text:
\section{Introduction}
\label{sec:intro}
The historical detection of gravitational waves (GWs) by the LIGO/Virgo Collaboration has marked the beginning of a new era in astrophysics
and the birth of GW astronomy~\cite{Abbott:2016blz}. The next generation of detectors will routinely observe the coalescence
of compact objects, such as black holes (BHs) and neutron stars. These observations will probe, for the first time, the highly dynamical
regime of strong-field gravity and may provide the answer to long-standing
issues~\cite{Cardoso:2016rao,Nakano:2016sgf,Yunes:2016jcc}. Is cosmic censorship
preserved in violent gravitational interactions? Do GW observations carry incontrovertible evidence for the event horizon of BHs? Can we pinpoint, in gravitational waveforms, the signature of the light ring or of ergosurfaces?

Simultaneously, the entire coalescence process can be used to constrain gravity theories in novel ways~\cite{Berti:2015itd,TheLIGOScientific:2016src,Yunes:2016jcc}. That general relativity (GR) is not the ultimate theory of gravity is a possibility that should be entertained
in the light of several observations (such as those related to the dark-matter and the dark-energy problems),
and of the difficulty to reconcile GR with quantum field theory~\cite{Berti:2015itd}.
Although such an extension of GR is unknown---and a robust spacetime parametrization in strong-field gravity is lacking---GW observations will help us to exclude or to strongly constrain
wide classes of alternative theories. The inspiral stage, for example, when the two objects are far apart, can teach us about possible extra radiation channels~\cite{TheLIGOScientific:2016src,Barausse:2016eii,Cardoso:2016olt,Yunes:2016jcc}, while the final ringdown stage---when the end-product is relaxing to its final state---provides for remarkable tests of GR, through the measurement of the characteristic quasinormal modes (QNMs)~\cite{Berti:2009kk}.
In GR, as well as in essentially any relativistic theory of gravity, BHs are extremely simple objects described by only a handful of parameters. Accordingly, their QNMs are completely characterized
by only a few parameters as well. For example, Kerr BHs in GR are characterized by their mass and angular momentum, and so are their QNMs.
In a nutshell, measurement of one single QNM (i.e, a ringing frequency and a decay time scale~\cite{Berti:2005ys,Berti:2009kk}) allows for a determination of the BH mass and angular momentum. The measurement of a second QNM tests GR~\cite{Berti:2005ys,Berti:2007zu,Gossan:2011ha,Meidam:2014jpa}. In the context of modified theories of gravity, a second QNM can be used to measure possible extra coupling parameters, as was shown recently
for a theory with an extra vector degree of freedom~\cite{Cardoso:2016olt}.

Some of the most viable and appealing modifications of gravity are those obtained via the inclusion of extra scalar fields---such as scalar-tensor theories of gravity---or of higher-curvature terms in the action, or both. Higher-order gravity is generically motivated by UV corrections, which also arise naturally in some low-energy truncations of string theories. The paradigmatic case, and the one we focus on here, is Einstein-dilaton-Gauss-Bonnet (EDGB) gravity, described by the action~\cite{Kanti:1995vq,Berti:2015itd}
\begin{equation}
S=\int d^4 x\frac{\sqrt{-g}}{16\pi}\left(R-\frac{1}{2}\partial_a\phi\partial^a\phi+\frac{\alpha}{4}e^{\phi}{\cal R}_{\rm GB}^2\right)+S_{m}\,,
\label{eq:action}
\end{equation}
where
\begin{equation}
{\cal R}_{\rm GB}^2={R_{abcd}} R^{abcd}-4 {R_{ab}} R^{ab}+R^2\,,
\end{equation}
is the Gauss-Bonnet topological term, $S_m$ represents the matter sector and we use (throughout this work) units for which the Newton's constant and the speed of light are unity, $G=c=1$.
Current best constraints on the coupling constant $\alpha$ are $\sqrt{\alpha}<10\,{\rm km}$~\cite{Yagi:2012gp,Berti:2015itd}\footnote{We note a typo in the review~\cite{Berti:2015itd}. In the notation used in the review, Eq.~(2.26) should read $\sqrt{|\alpha_{\rm GB}|}\lesssim 5\times10^5\,{\rm cm}$.}.

We will close two important gaps in the literature concerning BHs in this theory:
we first find strong evidence that static EDGB BHs are linearly stable,
and then compute gravitational waveforms from plunging particles,
which can be argued to be an indicator of how BH collisions proceed in this theory.
Finally, we discuss the constraints on the coupling parameter $\alpha$ from current and future ringdown observations.

\section{Framework}
\label{sec:review}
The equations of motion obtained by extremizing \eqref{eq:action} with respect to the metric and dilaton field are given by~\cite{Kanti:1995vq}
\begin{align}
\Box \phi&=\frac{\alpha}{4}e^{\phi}{\cal R}_{\rm GB}^2\label{eq:scalarphi}\,,\\
G_{ab}&=\frac{1}{2}\partial_a\phi\partial_b\phi-\frac{1}{4}g_{ab}
(\partial_c\phi) (\partial^c\phi)
-\alpha {\cal K}_{ab}+8\pi T_{ab} \label{eq:einsteineq}\,,
\end{align}
where $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$ is the Einstein tensor, $T_{ab}$ is the matter stress-energy tensor,
and
\begin{equation}
{\cal K}_{ab}=(g_{ac}g_{bd}+g_{ad}g_{bc})\epsilon^{idjk}\nabla_{l}\left(\tilde{R}^{cl}_{~~jk}\partial_i e^{\phi}\right)\,,
\end{equation}
where $\epsilon^{abcd}$ is the contravariant Levi-Civita tensor, $\tilde{R}^{ab}_{~~cd}=\epsilon^{abij}R_{ijcd}$.
\subsection{BH solutions in EDGB gravity}

BHs in EDGB gravity are scalar-vacuum solutions of the above equations, which were first constructed analytically in spherical symmetry, in the small-coupling regime~\cite{Mignemi:1992nt},
\begin{equation}
\zeta:=\frac{\alpha}{M^2}\ll1 \,,
\end{equation}
where $M$ is the BH mass. In spherical symmetry, the line element reads
\begin{equation}
ds^2=-A(r)dt^2+B(r)^{-1}dr^2+r^2d\Omega^2\,,
\label{eq:linel}
\end{equation}
where $d\Omega^2$ is the standard unit 2-sphere line element.
Both functions $A(r)$ and $B(r)$ and the scalar field $\phi$ can be expanded in powers of the coupling parameter $\zeta$, and the corresponding solutions can be found by solving the field equations~\eqref{eq:scalarphi}--\eqref{eq:einsteineq} perturbatively (see also Ref.~\cite{Yunes:2011we}).
Further details are given in Appendix \ref{app:motion}.

Nonperturbative solutions were investigated numerically in Ref.~\cite{Kanti:1995vq} for static geometries and in Ref.~\cite{Pani:2009wy} for slowly rotating BHs to first order in the spin.
It was shown that static BH solutions exist only up to a maximum value of $\zeta$, namely~\cite{Pani:2009wy}
\begin{equation}
0\leq \zeta \lesssim 0.691\,.
\end{equation}
Because $\zeta$ is strictly less than unity, higher-order perturbative expansions~\cite{Maselli:2015tta} are
accurate almost in the entire parameter space.

