.
\end{enumerate}

\begin{figure*}[htbp]
\centering
\includegraphics[width=0.4\textwidth]{images/final/Mat_SNR_M-y_099_Ms500_q1_o3_l1_zS01_3sig.png}~~~
\includegraphics[width=0.4\textwidth]{images/final/Mat_SNR_M-y_093_Ms500_q1_o3_l1_zS05_3sig.png}\\
\includegraphics[width=0.4\textwidth]{images/final/Mat_SNR_M-y_10_Ms500_q1_o3_l1_zS01_3sig.png}~~~
\includegraphics[width=0.4\textwidth]{images/final/Mat_SNR_M-y_10_Ms500_q1_o3_l1_zS05_3sig.png}\\
\includegraphics[width=0.4\textwidth]{images/final/Mat_SNR_M-y_101_Ms500_q1_o3_l1_zS01_3sig.png}~~~
\includegraphics[width=0.4\textwidth]{images/final/Mat_SNR_M-y_103_Ms500_q1_o3_l1_zS05_3sig.png}
\caption{Signal to noise ratio matrix for interference regime. \textit{(Left column.)} The signal (the same for all panels) corresponds to a source with a (rest) total mass of $M_{\rm tot} = 100~M_\odot$, $q=1$ at redshift $z_S=0.1$ and at $y=1$, and to a lens with (rest) mass $M_{L}=500~M_\odot$ at redshift $z_{L}=0.01$. The templates are calculated at the $\{y,M_L\}$ values shown in the axis, while the MSD parameter is fixed at $\lambda=\{0.99,1,1.01\}$ from top to bottom. The dark red $3\sigma$ confidence levels are calculated from $\rho_{opt}=55$ and threshold $\rho/\rho_{opt}=0.998$. \textit{(Right column.)} Same as left, but with $z_S=0.5$, and the templates calculated with the MSD parameter fixed at $\lambda=\{0.93, 1, 1.03\}$. The $3\sigma$ confidence levels are calculated from $\rho_{opt}=11.5$ and threshold $\rho/\rho_{opt}=0.955$.}
\label{fig:SNR_INT}
\end{figure*}

For a more detailed analysis, we explore the SNR mismatch in the $(M_L,y)$ plane. We present SNR calculations in form of matrices for a grid of $M_L$ and $y$.
Depending on the goal, this kind of representation: will ease the identification of the MSD range (i.e. in $\lambda$) where we can assess it is broken; will make us able to ``convert'' such $\lambda$-validity ranges into $y$ and $M_L$ ranges; and will ease the study of possible degeneracies among the parameters. We thus expect that $\rho/\rho_{opt}$ is larger than the $3\sigma$ threshold stated above not only when the template matches the signal, but also when a degeneracy is taking place. In other words, when different combinations of $y$, $M_L$ and $\lambda$ would give an indistinguishable waveform at $3\sigma$ confidence level, we may say that the MSD is on and identify the range of validity where the MSD is not broken. On the other hand, if $\rho/\rho_{opt}$ is smaller than the $3\sigma$ threshold, then we can state that, for the values of $y$ and $M_L$ for which this happens, the signal and the templates are intrinsically different, we can exclude the presence of degeneracy and we can assess that the MSD is finally broken, leading to unbiased estimation of the other parameters.

Since we have confirmed in Fig.~\ref{fig:SNR_regimes} that the MSD cannot be broken in the GO limit ($\rho(\lambda)/\rho_{opt}=1$), we focus for this analysis in the interference regime. Our results are given in Fig.~\ref{fig:SNR_INT}. The left column shows SNR calculation for this regime, when the source is at $z_S=0.1$ and has $q=1$. Each panel shows $100$ templates in correspondence of the $y$ and $M_L$ values shown on the axis. What is more important to say, is that the templates are different from one panel to the other because we consider different $\lambda$ values in each of them. In particular, $\lambda=\{0.99,1,1.01\}$, from top to bottom. The signal, instead, is always the same for each panel, and is given by $y=1$, $M_L=500~M_\odot$ and $\lambda=1$, i.e. our designed ``original'' waveform. The $3\sigma$ contours are shown as highlighted dark red boxes. We can see that when $\lambda=1$ (central left panel), we have a perfect match among templates and signal only in a very narrow region around the fiducial values corresponding to $\Delta y \sim 3\%$ and $\Delta M_L \sim 4\%$.

But if we aim at finding what could be the possible error on $y$ and $M_L$ associated to the impossibility of breaking the MSD, we must look at the panels where $\lambda = 0.99$ and $1.01$, which are the extreme cases where the $3\sigma$ confidence levels still overlap with the original signal. In that case, we can see how the most realistic $3\sigma$ uncertainty on the parameters would be $\Delta y \sim 5\%$ and $\Delta M_L \sim 6\%$.
Although we do not show other waveforms with $\lambda$ out of the range $[0.99,1.01]$, it is clear that we can now established them as safely breaking the MSD.

When we move the source further away, posing it at $z_S=0.5$, things change quantitatively, because the SNR is lower ($\rho_{opt}=11.5$), but, qualitatively speaking, we can still identify a quite narrow range under which MSD is not safely broken. From the right panel of Fig.~\ref{fig:SNR_INT} we can see how now the MSD can be considered broken for $\lambda<0.93$ and $>1.03$. The corresponding $3\sigma$ uncertainties on $y$ in this case grow up to $\Delta y \sim 16\%$ on the lower bound and $40\%$ on the upper one, while for $M_L$ we have $\Delta M_L \sim 35\%$.

Finally, we perform the same check for the WO case, which we omit to show in figures here. In this case $\rho_{opt}=61$ and the $3\sigma$ region is defined by $\rho/\rho_{opt}\geq0.998$. Here the MSD can be broken for $\lambda$ out of the range $[0.93,1.06]$, which corresponds to $3\sigma$ uncertainties of $\Delta y \sim 20\%$ and $\Delta M_L\sim 30\%$.

\section{Discussion and Conclusions}\label{ch:conclusions}

The mass-sheet degeneracy (MSD) is a well-known problem in gravitational lensing limiting the possibility of determining the lens model parameters from observations of lensed signals.
Although the implications of the MSD for lensed EM signals have been extensively studied, e.g. \cite{Falco85_MSD,Gorenstein88_MSD,Schneider:2013sxa,Schneider:2013wga,Kochanek:2019ruu,Wucknitz:2020spz,Tagore:2017pir,Blum:2020mgu}, the same is not true for lensed GWs. In this paper we have addressed this point. Understanding when the MSD can be broken is important since it will open new opportunities to infer more accurately cosmological and astrophysical parameters, e.g. H$_0$ \cite{Cremonese:2019tgb}, or lens parameters, like $M_L$ or $y$, from GWs astronomy.

We have studied how the MSD modifies the physics of the lens-GW source system, and we have shown, both qualitatively and quantitatively, how it affects all the optics regime which can be explored in GW astronomy. In fact, contrarily to EM events, which mostly happen in the geometrical optics regime, when dealing with GWs, depending on the lens-source geometry, on the GW frequency and on the mass of the lens, we might need to rely on geometrical or wave optics formulation. Indeed, we have shown here that the most interesting scenario is related to the interference regime, intermediate between wave- and geometric-optics.

Our first result was to illustrate the effects of the MSD on the lensed waveforms in the geometric-optics approximation. We find that the usual problem connected to the MSD in the EM persists also for GW lensing without any substantial modification. In fact, the MSD only acts on the magnification of the lensed waveforms and does not alter their shape.

The most interesting situation is related to the interference regime. Here, we demonstrate that the MSD is clearly solved because the waveform, both in the time and frequency domain, presents some characteristic features that depend on the MSD parameter, $\lambda$. In other words, the waveforms are intrinsically different if the source position and/or the lens mass are varied; thus, if the lensing analysis is included in the detection pipeline, we can infer unbiased estimation for the interested parameters.

For a full assessment of the MSD breaking it is important to determine how these signals would be heard in a given detector networks and the impact of observational errors. We have considered two different signals, both of them detectable by current LIGO/Virgo detectors, but with different SNR: one ``realistic'' case, with $\rho_{opt}=11.5$, has SNR comparable to the signals which have been detected so far; one ``optimistic'' case, with $\rho_{opt}=55$, has a SNR five times higher and can provide insights also on future GW detector sensitivities.

If we stick to the realistic case, we have found that the MSD is broken for $\lambda <0.93$ or $>1.03$. In terms of parameters which are measured, this converts in the $3\sigma$ errors $\Delta y \sim 16-40\%$ and $\Delta M_L \sim 35\%$. Thus, a rough estimation of the $1\sigma$ error would be $\Delta y \sim 5-15\%$ and $\Delta M_L \sim 12\%$. These values show that the MSD might still be a problem: in fact, realistic errors on the dispersion velocity of the lens system are of the order of $6-10\%$ \cite{Schneider:2013sxa,Kochanek:2019ruu}, which thus convert in $\Delta M_L \sim 12-20\%$.
This should be taken just as a rough estimation, considering that here we are considering point mass lenses, but it is indicative of a consistent role of the MSD in the error budget.

Instead, with higher SNRs, the MSD can be broken for $\lambda<0.99$ or $>1.01$, which implies $\Delta y \sim 5\%$ and $\Delta M_L \sim 6\%$ at $3\sigma$ or $\Delta y \sim 2\%$ and $\Delta M_L \sim 2\%$ at $1\sigma$. Thus, in this case, the role of the MSD would be clearly subdominant with respect to other known sources of uncertainty.

Finally, we studied the MSD in the wave-optics regime. We have demonstrated that, contrary to the geometric-optics case and similarly to the interference one, the MSD does not simply rescale the waveform, but intrinsically changes its shape, making it possible to break the degeneracy. However, the lensing effects being much weaker than in the interference regime, the MSD breaking does not perform as well as in the interference regime.

We plan to extend our analysis in several ways. First of all, one could improve the quantitative part, possibly through direct inference, on how well the MSD can be solved in all regimes. Then, we could expand the study to different lens models, while here we were restricted to a point mass only. One could also expand the study to different sources, considering BBHs with different masses and/or at higher redshifts, and possibly considering also binary neutron stars. And, finally, a deeper analysis of the WO regime should be performed, in order to define a better limit up to which the MSD can be solved.

\section*{Acknowledgments}

P.C. is supported by the project ``Uniwersytet 2.0 - Strefa Kariery, ``Miedzynarodowe Studia Doktoranckie Nauk Scislych WMF", nr POWR.03.05.00-00-Z064/17-00'' co-funded by the European Union through the European Social Fund.

J.M.E. is supported by NASA through the NASA Hubble Fellowship grant HST-HF2-51435.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. He is also supported by the Kavli Institute for Cosmological Physics through an endowment from the Kavli Foundation and its founder Fred Kavli.