Slowly rotating solutions were described numerically in Ref.~\cite{Pani:2009wy}, and analytically in the small-coupling regime in Refs.~\cite{Pani:2011gy,Ayzenberg:2014aka}, and were recently extended to higher order in the coupling and in the spin parameter~\cite{Maselli:2015tta}. In the latter case, the line element
(as well as the scalar field) can be expanded in a complete basis of orthogonal functions according to their symmetry properties~\cite{Maselli:2015tta}.

Numerical solutions describing rotating BHs for arbitrary coupling and spin were found in Ref.~\cite{Kleihaus:2011tg} and have been recently thoroughly discussed in Ref.~\cite{Kleihaus:2015aje}.

The linearized mode stability of spherically symmetric BHs against radial fluctuations was studied in Ref.~\cite{Kanti:1997br}, whereas axial gravitational perturbations were studied in Ref.~\cite{Pani:2009wy}. In the polar sector, the linear stability of EDGB BHs was analyzed in Ref.~\cite{Ayzenberg:2013wua}, focusing in the particular regimes where perturbations are dominantly gravitational or scalar, as well using a high-frequency analysis of the perturbations.
In addition, axial quasinormal modes of neutron stars in EDGB theory were studied in Ref.~\cite{Blazquez-Salcedo:2015ets}.

Due to the cumbersome field equations, there is currently no result concerning the stability of nonrotating EDGB BHs for the most relevant gravitational polar sector, and there is no stability analysis for rotating solutions.
In this work we partly fill this gap by performing a full linear (mode) stability analysis of static EDGB BHs.

\subsection{Perturbed BHs in EDGB}\label{subsec:pert}
We are interested in understanding how BHs in EDGB theory respond to small perturbations, as those induced by a
fluctuation of the metric or of the dilaton or by a small external perturbing object. We focus our attention in both
dilaton fluctuations in vacuum and those induced by a small pointlike particle plunging into the BH. The first case will describe the late-time behavior of perturbed EDGB
BHs, which also dominates the ringdown signal from a distorted BH formed in a coalescence. Pointlike particles, on the
other hand, are a good proxy for small BHs or neutron stars falling into massive BHs, but are also known to provide
reasonably accurate estimates even for equal-mass BH
collisions~\cite{Berti:2010ce,Sperhake:2011ik,Tiec:2014lba}.

Pointlike particles of mass $\mu\ll M$ are modeled by~\cite{Zerilli:1971wd,Poisson:2011nh,Breuer:1974uc}
\begin{align}
S_m=\mu \int d\tau\sqrt{-g}\,,
\end{align}
with $d\tau=d\lambda \sqrt{-g_{ab}\dot x^a\dot x^b}$. We will consider pointlike objects with a scalar charge which is entirely due to the Gauss-Bonnet coupling. In other words, we will investigate BH spacetimes of which the scalar charge arises purely from the Gauss-Bonnet term. The theory we consider contemplates no other couplings to matter, and therefore
pointlike particles follow geodesics. This need not be the case generically, and nontrivial couplings to matter can be envisioned~\cite{Quinn:2000wa,Poisson:2011nh}. These couplings will certainly influence the motion of particles and the radiation in collisions but will not affect the intrinsic ringdown properties of the spacetime.

We consider a spherically symmetric EDGB BH distorted by either the pointlike particle or through some fluctuation in the metric or scalar field. At the linearized level, the full geometry is described by
\begin{align}
g_{ab}&=g_{ab}^{(0)}+\varepsilon\, h_{ab}\,,\\
\phi&=\phi_0(r)+\varepsilon\,\delta\phi\,,
\end{align}
where $\varepsilon \ll 1$ is a bookkeeping parameter, $g_{ab}^{(0)}$ is described by \eqref{eq:linel} and $\phi_0(r)$ is the corresponding background scalar. As background solution, we consider both a perturbative solution~\cite{Maselli:2015tta} up to ${\cal O}(\zeta^6)$, and a numerical solution for arbitrary values of $\zeta$.

The fluctuations $h_{ab}$ and $\delta \phi$ are functions of $(t,r,\theta,\varphi)$. Einstein's equations can be further simplified by Fourier transforming these  quantities
and by expanding them in (tensor and scalar) spherical harmonics, e.g.,
\begin{equation}
\delta\phi(t,r)=\frac{1}{\sqrt{2\pi}}\int d\omega \frac{\phi_1(\omega,r)}{r}Y^{lm} e^{-i\omega t}\,,\label{Fourier}
\end{equation}
where $Y^{lm}$ are the standard spherical harmonics and $\omega$ is a Fourier frequency.

The metric perturbations can be decomposed in terms of tensorial spherical harmonics~\cite{Regge:1957td,Zerilli:1971wd}. By using this decomposition, perturbations naturally split into two sectors according to their parity (either axial or polar). Axial perturbations of EDGB BHs are simpler, because they are decoupled from the scalar-field perturbations~\cite{Pani:2009wy}. Here, we shall consider the two sectors of the perturbations

In the Regge-Wheeler gauge~\cite{Regge:1957td}, the polar sector of the metric perturbations is given by
\begin{align}
&h_{ab}=\left[
\begin{array}{cccc}
A H_0 &  H_1  & 0 & 0 \\
H_1 & H_2/B & 0 & 0\\
0 & 0 & r^2 K & 0\\
0 & 0 & 0 & r^2 \sin^2\theta K
\end{array}\right] Y^{lm} \,,
\label{hab13}
\end{align}
while the axial sector reads
\begin{align}
&h_{ab}=\left[
\begin{array}{cccc}
0                         & 0                         & 0 & \sin\theta~ h_0\partial_\theta\\
0                         & 0                         & 0 & \sin\theta~ h_1\partial_\theta\\
0                         & 0                         & 0 & 0\\
\sin\theta~ h_0\partial_\theta & \sin\theta~ h_1\partial_\theta & 0 & 0
\end{array}\right]\,Y^{lm} \,.
\end{align}
In the above definitions, the metric perturbations depend only on $t$ and $r$. Note that we have already specialized the spacetime to axial symmetry: for the cases handled here, one can always rotate the coordinate axis such
that the spacetime is axially symmetric.
We shall Fourier-decompose these perturbation functions
as, e.g., $X(t,r)=(2\pi)^{-1/2}\int d\omega X(\omega,r)e^{-i\omega t}$.

Likewise, the stress-energy tensor can be decomposed into spherical harmonics~\cite{Zerilli:1971wd,Sago:2002fe}. In the radial plunging case considered here, the particle only disturbs the spacetime in the polar sector, and its stress-energy tensor can be written as
\begin{equation}
T_{ab}=
\left[
\begin{array}{cccc}
A_{lm}^{(0)} & \frac{i}{\sqrt{2}}A_{lm}^{(1)} & 0 & 0 \\
 \frac{i }{\sqrt{2}}A_{lm}^{(1)} & A_{lm}& 0 & 0 \\
 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 \\
\end{array}
\right]Y^{lm}\,,
\end{equation}
where the functions $A_{lm}^{(0)}$, $A_{lm}^{(1)}$ and $A_{lm}$, like the dilaton and metric perturbations, can be Fourier decomposed, such that they depend only on $r$ and $\omega$. The explicit form of these functions is given in Appendix~\ref{app:line}.