\bibliographystyle{apsrev4-1}


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Posted 1 month ago by Austin Bumpus1 Comment on What's on each NFL team's holiday wish list as the end of the regular season nears?	Football
What's on each NFL team's holiday wish list as the end of the regular season nears?
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But, when you're in the NFL, things are a bit different. While some fans are looking to find the perfect gift from Santa under their tree, NFL teams are hoping for a few non-material gifts to fall into their lap ahead of the playoffs: a playoff spot, a high draft pick, a first-round bye.
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With 32 teams across the league, it's tough for Santa to deliver exactly what every team may be hoping for as the holiday season comes and goes. But, it helps to narrow things down when he has a list.
With that in mind, it's time to dive in with the items on every NFL team's wish list for the 2019 holiday season.
Arizona Cardinals: More offensive weapons for Kyler Murray to work with
Can you imagine how good Kyler Murray would be with an offense like Kurt Warner had during his tenure with the Cardinals? Get Murray a solid receiving corps and a consistent running back with good hands in the backfield, and you could have some potential for the 2020 season.
Atlanta Falcons: A time-travel machine to go back to the first half of Super Bowl LI
Seriously, what has happened to the Falcons since that Super Bowl collapse against the Patriots? The team went from playing with the league MVP under center and one of the game's best wide receivers to scratching and clawing through an entire season in a relatively weak NFC South. Maybe they'll find a special machine under the tree in a few weeks to help them get back on track.
Baltimore Ravens: The top seed in the AFC and home-field advantage
The Ravens are that kid in your class that already seems like it has it all, but what they have is never enough. Sure, the boys in Baltimore have the AFC North locked up, but they're out for more.
A first-round bye? The top seed with home-field advantage through the playoffs? A Super Bowl run? Let's see what Santa has in store for Lamar Jackson and Co. this winter.
Buffalo Bills: Their first AFC East title since 1993
Sure, the Bills are just hoping for a playoff spot at this point, fighting through what has turned into a jam-packed playoff race in the AFC. But, they could be in for a historic feat if they can pull off an impressive winning streak to end the regular season, possibly resulting in their first AFC East title in 26 years.
Even more shocking? They're technically still in good enough shape to finish with a first-round bye if a long list of miracles were to occur in the upcoming weeks.
Carolina Panthers: Just make it to 2020
The Panthers' playoff hopes are officially gone, and they're riding the Kyle Allen train at the quarterback position for the rest of 2019. Now, the goal is simple: Just stay healthy and head into 2020 with a new look.
Keep Christian McCaffrey out of harm's way, keep the core intact, and rebuild to win an NFC South in 2020 that could be anyone's ballgame.
Chicago Bears: More wins, baby
The Bears seem to be turning things around in the second half of the 2019 season, with a good shot at a spot in the postseason despite a loaded field of teams in the NFC.
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Mitchell Trubisky is hitting his stride, the defense (outside of a late Cowboys comeback on Thursday Night Football) looks good, and things are looking up. Could they pull off a shocking run to get some redemption in the playoffs this year? If they can keep up their winning ways, they've got a good shot.
Cincinnati Bengals: More losses and a No. 1 overall pick in the 2020 NFL Draft
While the Bears' message has to be "Just win, baby," the Bengals' motto has to be the opposite. Just "Bungle for Burrow" or "Crash for Chase" at this point, and get a fresh start as they continue to rebuild in 2020.
Cleveland Browns: A happy Odell Beckham, Jr.
The Browns' 2019 season didn't live up to the hype. I think everyone got over that fact early enough in the year to move on quickly, but that doesn't mean they should move on from any of the insane amount of talent they have on offense. Get Odell Beckham, Jr. back on track alongside Baker Mayfield and Jarvis Landry, and let everyone come back with some added motivation in 2020. It could be just what they need to compete with the Ravens next year.
Dallas Cowboys: An NFC East title for a first-round home game in the playoffs
I personally believe the Cowboys have all the pieces right now to earn a spot in the Super Bowl down in Miami, but the coaching just isn't there. Like Jerry Jones said, however, all it takes is a home playoff game after winning the NFC East to get this team going, and they could pull off some surprises in the postseason if they can squeeze in.
Denver Broncos: A face of the franchise under center
The Broncos have been through a lot on offense since their Super Bowl 50 win against the Panthers and Peyton Manning's retirement. They've gone through Joe Flacco, Brandon Allen and Drew Lock under center in the few months of the 2019 season, and they can't find an answer offensively. They'll need to figure out where to go next for 2020, and if they can get back to the top of the totem pole in the AFC West.
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Detroit Lions: Calvin Johnson? Barry Sanders?
The Lions always give fans hope at the beginning of the year that the upcoming season could be something special, and something always goes awry. Injuries derailed this team, and they'll need a miracle to bounce back against a loaded NFC North in 2020. Maybe Calvin Johnson suddenly wants to come out of retirement? Barry Sanders looks like he could still go, too…
Green Bay Packers: Keep Aaron Rodgers healthy
The Packers are finally back to the success that many thought they'd have throughout the 2010s, and now it's time to finish the job. Can they keep their team healthy and have a full-strength Aaron Rodgers for the postseason? If that happens, they could be in line for a strong playoff run in January.
Houston Texans: A comeback from the Patriots hangover and an AFC South title
Every team that manages to beat the Patriots in the regular season seems to fall apart in the next few weeks. In 2018, the Titans, Lions, Dolphins and Lions all experienced it, and none managed to make the playoffs as a result. The Texans need to avoid the Patriots hangover, and finish the year strong en route to an AFC South title.
Indianapolis Colts: Consistency
The Colts know all about how tough it is to have consistency in the NFL, especially after the last-second retirement of Andrew Luck before the season. Now, they just need to hit their stride and stop hitting the brakes in 2019, and hope for a spot in the AFC playoff seeding. After that, it's time to find one more star to help Jacoby Brissett in the Colts offense.
>>RELATED: The Patriots Wolf Pack: Evaluating Tom Brady, Jacoby Brissett and Jimmy Garoppolo as they look to take over the NFL in 2019
Jacksonville Jaguars: A new defense
The Jaguars shook everything up when they shipped cornerback Jalen Ramsey off to Los Angeles after a rollercoaster of a tenure with the Jaguars in the first half of 2019. Now, it's time to shake things up in a different direction and get back to the winning ways that led them to an AFC Championship game appearance a few years ago. Defensive line help would be a good start, and then it's time to rebuild the secondary.
Kansas City Chiefs: The second seed in the AFC and a road through Arrowhead
The Chiefs already beat the Patriots on the road, albeit among some controversy at Gillette Stadium as it relates to the officiating. That is undoubtedly one item crossed off their holiday wish list, and they now just have to finish the job over the last few weeks of the season.
If they can win out, they'll just be rooting for some bad Patriots losses for a first-round bye in the playoffs. That would give them some extra time to get Patrick Mahomes healthy, and would give them some home-field advantage until a potential AFC Championship game against the Ravens. Could Arrowhead Stadium prove to be the difference-maker in the 2019 playoffs?
Los Angeles Chargers: A perfect 2020 NFL Draft
The Chargers haven't been the same since January 2018, when they were trounced on the road in the postseason against the Patriots. They seem to invent a new way to lose as each game passes in the 2019 season, and now it's time to rethink everything as they look towards the new decade. Now, it's time for a perfect storm to get back on track before next September hits.
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Do they draft a quarterback in the first round as a succession plan for Philip Rivers? Do they go with a defensive stud early to build up that side of the ball and go with offense in the second round? Do they try to trade up for a big splash early in the first round? It's pivotal that the Chargers nail this draft with every pick they have, or else they may continue to spiral downwards in the AFC West.
Los Angeles Rams: An all-NFC West Wild Card round
The Rams may just be the unluckiest team in the NFL at this point. They finally get back to a Super Bowl after a high-scoring 2018 season, only to run into Bill Belichick and one of the best defenses the Patriots head coach has ever put together. Now, they're having a solid season, only to fall behind the best two teams in the entire conference, who happen to be in their own division.
Could the NFC see three NFC West teams in the mix in the playoffs? If the Rams have it their way and their wish comes true, the conference could be in for a rude awakening in January.
Miami Dolphins: Big-time defensive picks in the first round of the 2020 NFL Draft
Initially, it seemed like the Dolphins were "tanking for Tua," as they searched for their franchise quarterback. Now, they find themselves holding a hodgepodge of first-round picks for the 2020 NFL Draft, and it's just about using them properly if they hope to take advantage of Tom Brady's eventual departure from the AFC East.
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Brian Flores was a great defensive coach for Brady's Patriots under the tutelage of Bill Belichick, and that's where the Dolphins need to strike in the upcoming draft. They can't expect to build up a divisional contender by only prioritizing offense, and they'll need to play to Flores' strengths if they want a chance to swim back to the top of the AFC East.
Minnesota Vikings: A primetime win for Kirk Cousins and a playoff spot
Kirk Cousins is on fire in the latter half of 2019, and he could be the key to a surprising playoff performance for the Vikings if the NFC North squad can pull into the postseason. But, he'll have to shine when the lights are brightest in a big primetime game against the Packers in Week 16 first.
If Cousins can get over that hump (he's 0-8 in Monday night games throughout his NFL career) and if the Vikings can win out, they could be a real sleeper pick in a stacked NFC.
New England Patriots: A new receiver, offensive line help and a first-round bye
The Patriots went from having the most loaded receiving corps of any team in 2019 to struggling to find any receivers for Tom Brady to throw to. From a group that included Josh Gordon, Demaryius Thomas and Antonio Brown to a squad of backups including Mohamed Sanu, N'Keal Harry and Jakobi Meyers.
>>RELATED: Brady's Bunch: A compilation of what Tom Brady thinks about each member of the 2019 Patriots offense
Somehow, they're still a top-two seed in the AFC seed, but they may need some help to get the job done if they want to repeat as Super Bowl champions. They'll be asking for some receiving help to get some of the stress off of Julian Edelman, and some offensive line help to get pressure off of Brady under center. Add it all up, and they'll maybe get their wish of a seemingly-annual first-round bye in the postseason.
New Orleans Saints: An NFC West collapse
The Saints already have the NFC South locked up, but now their focus has to be on the NFC West. They'll need an NFC West collapse of some sorts, whether it's the 49ers, Seahawks or both, if they want to lock up the top seed in the NFC and home-field advantage until a potential Super Bowl appearance.
New York Giants: A healthy Daniel Jones
When you're not even competing in a division where the current leader has a 6-7 record, the season was probably over a long time ago. Just get Daniel Jones healthy, keep Saquon Barkley intact and get on to 2020 in a hurry.
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New York Jets: A Ghost of Christmas Past
Sam Darnold may have been ridiculed for thinking he was seeing ghosts on Monday Night Football against the Patriots earlier this season, but the whole organization may wish they could start seeing some Ghosts of Christmas Past to revert back to some past success. They're consistently a bad team, and fans will just have to hope that the new decade can bring a new look to the Jets offense.
Oakland Raiders: A win in the final game at the Coliseum
The Raiders were so close to really making a push for an AFC West title, but couldn't get the job done when the Chiefs were on the other side of the field. Who knows what would've happened with a healthy Johnathan Abram on defense, or a certain wide receiver on offense, but it's now just time to send the fans home happy from the Coliseum one last time before the ride's over (again).
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Philadelphia Eagles: An NFC East title
Somehow, someway, the Eagles are still in a race for an NFC East title with the Cowboys. Yes, you read that correctly. The 6-7 Eagles are still in the hunt for a division title, battling against the 6-7 Cowboys the rest of the way.
It's like the horrible movie that you went to the theaters to see, and you just want it to end so you can move on with your life. If the Eagles can win the division, they can head into the playoffs with a fresh start and treat it as a whole new season.
Pittsburgh Steelers: A Duck-led surge into the postseason
Quack. Quack. Quack. Quack. Quack.
The Steelers are the modern-day Mighty Ducks, led by Devlin "Duck" Hodges under center. They're surging in the AFC standings, and somehow could squeak right into a Wild Card spot after a rough start to the year and three quarterbacks throughout 2019. Could they be the team to shock the system in the NFL this year?
San Francisco 49ers: A top seed in the NFC and a healthy roster
The Niners, much like the Ravens, have it all at this point. They're in control of their own destiny, and just need to keep up their high-scoring pace for the rest of the year. If they stay healthy and keep the foot on the gas pedal, they'll be a force to be reckoned with as the road to the Super Bowl heads through the Bay Area.
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Seattle Seahawks: Consistency and a win over the 49ers
The Seahawks seem to have the weapons on both sides of the ball to make a serious run in the playoffs, but the biggest issue they've run into relates to their own consistency. Russell Wilson is playing at an MVP level, and they have plenty of talented receivers to work with offensively.
But, it's been a story of Dr. Jekyll and Mr. Hyde at times this year, evident by their loss against the Rams on Sunday Night Football. If the right version of the Seahawks shows up the rest of the way in 2019, they'll be in good shape. If not, say hello to the fifth or sixth seed in a stacked NFC.
Tampa Bay Buccaneers: Defensive help
You can probably add a miraculous recovery for Mike Evans to the Buccaneers' holiday wish list, but at this point, it's time to start prioritizing for 2020. To do that, it's time to return to what won them a Super Bowl in the early part of the 2000s, and get some defensive help to go with the already-stellar one-two punch of Evans and Chris Godwin on offense.
Tennessee Titans: A contract extension for Ryan Tannehill and business as usual in the second half of the season
Ryan Tannehill has been lights out for the last few weeks for Tennessee, and the priority has become signing him long-term to keep him around as the new franchise quarterback (Sorry, Marcus Mariota). Now, they need to keep riding this momentum to a potential AFC South title, and a home game in the playoffs.
Washington Redskins: A new offensive line
The Redskins seem to have the quarterback that they plan to build their offense around for the near future, but now it's time to get him some protection.
The Redskins' offense has all sorts of issues to work through as they prepare for the 2020 season, but building an offensive line that actually respects their quarterback is a solid start. The Colts helped Jacoby Brissett a ton by providing a stellar offensive line to work with after Andrew Luck's retirement, and the Redskins could learn a lot from what Indianapolis has done to build up a divisional contender in the AFC.
It may take a few years to get done, but getting a good start on the process in the upcoming offseason would be the perfect wish list item for the NFC East squad.
Like this story? What do you think should be on your favorite team's wish list? Let us know by following @SOTSports on Twitter or by liking our Facebook page!
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text:
Fall 2011 : A Firm Foundation
Education Continues a Strong Emphasis