The linearized dynamics is governed by a coupled system which can be obtained by expanding the field equations \eqref{eq:scalarphi}--\eqref{eq:einsteineq} up to first order in the perturbation functions $h_{ab}$ and $\delta\phi$. The equations take the schematic form (no sum on $j$)
\begin{equation}
\frac{d}{dr}{\bm{\Psi}}_j+\bm{V}_{j}\bm{\Psi}_j=\bm{S}_j\,,
\label{eq:perturbations}
\end{equation}
where $j=({\rm p},{\rm a})$ for the polar and axial sectors, respectively, $\bm{\Psi}_j$ is a column vector, the components of which are $\bm{\Psi}_{\rm p}\equiv (H_1,K,\phi_1,\phi_1')$ for the polar sector and $\bm{\Psi}_{\rm a}\equiv (h_1,h_0)$ for the axial sector, respectively. The matrices $\bm{V}_{j}$ describe the coupling among the perturbations and depend on the background fields. The vectors $\bm{S}_j$ represent the source terms associated to the point-particle stress-energy tensor. The explicit forms of $\bm{V}_{j}$ and $\bm{S}_j$ are derived in Appendix \ref{app:line}, where we also show that the functions $H_2$ and $H_0$ can be completely determined in terms of $\bm{\Psi}_{\rm p}$. For the axial sector, the source terms vanish identically and we can write the coupled first-order equations into a second-order Schr\"odinger-like equation with an effective potential.

\subsection{Computation of the QNMs}\label{sec:qnms}
The late-time gravitational signal from a perturbed BH is dominated by a sum of exponentially damped sinusoids, the QNMs, which correspond to the characteristic vibration modes of the spacetime~\cite{Nollert:1999ji,Berti:2009kk}.
The QNMs are solutions of the sourceless wave equations~\eqref{eq:perturbations}
(i.e., $\bm{S}_j=0$), along with proper boundary conditions---purely outgoing
waves at infinity and ingoing waves at the horizon.
The latter correspond to the following behavior of the wave function at the
boundaries
\begin{equation} \bm \Psi_j\propto \left\{ \begin{array}{ll} e^{-i\omega r_*}\,,&r\sim r_h \,,\\
e^{i\omega r_*}\,, &r\to\infty\,, \end{array}\right.  \end{equation}
where $r_h$ is the horizon radius in these coordinates and the ``tortoise''
coordinate $r_*$ is defined by
\begin{equation} \frac{dr_*}{dr}=\frac{1}{\sqrt{A B}}\,.  \end{equation}

To compute the QNMs, we use a direct-integration method. We construct a square matrix $\bm X$ (four and two dimensional in the polar and axial case, respectively) of which the columns are independent solutions of Eq.~\eqref{eq:perturbations}. This matrix can be constructed by a certain combination of the solution which is regular at the horizon and the one which is regular at infinity (cf. Refs.~\cite{Pani:2013pma,Macedo:2016wgh} for details). For the polar sector, we note that the boundary conditions defining the QNM eigenvalue problem depend on two parameters, related to the amplitudes of the scalar and gravitational perturbations.
Two of the columns of $\bm X$ can be constructed by integrating the equations from the horizon outward. Likewise, two other solutions can be constructed by integrating the equations from infinity inward. In general, the solutions integrated from the horizon are linearly independent from the ones integrated from infinity, unless $\omega$ is the QNM frequency. In other words, the QNM frequencies are obtained by imposing
\begin{equation}
{\rm det} (\bm X)|_{r=r_{\rm m}}=0\,,
\end{equation}
where $r_{\rm m}$ is an arbitrary matching point of the order of the horizon
radius. The same procedure can be done for the axial modes, with the difference
that the boundary conditions apply only to the gravitational amplitude and therefore the problem is technically
less involved.

We have computed the lowest QNMs of static BHs using both the full numerical background and the perturbative analytical solution given by Eqs.~\eqref{eq:expme}--\eqref{eq:expphi}, up to $N=4$, i.e. up to ${\cal O}(\zeta^6)$. We checked the numerical stability of the QNM frequencies against changes in the values of the numerical horizon $r_h$, numerical infinity and the matching radius $r_{\rm m}$ (typically, we use $r_{\rm m}\sim 4 r_h$).
One of the advantages of the above procedure is that it can be applied to both the perturbative solution and the full numerical one.

\subsection{Plunging particles}\label{sec:plunge}

To obtain the metric and dilaton perturbations due to a particle plunging into a BH, we need to solve the inhomogeneous system~\eqref{eq:perturbations}. Two common methods used in the literature to solve this problem are a direct integration and a Green's function approach (see, e.g., Ref.~\cite{Cardoso:2016olt}).

The Green's function method relies on the fundamental matrix $\bm X$, as constructed by using the homogeneous solutions as discussed in the previous section. The formal solution of \eqref{eq:perturbations} can be written as \cite{boyce2008elementary}
\begin{equation}
\bm \Psi_j=\bm X \bm \beta +\bm X\int dr \bm X^{-1}\bm S\,,
\label{eq:formal_sol}
\end{equation}
where $\bm \beta$ is a constant vector to be determined by imposing the proper boundary conditions. Thus, once the fundamental matrix $\bm X$ of the homogeneous problem is computed, its convolution with the source term $\bm S$ in \eqref{eq:formal_sol} yields the solution of the inhomogeneous problem. However, in many problems, the source term might converge slowly at the boundaries or even diverge, as in our case in which the source term diverges at the event horizon. These problems can be avoided by performing a suitable nontrivial transformations of the perturbation functions~\cite{Sasaki:1981kj,Cardoso:2002jr}, or with a careful choice of Green's function~\cite{Poisson:1996ya}.

A different scheme which avoids this problem consists in integrating directly the full inhomogeneous system, by imposing the proper boundary conditions for the full solution. First, we expand the perturbation functions near the horizon as
\begin{equation}
\bm \Psi_j(r\to r_h)\approx \sum_{k=0}^{N}(r-r_h)^{k+p}\bm \psi_{j,k}e^{-i\omega r_*} +\bm \Psi_{j,H}\,,
\end{equation}
where $p$ is a constant chosen such that the above expansion satisfies the inhomogeneous equations near the horizon and $\bm \Psi_{j,H}$ is the (ingoing) solution of the homogeneous equation. The above expansion is solved iteratively near the event horizon for the coefficients $\bm \psi_{j,k}$ up to $k=N$, and they generically depend on constants (say $\bm \psi_{j,0}$) which are related to the amplitude of the fields at the horizon. The amplitudes at the horizon are then used as shooting parameters; i.e. we chose them such that the numerical solution satisfies the proper boundary conditions also at infinity. For arbitrary amplitudes at the horizon, the numerical solution far from the BH is a combination of ingoing and outgoing waves, i.e.,
\begin{equation}
\bm \Psi_j=\bm\Psi_j^{out}+\bm\Psi_j^{in}\,,
\end{equation}
and the required solution is obtained by setting the amplitude of the ingoing waves to zero.  In the EDGB case, the amplitudes are related to the gravitational and dilaton perturbations, and therefore the problem is a two-parameter shooting problem for the amplitudes at the horizon. Note that for radial plunging, since the source terms for the axial sector are zero, the particle only induces perturbations in the polar sector.

With the numerical solution at hand, one can compute the gravitational and scalar energy spectra at infinity. This is achieved through the effective stress-energy tensor for the gravitational perturbations (i.e., the Isaacson tensor) and the stress-energy tensor of the dilaton field \cite{Isaacson:1968zza}. As shown in Ref.~\cite{Stein:2010pn}, the Isaacson tensor is the same in GR and in a class of theories including EDGB gravity. The gravitational flux, for a given multipole $l$, can be written as \cite{Zerilli:1971wd}
\begin{equation}
\frac{dE_g}{d\omega}= \frac{1}{32\pi} \frac{(l+2)!}{(l-2)!}|K(r\to\infty)|^2\,,
\end{equation}
where $K(r)$ is the polar perturbation given by Eq. (\ref{hab13}).~\footnote{Note that, formally, $K$ is a gauge-dependent quantity whereas the use of gauge-independent quantities is obviously desired. Moreover, at large distances the Regge-Wheeler gauge is not well defined.
However, in this limit the function $K$ yields the so-called Zerilli function $Z(r)$~\cite{Zerilli:1971wd} (which is a gauge-invariant quantity~\cite{Moncrief:1974am}) since $K(r\rightarrow\infty)=i\omega Z(r\rightarrow\infty)$.}

The dilaton flux reads
\begin{equation}
\frac{dE_\phi}{d\omega}=\frac{\omega^2}{16\pi}|\phi_1(r\to\infty)|^2\,,
\end{equation}
where $\phi_1(r)$ is given by the dilaton perturbation in Eq. (\ref{Fourier}).

As shown in Appendix~\ref{app:line}, since the particle does not have a direct coupling to the dilaton field, dilaton radiation only exists due to the gravitational perturbations. In this sense, the gravitational perturbations work as a ``source'' for the dilaton radiation. Since in the dilaton equation they are proportional either to the parameter $\zeta$ or the background scalar field (and derivative), it is natural to expect that the dilaton radiation scales with $\zeta^2$. Additionally, for $l=0$ and $l=1$ the perturbations of EDGB BHs can be written as a single second-order ordinary differential equation, which represents the dilaton perturbation (see Appendix~\ref{app:line}).

\section{Stability and QNMs of EDGB black holes}
\begin{figure*}
\includegraphics[width=\columnwidth]{axial_real_finite.pdf}\includegraphics[width=\columnwidth]{axial_imaginary_finite.pdf}%
\caption{Real (left) and imaginary (right) parts of the axial $l=2$ fundamental mode, normalized by the Schwarzschild ($\zeta=0$) values. We compare three different approaches: the geodesic (black dotted line), the small-coupling limit (red dashed line), and the finite $\zeta$ regime (blue solid line). In the inset we show that for small values of $\zeta$ the small-coupling limit (red squared) agrees well with the finite-coupling result up to $\zeta\approx 0.3$.}%
\label{fig:axial_modes}
\end{figure*}
As previously discussed, the QNMs govern the late-time behavior of any small fluctuation away from axisymmetry of the BH. Since the time dependence is of the form $e^{-i\omega t}$, the {\it absence} of a QNM frequency with the {\it positive imaginary} part in the spectrum implies that all fluctuations decay exponentially with time. Thus, a criterion for linearized mode stability of the spacetime is that all its QNM frequencies have a negative imaginary part~\cite{Berti:2009kk}. In addition, the late-time dynamics is controlled by the fundamental QNM, i.e., the mode with the smallest imaginary component (equivalently, with the longest decay time).

The Schwarzschild spacetime is stable, and its fundamental QNM frequency, $\omega^{S}=M\omega_{R}^{S}+iM\omega_{I}^{S}$, for the $l=2$ mode reads~\cite{Chandrasekhar:1975zza,Berti:2009kk,ref:webpage}
\begin{eqnarray}
 M\omega^{S}&\approx&0.3737-i\,0.08896 \quad {\rm gravitational}\label{eq:qnm_s}\\
 M\omega^{S}&\approx&0.4836-i\,0.09676 \quad {\rm scalar}\label{eq:qnm_ss}
\end{eqnarray}
for the gravitational and scalar fundamental modes, respectively, where $M$ is the BH mass. In EDGB gravity, because the coupling $\zeta$ is smaller than unity, we expect that the fundamental QNM frequencies\footnote{In the $\zeta\to0$ limit the scalar sector is decoupled and we expect to recover the scalar QNMs of a Schwarzschild BH. This is confirmed by the computation presented in this section.} are only slightly different from Eqs.~\eqref{eq:qnm_s} and \eqref{eq:qnm_ss}. In other words, we expect EDGB BHs to be stable for sufficiently small coupling.
We will now study if these BHs are stable throughout all values of $\zeta$, and also quantify the deviation from the corresponding Schwarzschild value.

We have computed the QNMs of EDGB BHs with two independent codes, both in a small-$\zeta$ expansion and using the full numerical background.
In the small-coupling limit, where the expressions can be expanded in powers of $\zeta$ (see Appendix~\ref{app:motion}), the real and imaginary parts of the QNM frequencies can be written as
\begin{equation}
\frac{\omega_{R}}{\omega_{R}^{S}}=1+\sum_{j=1}^{N}R_{j}\zeta^j \,,\quad\frac{\omega_I}{\omega_{I}^{S}}=1+\sum_{j=1}^{N}I_{j}\zeta^j \,,
\label{eq:expan_modes}
\end{equation}
where $\omega_{R}^{S}$ and $\omega_{I}^{S}$ are, respectively, the real and imaginary parts of the modes of Schwarzschild BH with the same mass $M$.
The coefficients $R_j$ and $I_j$ are obtained by fitting the numerical data with the above expression and depend on $l$ and on the nature of the mode

\subsection{QNMs, light ring, and geodesic correspondence} \label{sec:geodesics}
Computing the QNMs of spacetimes known only numerically can be challenging. For
spherically symmetric spacetimes, a WKB-type analysis has shown that, in the
eikonal ($l\gg 1$) limit, the QNMs can be obtained using only properties of the
light ring (which defines the radius of the photon sphere of the BH). This ``null geodesic
correspondence''~\cite{Ferrari:1984zz,Cardoso:2008bp,Yang:2012he} is useful as
it requires only manipulation of background quantities which are easy to obtain,
and provides a clear physical insight into the QNMs of BHs: they correspond to
waves trapped near the peak of the potential barrier for null particles (i.e., within the photon sphere), slowly
leaking out on a time scale given by the geodesic instability time scale.

The geodesic correspondence only works, formally, in the $l\gg 1$ regime, but can be used even at low $l$ using an appropriate calibration. In fact, this approach has proven to provide reliable results also for low-$l$ modes for a variety of BH spacetimes~\cite{Cardoso:2008bp}, including Kerr-Newman BHs~\cite{Cardoso:2016olt}.