Education has formed the foundation for MD Anderson since it was established in 1941 as part of The University of Texas System.
BY Mary Jane Schier
Michael Ahearn, Ph.D., founding dean of the School of Health Professions, congratulates Melinda Garza, who graduated summa cum laude and received the Outstanding Student Award in the Clinical Laboratory Science Program. Photo: Kevin McGowan
Its first formal training program began in 1949 with four histotechnology students, whose education taught them to prepare human or animal tissue samples for microscopic examination. Courses for other allied health disciplines were added over the years.
Today, the School of Health Professions offers eight degree-granting allied health programs.
"The Bureau of Labor Statistics predicts that within a few years we will have a greater shortage of allied health professionals than nurses and doctors. Our school's goal is to prepare more allied health specialists who are critical to MD Anderson's continuing to lead the world in excellent cancer care and research," says Michael Ahearn, Ph.D., the school's founding dean.
He notes that enrollment has increased more than 10-fold — from 35 allied health students a decade ago to 367 for fall 2011.
Ahearn says one reason for this growth is more donors supporting scholarships. About 15% of current students receive financial support.
In recent years, more than 40% of the school's graduates have been hired at MD Anderson, another 25% went to other Houston-area hospitals and the others were in high demand across the country.
Training for students interested in allied health careers is in short supply.MD Anderson's school offers the only baccalaureate molecular genetics program in Texas and one of six accredited in the United States. The diagnostic imaging program is one of only two that award degrees in Texas.
"Our students have access to an unparalleled array of equipment, laboratories and expert clinical and teaching faculty. In addition to MD Anderson's extensive resources, students can take clinical rotations at several other Houston-area hospitals and diagnostic laboratories," says Ahearn, who retired at the end of August.
Two educators leave their mark
The most important educational advance for MD Anderson was gaining approval to award bachelor's degrees to allied health students in 2001 and to jointly confer master's and doctoral degrees in the biomedical sciences in 2002.
"After becoming a degree-granting institution, the next major milestone was earning an initial five-year accreditation in 2005 from the Commission on Colleges of the Southern Association of Colleges and Schools (SACS) that reaffirmed MD Anderson as a leading academic institution. SACS approved re-accreditation last year," notes Stephen Tomasovic, Ph.D., former senior vice president for academic affairs.
THE FULL STORY:
3-D Isn't Just for the Movies
Memorable Day, Rewarding Career
Tomasovic and Ahearn have shared leadership in much of MD Anderson's educational progress over the years. Before both retired recently, they reflected on their highly satisfying careers.
After coming to MD Anderson in 1965, Ahearn directed the diagnostic ultrastructure hematology laboratory for 23 years and then joined the Office of Academic Programs with increasing administrative responsibilities.
In addition to being the founding dean of the School of Health Professions, he has directed the Carl B. and Florence E. King Foundation Summer Programs in Biomedical Sciences for graduating high school students. He also started a summer workshop for high school science educators now named for him.
His passion for preventing cancer led to developing Project S.A.F.E.T.Y. (Sun Awareness for Educating Today's Youth), a skin cancer awareness curriculum now in nearly 42,000 public schools nationwide.
Advice from someone who's been there
Tomasovic was recruited to MD Anderson in 1980 as an assistant professor in the Department of Tumor Biology and began teaching in the Graduate School of Biomedical Sciences (GSBS) the next year.
Stephen Tomasovic, Ph.D., prepares to give his keynote address at
the 2011 commencement for the School of Health Professions,
where he shared advice he had used during 31 years at MD Anderson.
Photo: Kevin McGowan
He developed the cancer biology program at GSBS, was honored as the John P. McGovern Outstanding Teacher, was elected president of the graduate school faculty and, in 1999, was named associate dean.
For MD Anderson, he served as founding chair of the Institutional Animal Care and Use Committee and the Faculty Senate. Since 1998, he has held key administrative appointments that culminated in 2006, when he was appointed MD Anderson's senior vice president for academic affairs.
In that position, he has been responsible for coordinating policies and programs affecting more than 7,000 trainees, 1,500 faculty members and 250 support staff.
Among their many honors for educational contributions, Tomasovic and Ahearn both were inducted into The University of Texas System's Academy of Health Science Education and named UT System Distinguished Teaching Professors.
They also shared the podium at the August graduation for 120 students at the School of Health Professions.
In his keynote address, Tomasovic told the students they will have "many opportunities and challenges as well as stress and worry" in their chosen careers.
His advice included to "listen well, think critically, maintain balance, be adaptable, problem solve not problem create, and deliver dependably."
That advice obviously helped Ahearn and Tomasovic succeed.
New Graduate, Undergraduate Leaders
Two new leaders began directing educational programs on Sept. 1:
Oliver Bogler, Ph.D., who joined MD Anderson's Department of Neurosurgery in 2005, was named senior vice president for academic affairs. He will continue as vice president for Global Academic Programs until a successor is selected. Bogler also directs the Odyssey Program
that supports outstanding postdoctoral fellows and chairs the National Brain Tumor Society's Scientific Advisory Council.
Shirley Richmond, Ed.D., former dean of the College of Health and
Human Sciences at Northern Illinois University (NIU), was appointed
dean of MD Anderson's School of Health Professions. Prior to going to
NIU a decade ago, she spent 15 years at The University of Texas Medical Branch at Galveston in several leadership positions. She is a past president of the Texas Society of Allied Health Professions.
Latest Research Advances
Lung cancer screenings now available
Addresses a dangerous drug shortage
New drug shows great promise for all
Vaccine Improves Response Rates, Survival
Offers Benefits for Breast Cancer Patients
Some Head and Neck Cancer Patients Could Benefit
Could Guide Treatment for Breast Cancer Patients
MD Anderson's fourth full-time president
Ronald DePinho, M.D., has a mission — to impact the human condition.
Among Leaders of Cancer Genome Atlas Project
As a busy medical student in 1978 at Albert Einstein College of Medicine in New York, Ronald DePinho found time to follow another passion — martial arts.
DePinho Focuses on Molecular Basis of Link
Although presidential responsibilities take most of his time, Ronald DePinho, M.D., continues his research efforts to bring scientific knowledge to cancer patient care.
Four Mentors Help Shape DePinho's Career
Though a grand master would seem a non-traditional mentor for a physician-scientist, Ronald DePinho, M.D., credits Ik Jo Kang and martial arts with his being a good student.
'Colbert Report' Allows DePinho to Connect With Young Audience
Last winter was an unusually busy time for Ronald DePinho, M.D.
Cancer Patients Bank on the Kindness of Donors
Most people go through life never having to receive a blood transfusion. But the same can't be said for cancer patients.
Bus Rounds Reveals Home Hospice Care Landscape
As a bright red bus pulls away from the curb in front of MD Anderson, Eduardo Bruera, M.D., begins to share his extensive experience of patients in the palliative setting.
Lines Between Medical, Geriatric Oncology Blur
Elba Circelli is a sharp, energetic 86-year-old living in Friendswood, a Houston suburb.
Group Addresses Concerns, Shares Resources
Physician Tracks Real-Life Patterns of Care
Virtual Environment Great Teaching Tool for Radiation Therapy Students
School of Health Professions Key for Coleman
Khalifa Bin Zayed Al Nahyan Foundation Provides MD Anderson With $150 Million Grant
Latest MD Anderson News
A piece of MD Anderson heads to space
Astronaut honors institution with rare opportunity
Banner MD Anderson Cancer Center Opens
New Administrative Facility Helps Unite Staff
Leading Hospital Again in U.S. News Survey
By Julie Penne
Young patients 'Empowered' by opportunity to create
Karissa Ma
October holds special meaning for Karissa Ma.