By extending the analysis of Refs.~\cite{Ferrari:1984zz,Cardoso:2008bp,Yang:2012he}, Ref.~\cite{Cardoso:2016olt} recently showed that the complex QNMs of a stationary and axisymmetric BH in the eikonal limit can be written as
\begin{equation}
 \omega_R+i\omega_I \sim \Omega l-i(n+1/2)|\lambda|\,, \label{eikonal}
\end{equation}
where $n$ is the overtone number,
\begin{eqnarray}
 \Omega &=&\frac{-g_{t\varphi}'+\sqrt{{g_{t\varphi}'}^2-g_{tt}' g_{\varphi\varphi}'}}{g_{\varphi\varphi}'}\,,
\end{eqnarray}
is the orbital frequency at the light ring on the orbital plane, and
\begin{eqnarray}
 \lambda = -\frac{1}{\dot t}\sqrt{\frac{V''}{2}}\,, \qquad \dot t=-\frac{E^2 g_{\varphi\varphi}+L g_{t\varphi}}{g_{t\varphi}^2-g_{tt}g_{\varphi\varphi}}
\end{eqnarray}
is the Lyapunov coefficient evaluated at the light-ring location on the equatorial plane. In the above expression, a prime denotes radial derivative, whereas $E$ and $L$ are the (conserved) specific energy and angular momentum of the geodesic, and $V$ is the effective radial potential. The expression~\eqref{eikonal} is valid for $l=m\gg1$ modes. A more involved result for the QNMs of a Kerr BH with generic $l\gg1$ and $|m|\leq l$ is derived in Ref.~\cite{Yang:2012he}.

The geodesic correspondence has been formally proven for Kerr BHs and for a variety of spacetimes, but it has never been checked for BH solutions in modified gravity. This is particularly interesting in light of the breaking of the isospectrality of the axial and polar QNMs of BHs in EDGB theory, as we discuss in the next section. It is important at this stage to point out that the geodesic approximation must fail to capture some of the features of the full problem. It is well adapted, in principle, to describe
the effects of rotation, but cannot take into account (at least not blindly) the presence of extra degrees of freedom like scalars in EDGB. Thus, using this correspondence to extract the QNMs of BHs in modified gravity should be done carefully.

As previously discussed, the background solution and the metric of a spinning BH in EDGB theory is known analytically up to ${\cal O}(\chi^5,\zeta^7)$~\cite{Maselli:2015tta} and numerically for any value of $\chi$ and $\zeta$~\cite{Kleihaus:2011tg,Kleihaus:2015aje}, where $\chi$ is the dimensionless angular momentum parameter,
\begin{equation}
\chi=J/M^2\,.
\end{equation}
We will use this result in Sec. \ref{detectability}, to estimate the modes of spinning EDGB BHs.

\subsection{Axial modes}

\begin{table*}%
\caption{Numerical value of the coefficients $R_j$ and $I_j$ for the expansions in the small-coupling limit, cf.~\eqref{eq:expan_modes} for the axial QNMs. The geodesic coefficients are computed from the exact analytical solution for small $\zeta$ limit, while the QNM frequencies coefficients are obtained through a polynomial fit with the data.}
\label{tab:coeff_axial}
\begin{tabular}{ccccc}
\hline\hline
  & $j$  & $l=2$  & $l=3$ & Geodesic \\
\hline
\multirow{4}{*}{$R_j$}
 & 1    &  0 					& 0 & 0 \\
 & 2    &  $1.002\times10^{-3}$  & $1.173\times 10^{-2}$  & $1.257\times 10^{-2}$  \\
 & 3    &  $1.906\times 10^{-3}$ & $5.035\times 10^{-3}$  & $6.872\times 10^{-3}$  \\
 & 4    &  $1.131\times 10^{-3}$ & $1.353\times 10^{-2}$  & $5.537\times 10^{-3}$ \\ \hline
\multirow{4}{*}{$I_j$}
 & 1    &	 0  & 0 & 0\\
 & 2    & $-5.174\times 10^{-3}$ & $-4.774\times 10^{-3}$  & $-5.267\times 10^{-3}$ \\
 & 3    & $5.766\times 10^{-3}$  & $7.590\times 10^{-4}$   & $-7.184\times 10^{-3}$ \\
 & 4    & $-7.091\times 10^{-3}$ & $-3.282\times 10^{-3}$  & $-7.822\times 10^{-3}$ \\
\hline \hline
\end{tabular}
\end{table*}

The axial sector of gravitational perturbations is decoupled from the scalar-field perturbations, and hence is simpler to study. Using the direct-integration procedure described in Sec.~\ref{sec:qnms}, we have computed the axial QNMs both in a small-$\zeta$ expansion and in the full numerical background. Our results are summarized in Fig.~\ref{fig:axial_modes} and in Table~\ref{tab:coeff_axial}.

In Fig.~\ref{fig:axial_modes}, we show the behavior of the axial $l=2$
fundamental mode as a function of the coupling $\zeta$, normalized by the
corresponding Schwarzschild quantity.  In most of the range of $\zeta$, the
behavior of the modes is smooth and given by a corresponding small deformation
of the Schwarzschild QNMs. The only exception occurs close to the critical value
of the coupling constant $\zeta \approx 0.691$, where the QNMs have a very
sensitive dependence on $\zeta$. For small values of $\zeta$, say
 for $\zeta\lesssim 0.4$,  analytically
expanded backgrounds [up to ${\cal O}(\zeta^6)$] yield QNMs which are in very
good agreement with the full numerical solution. This provides a nontrivial
check for both our (independent) codes.

Figure~\ref{fig:axial_modes} also shows the result of the geodesic algorithm
described in Sec.~\ref{sec:geodesics}.
For the axial modes, which are decoupled from the scalar perturbations,
the geodesic predictions are in good agreement with the results of
the full numerical solution.
Although not shown in Fig.~\ref{fig:axial_modes}, the agreement is better for higher multipoles, as
expected (cf. Table~\ref{tab:coeff_axial}).

Finally, Table~\ref{tab:coeff_axial} shows the results of the polynomial
fit~\eqref{eq:expan_modes} to the axial QNM of EDGB BHs, which is specially
accurate for small $\zeta$. By analyzing the perturbation equations, it is easy
to show that $R_1=I_1=0$ in the expansion~\eqref{eq:expan_modes}. The full
numerical results are available online~\cite{ref:webpage}.
In Table~\ref{tab:coeff_axial}, we also give the coefficients obtained by the geodesic algorithm, which help to quantify the accuracy of the geodesic approximation; for instance at $\zeta=0.5$ the difference in percentage between the real and the
imaginary parts of the (not-normalized) frequency, relative to the numerical result,
is, respectively, less than $3\%$ and less than $8\%$ for $l=2$, and improves for $l>2$. Similar deviations are obtained in the GR limit, $\zeta=0$.

As explained in Appendix~\ref{app:line}, similarly to what happens in GR, there are no axial QNMs for $l=0$ and $l=1$.

\begin{figure*}[ht]%
\includegraphics[width=\columnwidth]{polar_grav}\includegraphics[width=\columnwidth]{polar_scal}%
\caption{Real (left) and imaginary (right) parts of the polar quasinormal modes for $l=2$, for the gravitational- and scalar-led modes, as functions of the coupling $\zeta$, normalized by the Schwarzschild-limit quantities. The insets show a closeup in order to see the comparison for small values of $\zeta$.

}%
\label{fig:polar_modes}%
\end{figure*}
\subsection{Polar modes}

\begin{table}%
\caption{Numerical value of the coefficients $R_j$ and $I_j$ for the polar gravitational-led and scalar-led modes.
}
\label{tab:coeff}
\begin{tabular}{c c c c}
\hline\hline
  & $j$ & {\footnotesize Polar, gravitational $l=2$} & {\footnotesize Polar, scalar $l=2$}  \\
\hline
\multirow{4}{*}{$R_j$}
 & 1 & 0                     & $-1.408\times 10^{-2}$ \\
 & 2 & $-3.135 \times 10^{-2}$ & $1.127\times 10^{-1}$  \\
 & 3& $-9.674\times 10^{-2}$ & $-1.462\times 10^{-1}$  \\
 & 4 & $2.375\times 10^{-1}$ & $5.334\times 10^{-1}$\\ \hline
\multirow{4}{*}{$I_j$}
 & 1 & 0                  &$5.580\times 10^{-2}$\\
 & 2  & $4.371\times 10^{-2}$ &$-6.780\times 10^{-2}$ \\
 & 3  & $1.794\times 10^{-1}$ &$1.042\times 10^{-1}$\\
 & 4  & $-2.947\times 10^{-1}$ & $-2.868\times 10^{1}$\\
\end{tabular}
\begin{tabular}{c c c c}
\hline\hline
  & $j$ & {\footnotesize Polar, gravitational $l=3$}  & {\footnotesize Polar, scalar $l=3$}   \\
\hline
\multirow{4}{*}{$R_j$}
 & 1 & 0                     & $-6.361 \times 10^{-3}$ \\
 & 2 & $-9.911\times 10^{-2}$ & $1.442\times 10^{-1}$  \\
 & 3& $-4.907 \times 10^{-2}$ & $1.168\times 10^{-1}$  \\
 & 4 & $9.286\times 10^{-2}$ & $-1.803\times 10^{-1}$\\ \hline
\multirow{4}{*}{$I_j$}
 & 1 &  0                  &$2.906\times 10^{-3}$\\
 & 2  & $7.710\times 10^{-2}$ &$-5.670\times 10^{-2}$ \\
 & 3  & $1.399\times 10^{-1}$ &$-1.445\times 10^{-1}$\\
 & 4  & $-3.450\times 10^{-1}$ & $2.105\times 10^{-1}$\\
\hline \hline
\end{tabular}
\end{table}

Unlike the axial sector, the polar gravitational sector of the metric perturbations couples to the scalar-field perturbations.
The system of ODEs is more complex and finding the QNM frequencies is therefore more challenging. Even for arbitrarily small values of $\zeta$ the QNMs contain two families: (i) \emph{gravitational-led} modes, which reduce to the gravitational QNMs of Schwarzschild BHs in the $\zeta\to0$ limit, and (ii) \emph{scalar-led} modes, which reduce to the QNMs of a test scalar field on a Schwarzschild metric when $\zeta\to0$ (see  Ref.~\cite{Molina:2010fb} for a similar situation in another theory). The extent to which each of these modes is excited in actual physical setups
is discussed in the next section.

Our results for the quadrupole modes ($l=2$) are summarized in Fig.~\ref{fig:polar_modes} and
Table~\ref{tab:coeff}, where we show the fundamental gravitational-led and
scalar-led QNMs. The deviations from the GR case are larger than in the axial
case. This is probably due to the extra coupling between gravitational and
scalar degrees of freedom.  From the fits given in Table~\ref{tab:coeff}, it is
interesting to note that the leading-order correction to the $l=2$
gravitational-led mode has an opposite sign compared to the axial mode. In
particular, since the geodesic correspondence predicts only one type of modes
and the latter are in good agreement with the axial modes, we find that the
behavior of the polar modes is not captured by the geodesic correspondence; therefore, we do not plot the geodesics results in the figure.
This qualitative difference is
expected, because the axial potential resembles the geodesic potential at large
$l$, whereas in the polar sector the coupling to scalar perturbations
drastically changes the dynamics of the perturbations. Likewise, there is no
reason to expect that the behavior of scalar-led perturbations is well captured
by the geodesic correspondence, at least for small values of $l$.

Due to the coupling between the dilaton and gravitational perturbations, there are also nontrivial $l=0,\,1$ scalar-led modes for EDGB BHs. These reduce to their respective scalar modes in the Schwarzschild spacetime when $\zeta\to 0$. The results at finite coupling follow the trend of higher multipoles.

\subsection{Mode stability}

From the above results, it is clear that the fundamental QNMs of an EDGB BH change at most by a few percent relative to the Schwarzschild case. As a consequence, these modes are stable for any value of $\zeta$ in the domain of existence of static EDGB BHs. We have investigated this issue also for higher multipoles ($l\geq2$) and our numerical search has found no unstable modes in the entire parameter space. This strongly indicates that static EDGB BHs are linearly mode stable, just like Schwarzschild BHs.

\section{Radial plunge}
In this section, by using the procedures depicted in Sec.~\ref{sec:plunge}, we discuss the gravitational and dilaton radiation emitted by a particle plunging radially into an EDGB BH. For practical reasons, this computation was done within the small-$\zeta$ approach for the background, and the equations are expanded up to order ${\cal O} (\zeta^6)$. As discussed in the previous section, this higher-order perturbative approach gives precise results even for relatively large values of $\zeta$.

\begin{figure}%
\includegraphics[width=\columnwidth]{flux_g.pdf}%
\caption{Gravitational quadrupolar flux for radial plunges into an EDGB BH with different boosts for $\zeta=0.1$. For small couplings the changes in the gravitational fluxes are very small. In the inset, we plot the ratio of the fluxes for $\zeta=0.1$ and for the Schwarzschild spacetime. The vertical dotted lines in the inset are the values of the gravitational- and scalar-led QNMs.}
\label{fig:flux_g}%
\end{figure}

\begin{figure}%
\includegraphics[width=\columnwidth]{flux_phi.pdf}
\caption{Dilaton quadrupolar flux for radial plunges into an EDGB BH with different boosts for $\zeta=0.1$. As expected, the scalar flux starts to present an exponential suppression for frequencies roughly larger than the dilaton mode. In the inset, we plot the ratio between the fluxes for $\zeta=0.1$ and $\zeta=0.05$, which confirms that the scalar flux scales dominantly with $\zeta^2$.
}%
\label{fig:flux_phi}%
\end{figure}

In Fig.~\ref{fig:flux_g} we show the gravitational flux, by considering different
initial boosts for the particle (see Appendix~\ref{app:motion}).
The deviations from the GR case are very small,
of order of $\sim 1\%$, at least for $\zeta\lesssim0.1$.
Moreover, we note that all additional terms appearing in the metric perturbation
equations (see Appendix~\ref{app:line})---sources included---are at least of
order $\sim\zeta^2$, and therefore the corrections to the flux are proportional
to $\sim\zeta^2$. Therefore, in the small-coupling limit, the gravitational flux
can be written as
\begin{equation} \frac{dE_g}{d\omega}\sim \frac{dE_g^{S}}{d\omega}\left[1+{\cal
O}(\zeta^2)\right]\,, \end{equation}
where $dE_g^{S}/d\omega$ is the corresponding Schwarzschild flux.

Although the overall gravitational corrections are small, the dilaton perturbations can also be radiated by the plunging particles, similarly to the case of a neutral particle plunging into a charged BH~\cite{Johnston:1973cd,Johnston:1974vf,Cardoso:2016olt}. This is due to the coupling between the gravitational and dilaton perturbations, cf. Eq.~\eqref{eq:dilaton_per}. We show the dilaton flux for $\zeta=0.1$ in Fig.~\ref{fig:flux_phi}. Note that the flux displays a cutoff roughly at $\omega\sim\omega_R^{\phi}$, where $\omega_R^{\phi}$ is the scalar-led QNM. Also in this case the dilaton flux scales dominantly as $\zeta^2$. Additionally, because the source terms in the dilaton field are only due to the gravitational perturbations, the dilaton radiation is also dominantly quadrupolar (see Ref.~\cite{Johnston:1973cd} for a similar setup with perturbations in Reissner-Nordstr\"om BHs).