text:
Total anomalous of pulmonary venous connection in infants less than 3 months old
M. Hamawaki, T. Tomino, H. Sato, M. Imura, T. Suzuki, Narutoshi Hibino, M. Inukai, T. Kuji, T. Konuma
We examined the surgical results of total anomalous pulmonary venous connection (TAPVC) retrospectively in 6 infants, who were less than 3 months old and underwent a total repair at Ehime Prefectural Central Hospital between May, 1993 through May, 1998, in terms of the pre, peri, and postoperative management, the site of connection, and the surgical procedures. Aged at operation ranged from 1 day to 86 days (mean 39 days), and body weight ranged from 2.4 kg to 5.5 kg (mean 3.4 kg). All 6 patients had echocardiographic diagnosis and cardiac catheterization but one. In operative procedure, cut back method was done in a patient of paracardiac type of Darling's classification and posterior approach was used in total correction for 4 supracardiac and 1 infracardiac type. There were 3 hospital deaths who had poor conditions before operation, but no late deaths. Surgical results of TAPVC might have been improved with advances in non-invasive diagnosis by echocardiography, and pre and perioperative management. And we should take care of these patients of TAPVC in long term period to make sure that they have no pulmonary venous obstruction.
Kyobu geka. The Japanese journal of thoracic surgery
Dive into the research topics of 'Total anomalous of pulmonary venous connection in infants less than 3 months old'. Together they form a unique fingerprint.
Lung Medicine & Life Sciences 100%
Operative Surgical Procedures Medicine & Life Sciences 51%
Cardiac Catheterization Medicine & Life Sciences 48%
Echocardiography Medicine & Life Sciences 39%
Patient Care Medicine & Life Sciences 36%
Body Weight Medicine & Life Sciences 33%
Hamawaki, M., Tomino, T., Sato, H., Imura, M., Suzuki, T., Hibino, N., Inukai, M., Kuji, T., & Konuma, T. (1999). Total anomalous of pulmonary venous connection in infants less than 3 months old. Kyobu geka. The Japanese journal of thoracic surgery, 52(7), 559-562.
Total anomalous of pulmonary venous connection in infants less than 3 months old. / Hamawaki, M.; Tomino, T.; Sato, H. et al.
In: Kyobu geka. The Japanese journal of thoracic surgery, Vol. 52, No. 7, 1999, p. 559-562.
Hamawaki, M, Tomino, T, Sato, H, Imura, M, Suzuki, T, Hibino, N, Inukai, M, Kuji, T & Konuma, T 1999, 'Total anomalous of pulmonary venous connection in infants less than 3 months old', Kyobu geka. The Japanese journal of thoracic surgery, vol. 52, no. 7, pp. 559-562.
Hamawaki M, Tomino T, Sato H, Imura M, Suzuki T, Hibino N et al. Total anomalous of pulmonary venous connection in infants less than 3 months old. Kyobu geka. The Japanese journal of thoracic surgery. 1999;52(7):559-562.
Hamawaki, M. ; Tomino, T. ; Sato, H. et al. / Total anomalous of pulmonary venous connection in infants less than 3 months old. In: Kyobu geka. The Japanese journal of thoracic surgery. 1999 ; Vol. 52, No. 7. pp. 559-562.
@article{7edd18fdd0c646c6b86e8a218baf58dc,
title = "Total anomalous of pulmonary venous connection in infants less than 3 months old",
abstract = "We examined the surgical results of total anomalous pulmonary venous connection (TAPVC) retrospectively in 6 infants, who were less than 3 months old and underwent a total repair at Ehime Prefectural Central Hospital between May, 1993 through May, 1998, in terms of the pre, peri, and postoperative management, the site of connection, and the surgical procedures. Aged at operation ranged from 1 day to 86 days (mean 39 days), and body weight ranged from 2.4 kg to 5.5 kg (mean 3.4 kg). All 6 patients had echocardiographic diagnosis and cardiac catheterization but one. In operative procedure, cut back method was done in a patient of paracardiac type of Darling's classification and posterior approach was used in total correction for 4 supracardiac and 1 infracardiac type. There were 3 hospital deaths who had poor conditions before operation, but no late deaths. Surgical results of TAPVC might have been improved with advances in non-invasive diagnosis by echocardiography, and pre and perioperative management. And we should take care of these patients of TAPVC in long term period to make sure that they have no pulmonary venous obstruction.",
author = "M. Hamawaki and T. Tomino and H. Sato and M. Imura and T. Suzuki and Narutoshi Hibino and M. Inukai and T. Kuji and T. Konuma",
journal = "Japanese Journal of Thoracic Surgery",
publisher = "Nankodo Co., Ltd.",
T1 - Total anomalous of pulmonary venous connection in infants less than 3 months old
AU - Hamawaki, M.
AU - Tomino, T.
AU - Sato, H.
AU - Imura, M.
AU - Suzuki, T.
AU - Hibino, Narutoshi
AU - Inukai, M.
AU - Kuji, T.
AU - Konuma, T.
N2 - We examined the surgical results of total anomalous pulmonary venous connection (TAPVC) retrospectively in 6 infants, who were less than 3 months old and underwent a total repair at Ehime Prefectural Central Hospital between May, 1993 through May, 1998, in terms of the pre, peri, and postoperative management, the site of connection, and the surgical procedures. Aged at operation ranged from 1 day to 86 days (mean 39 days), and body weight ranged from 2.4 kg to 5.5 kg (mean 3.4 kg). All 6 patients had echocardiographic diagnosis and cardiac catheterization but one. In operative procedure, cut back method was done in a patient of paracardiac type of Darling's classification and posterior approach was used in total correction for 4 supracardiac and 1 infracardiac type. There were 3 hospital deaths who had poor conditions before operation, but no late deaths. Surgical results of TAPVC might have been improved with advances in non-invasive diagnosis by echocardiography, and pre and perioperative management. And we should take care of these patients of TAPVC in long term period to make sure that they have no pulmonary venous obstruction.
AB - We examined the surgical results of total anomalous pulmonary venous connection (TAPVC) retrospectively in 6 infants, who were less than 3 months old and underwent a total repair at Ehime Prefectural Central Hospital between May, 1993 through May, 1998, in terms of the pre, peri, and postoperative management, the site of connection, and the surgical procedures. Aged at operation ranged from 1 day to 86 days (mean 39 days), and body weight ranged from 2.4 kg to 5.5 kg (mean 3.4 kg). All 6 patients had echocardiographic diagnosis and cardiac catheterization but one. In operative procedure, cut back method was done in a patient of paracardiac type of Darling's classification and posterior approach was used in total correction for 4 supracardiac and 1 infracardiac type. There were 3 hospital deaths who had poor conditions before operation, but no late deaths. Surgical results of TAPVC might have been improved with advances in non-invasive diagnosis by echocardiography, and pre and perioperative management. And we should take care of these patients of TAPVC in long term period to make sure that they have no pulmonary venous obstruction.
JO - Japanese Journal of Thoracic Surgery
JF - Japanese Journal of Thoracic Surgery