Although the total radiated energy due to the dilaton radiation scales as
$\zeta^2$, it can still be considerably high, depending on the radiating source. For
instance, for the source GW150914 the luminosity due to the GWs was
${dE_g}/{dt}\approx 3.6 \times 10^{56}{\rm erg/s}$~\cite{Abbott:2016blz}.
Therefore, even if the scalar radiation is small compared to the gravitational
one, it can still have a considerable value. The implication of a burst of
dilaton radiation depends on how the environment (plasma, surrounding stars,
 etc.) interacts with the dilaton field.

\section{Constraints on the EDGB coupling from ringdown observations\label{detectability}}
The results of the previous section show that plunges of point particles do not excite considerably the scalar-led QNMs. This suggests that such modes might be only mildly excited during the coalescence of two BHs of equal mass and that the main signature of EDGB theory would be a shift of the ringdown frequencies, the latter being governed by the fundamental gravitational-led modes.
Thus, the use of ringdown measurements in EDGB theory to estimate the magnitude of the coupling
$\zeta$ would rely only on the deviation of the gravitational-led modes from their GR counterpart. These deviations are parametrized in terms of the fit~\eqref{eq:expan_modes} of which the coefficients are given in Tables~\ref{tab:coeff_axial} and \ref{tab:coeff} for the $l=2,3$ fundamental axial and polar modes, respectively.

The results of the previous sections refer to nonspinning BHs, whereas the end product of the coalescence is a spinning compact object. Thus, ringdown tests require the knowledge of the first dominant modes of a spinning BH as a function of the spin $\chi$ and of the coupling constant of the modified theory of gravity. Computing the QNMs of generic spinning BHs in modified gravity is a very challenging task\footnote{The task of computing the QNMs of a Kerr BH is enormously simplified by the fact that the gravitational perturbation equations are separable, due to
special properties of the background Kerr geometry which does not necessarily hold for other spinning BH solutions.}, which has witnessed some developments only recently (cf. Ref.~\cite{Pani:2013pma} for an overview). Nonetheless, most of the results are obtained within a perturbative expansion valid for $\chi\ll1$ and quickly become intractable at the higher perturbative order. The latter is required to extrapolate the perturbative result up to $\chi\approx 0.7$, which is roughly the spin of the final BH measured in the two coalescence events  detected by aLIGO to date~\cite{Abbott:2016blz,Abbott:2016nmj}.

To overcome this limitation and estimate how rotation affects the QNMs of an EDGB BH, we rely on the geodesic correspondence and on our knowledge of the metric of a spinning BH in EDGB theory. As previously mentioned, the latter is known analytically up to ${\cal O}(\chi^5,\zeta^7)$~\cite{Maselli:2015tta} and numerically for any value of $\chi$ and $\zeta$~\cite{Kleihaus:2011tg,Kleihaus:2015aje}.
In the previous section we have checked that the geodesic correspondence works reasonably well for axial modes, so in this section we will focus on the latter. However, we may argue that the order of magnitude of our estimates should be correct also for the more relevant polar modes.

By using the geodesic correspondence for $l=m$ modes described in Sec.~\ref{sec:geodesics}, we obtain the following result
\begin{widetext}
 \begin{eqnarray}
 \frac{\omega_R}{\omega_R(\chi=0)} &=& 1\, +\left(0.3849\, +0.0326 \zeta ^2\right) \chi +\left(0.2038\,
   +0.0264 \zeta ^2\right) \chi ^2+\left(0.1283\, +0.0169 \zeta^2\right) \chi ^3\nonumber\\
   &&+\left(0.0897\, +0.0105 \zeta ^2\right) \chi^4+\left(0.0671\, +0.0054 \zeta ^2\right) \chi ^5+{\cal O}(\chi^6,\zeta^3)\,,\\
 \frac{\omega_I}{\omega_I(\chi=0)} &=& 1\, -0.0059 \zeta ^2 \chi -\left(0.0741+0.0066 \zeta ^2\right)
   \chi ^2-\left(0.0713-0.0002 \zeta ^2\right) \chi^3\nonumber\\
  &&-\left(0.0604-0.0050 \zeta ^2\right) \chi^4-\left(0.0504-0.0079 \zeta ^2\right) \chi^5 +{\cal O}(\chi^6,\zeta^3)\,,\label{geodesic_correspondence}
\end{eqnarray}
\end{widetext}
where $\omega_{R,I}(\chi=0)$ are the corresponding axial modes for a nonspinning EDGB BH shown in Table~\ref{tab:coeff_axial}.
The above expressions are valid up to ${\cal O}(\chi^5,\zeta^2)$ and for any $l=m\gg1$; the only difference enters in the normalization factors on the left-hand side.

The detection of two ringdown modes is necessary to estimate the mass, spin and
coupling $\zeta$.  Here we adopt the same Fisher-matrix technique presented in
Ref.~\cite{Cardoso:2016olt}. Let us consider first the axial modes, which are, generically,
excited during the merger phase.
The measurement of the two most dominant modes (which we take to be $l=m=2,3$, excited with a relative amplitude of $3:1$~\cite{Berti:2007zu}) gives us sufficient information to extract the mass and spin of the BH, as well as the coupling parameter of the theory. To do this, we use the geodesic correspondence, Eq.~\eqref{geodesic_correspondence}. {\it Assuming} that both modes are excited to detectable amplitude (this can be quantified using the methods of Ref.~\cite{Berti:2007zu}), then a Fisher-matrix computation allows us to estimate the uncertainties in the parameters determining the ringdown~\cite{Cardoso:2016olt}: $M, \chi, \zeta$. This then allows us to constrain the magnitude of the coupling parameter $\zeta$,
\begin{equation} \zeta \lesssim \frac{3.98 -0.718 \chi+0.181\chi^2-0.045
\chi^3}{\sqrt{\rho}}\,, \label{boundzeta} \end{equation}
where $\rho$ is the signal-to-noise ratio in
the ringdown waveform and the ${\cal O}(\chi^4)$ and ${\cal O}(\chi^5)$ terms
are negligible. The numerator in the expression above is a (mildly) decreasing
function of the spin and ranges from $\approx 4$ to $\approx 3.5$ in the region
$0\leq\chi\approx0.8$.

This analysis can be extended to polar modes. In fact, one might even argue that corrections to polar modes are higher, since the polar QNMs of nonrotating BHs are more affected than the axial, cf. Tables~\ref{tab:coeff_axial} and \ref{tab:coeff}.
However, there is little evidence that they follow the geodesic correspondence,
but we expect that the order of magnitude change in the modes remains the same.
Therefore, it is reasonable to believe that, even when polar modes are excited, a mode analysis of the ringing signal can set constraints on $\zeta$ of the order of those given by Eq.~(\ref{boundzeta}).