text:
NASCAR announces safety additions, competition changes
By Dustin LongMay 1, 2020, 4:54 PM EDT
NASCAR issued a rules bulletin Friday to teams that included safety enhancements developed from the investigation of Ryan Newman's car after his Daytona 500 crash.
Newman suffered a brain bruise in a last-lap crash of the Feb. 16 event. He missed three races before the season was suspended by the coronavirus pandemic. He has been medically cleared by NASCAR to race.
NASCAR also announced competition adjustments Friday.
"As teams prepare for the return to racing, we want to provide as much advance notice as possible for upcoming technical changes," said John Probst, NASCAR senior vice president of racing development, in a statement. "Some of these updates stem from the investigation into the (Newman) incident at Daytona, and all are intended to produce a safe and competitive race at all venues. We look forward to providing more details in the near future."
The safety enhancements include:
Elimination of aero ducts at superspeedway tracks.
Reduction in size of throttle body from 59/64" to 57/64" (superspeedways only).
Updated roll bar padding specifications (mandatory at all tracks beginning June 1).
Oil reservoir tank or overflow expansion tank must contain a check valve (mandatory at all tracks beginning with Talladega).
Slip tape must be applied along the entire length of the lower rearward facing surfaces of the rear bumper cover and extension (superspeedways only).
Addition of a lower main roll bar support bar #20 / intrusion plate and upper main roll bar support bar #21 (mandatory superspeedways, optional elsewhere).
Click here for image of new roll bars and plates (shown in yellow)
The competition changes include:
The temporary ban on most testing will be lifted Monday (May 4), however on-track testing will not be allowed in the Cup, Xfinity or Gander Truck Series for the remainder of the 2020 season.
Organizations are allocated a total of 150 hours in the wind tunnel through Dec. 31, 2021 with a maximum usage of 70 hours in 2020 and 90 hours in 2021. Wind tunnel testing of Next Gen vehicles by individual organizations is not permitted.
All remaining parts submission meetings for 2020 have been cancelled.
Minimum number of short block sealed engines changes from 13 to eight.
In regards to the testing issues, NASCAR banned all testing not related to the Next Gen car on March 17. That included but was not limited to wind tunnels (full and scale models), climatic tunnels, 7/8 posters, K&C rigs (static and dynamic) and driver simulators, etc.


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text:
Rumänien nahm an den Olympischen Winterspielen 1968 im französischen Grenoble mit 30 Athleten, 29 Männer und eine Frau, teil.

Seit 1928 war es die achte Teilnahme Rumäniens bei Olympischen Winterspielen.

Flaggenträger
Die Eiskunstläuferin Beatrice Huștiu trug die Flagge Rumäniens während der Eröffnungsfeier.

Medaillen
Mit einer gewonnenen Bronzemedaille belegte das rumänische Team Platz 15 im Medaillenspiegel.

Ion Panțuru und Nicolae Neagoe gewannen mit Bronze im Zweierbob die erste rumänische Medaille überhaupt bei Olympischen Winterspielen.

Bronze
 Ion Panțuru und Nicolae Neagoe: Zweierbob

Teilnehmer nach Sportarten

Biathlon
 Nicolae Bărbășescu
 Einzel (20 km): 29. Platz
 Staffel (4 × 7,5 km): 7. Platz
 Gheorghe Cimpoia
 Einzel (20 km): 23. Platz
 Staffel (4 × 7,5 km): 7. Platz
 Vilmoş Gheorghe
 Einzel (20 km): 22. Platz
 Staffel (4 × 7,5 km): 7. Platz
 Constantin Carabela
 Einzel (20 km): 14. Platz
 Staffel (4 × 7,5 km): 7. Platz

Bob
 Ion Panțuru
 Zweierbob:  Bronze
 Viererbob: 4. Platz
 Nicolae Neagoe
 Zweierbob:  Bronze
 Viererbob: 4. Platz
 Romeo Nedelcu
 Zweierbob: Qualifikation
 Gheorghe Maftei
 Zweierbob: Qualifikation
 Viererbob: 4. Platz
 Petre Hristovici
 Viererbob: 4. Platz

Eiskunstlauf
 Beatrice Huștiu
 29. Platz

Eishockey
 Männer: 12. Platz

Ski Alpin
 Dorin Munteanu
 Riesenslalom, Männer: 55. Platz
 Abfahrt, Männer: 65. Platz
 Slalom, Männer: Qualifikation
 Dan Cristea
 Riesenslalom, Männer: 49. Platz
 Abfahrt, Männer: 62. Platz

Weblinks

Rumänien
Winterspiele 1968
Wintersport (Rumänien)

redpajama_set_name: RedPajamaWikipedia


My love affair with Paris began when I first traveled to Europe at age 17.
I have not been the same since.
Paris was where I spent my senior year of college as a student at the Sorbonne.
I am convinced, however, that I learned more in my experience as an au pair girl for a delightful French family with three toddlers. When we all grew up, I photographed the weddings of the two girls for whom I was the au pair.
photos give a glimpse of apartment life while others show what enchanted me as I ambled on foot or made my way around Paris by bus or metro.