As a reference, the final BH spin of GW150914 was $\chi\approx0.67$~\cite{Abbott:2016blz} and $\rho\approx7.7$ in the ringdown part~\cite{TheLIGOScientific:2016src}.
This [assuming, as a first approximation, that Eq.~(\ref{boundzeta}) also holds for polar modes] yields roughly $\zeta\lesssim 1.3$, which is weaker than the theoretical bound~\cite{Pani:2009wy} for the finite-$\zeta$ solution, $\zeta\lesssim0.691$, and therefore meaningless.
This constraint is nevertheless comparable with that derived from the orbital decay rate of low-mass x-ray binaries~\cite{Yagi:2012gp}
and it is slightly larger than the projected bound achievable in the near future from the measurements of quasiperiodic oscillations in the spectrum of accreting BHs~\cite{Maselli:2014fca}, although the latter might be affected by astrophysical systematics.
Our estimate suggests that $\rho\gtrsim25$ in the ringdown waveform is needed to obtain an upper limit which is more stringent than the theoretical bound using GW ringdown detections.
Of course, in order to set these upper limits, an accurate computation of both polar and axial modes of rotating EDGB BHs will be needed.
\section{Discussion and conclusions}
EDGB gravity is a simple and viable higher-curvature correction to GR which predicts BH solutions with scalar charge. Since strong-curvature corrections are suppressed at large distance, it is natural to expect that the most stringent constraints on this theory come from the strong-curvature, highly dynamical regime as the one involved in a BH coalescence. Furthermore, compact stars in this theory possess only a very small scalar charge~\cite{Yagi:2011xp,Kleihaus:2016dui} and therefore EDGB gravity evades the stringent constraints on the dipole radiation coming from current binary-pulsar systems~\cite{Berti:2015itd}.

The estimate~\eqref{boundzeta} translates to the following upper bound on the dimensionful EDGB coupling\footnote{Since one of the parameters of our Fisher-matrix analysis is $\zeta=\alpha/M^2$, propagation of errors implies a relative uncertainty $\delta\alpha/\alpha=\delta\zeta/\zeta+2\delta M/M$. However, in the large-$\rho$ limit the term $\delta M/M$ is negligible because it scales as $1/\rho$, compared to the $1/\sqrt{\rho}$ behavior of the error on $\zeta$ [cf. Eq.~\eqref{boundzeta}]. Therefore, in this limit $\delta \alpha\lesssim \delta \zeta M^2$.},
\begin{equation}
 \alpha^{1/2}\lesssim	11 \left(\frac{50}{\rho}\right)^{1/4}\left(\frac{M}{10 M_\odot}\right)\,{\rm km}\,, \label{boundalpha}
\end{equation}
where the prefactor changes by less than $10\%$ depending on the final BH spin. This result is in agreement with the simple estimates derived in Ref.~\cite{Barausse:2014tra}. As a consequence, our analysis also confirms that in most cases modified-gravity effects can be distinguished from environmental effects~\cite{Barausse:2014tra}.

Future GW detectors will greatly increase the signal-to-noise ratio, a large value of which is necessary to perform ringdown tests of the Kerr metric~\cite{Berti:2016lat}. The signal-to-noise ratio of a ringdown waveform scales approximately (among its dependence on other quantities not shown here) as $\rho\sim M^{3/2}/S_n(f)^{1/2}$~\cite{Berti:2005ys}, where $M$ is the final BH mass and $S_n(f)$ is the detector noise power spectral density at a given frequency $f$. The best sensitivity of the future Voyager~\cite{Voyager} and Einstein Telescope~\cite{ET} detectors will be, respectively, roughly a factor of $10$ and a factor of $100$ better than in the first aLIGO observing run at the same optimal frequency $f\sim 10^2\,{\rm Hz}$~\cite{Berti:2016lat}.
Thus, the Einstein Telescope with an optimal design can achieve a signal-to-noise ratio of roughly $\rho\approx 100$ for the ringdown signal of a GW150914-like event.
From Eqs.~\eqref{boundalpha} and~\eqref{boundzeta}, this would translate into the bound $\alpha^{1/2}\lesssim 8 \left(\frac{M}{10 M_\odot}\right)\,{\rm km}$ and $\zeta\lesssim 0.4$. As expected, lighter BHs would provide a significantly more stringent constraint on $\alpha$, although their ringdown frequency might not fall into the optimal frequency range for ground-based detectors. Due to the small exponent of $\rho$ in Eq.~\eqref{boundalpha}, even an increase of $\rho$ of 1 order of magnitude will not provide a significantly more stringent constraint on the EDGB coupling. A stronger constraint may be set if future observations detect a light BH with a very large signal-to-noise ratio.

Given this scenario, electromagnetic observations of accreting BHs (like the one discussed in Ref.~\cite{Maselli:2014fca}) might provide more stringent constraints in the future, although the latter are affected by astrophysical systematics that are absent in the ringdown case.

Our estimates in the case of spinning BHs rely on the geodesic analogy for QNMs, which we verified only for axial modes in the static case and for Kerr BHs with any spin~\cite{Cardoso:2016olt}. It would be interesting to compute the modes of slowly rotating EDGB BHs (e.g. by adapting the methods discussed in Ref.~\cite{Pani:2013pma}) and to check the geodesic approximation in the spinning case. This computation will be required to place precise constraints on the EDGB coupling through future detections of BH ringing with high signal-to-noise ratio.

Another interesting extension of our work concerns the scalar waves emitted during the coalescence. Although the luminosity in scalar waves is significant, this radiation may be possibly detected only if the dilaton is coupled to matter. Such coupling is presumably small and would not give rise to any effects in the detectors. Nonetheless, if the dilaton-matter coupling is non-negligible, the scalar radiation might be investigated through the same techniques developed to study the scalar emission in scalar-tensor theories, e.g. by using a network of ground-based detectors~\cite{hayama}.

\begin{acknowledgments}
V.C. acknowledges financial support provided under the European Union's H2020 ERC Consolidator Grant ``Matter and strong-field gravity: New frontiers in Einstein's theory'' Grant No. MaGRaTh--646597.
C.M. acknowledges financial support from Conselho Nacional de Desenvolvimento Cient\'ifico e Tecnol\'ogico through Grant No.~232804/2014-1.
J.L.B.S., F.S.K. and J.K. gratefully acknowledge support by the DFG Research Training Group 1620 ``Models of Gravity''.
J.L.B.S. and J.K. gratefully acknowledge support by the grant FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-606096).
Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.
This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant No.~690904 and from FCT-Portugal through the Project No. IF/00293/2013.
This work was supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


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WaterFX's design has another major advantage over traditional desalination: It runs fully on solar power, drastically shrinking the carbon footprint. When it removes the salt from the water, it turns it into usable products; traditional desalination dumps salty brine back into the sea, which can harm marine ecosystems.
Anyone can invest in the startup's first desalination plant, which will be the largest solar desalination plant in the country. (The company already successfully tested a pilot version). But this isn't just another crowdfunding campaign–if you invest in the plant, you can actually make money from it.
On September 8, the company filed a direct public offering under a subsidiary called HydroRevolution. Both accredited investors and the public can buy shares (at a minimum of $2, 000), and later get paid in dividends as the water is sold. By the close of the campaign, the plant will be majority-owned by California residents.
Source: www.fastcompany.com
- Solar Power Company in California
- Which Solar Company is the Best?
- Solar Company Reviews
- solar panel building kit

redpajama_set_name: RedPajamaCommonCrawl


text:
CATHOLIC ARCHIVES OF TEXAS
Sister Dolores Kasner, O.P.
CATHOLIC ARCHIVES OF TEXAS. The Catholic Archives of Texas, in Austin, serves as a central depository for records and documents pertaining to the activities of the Catholic Church in the state. The holdings of the collection range from earliest sixteenth-century Spanish explorations to present-day events. The depository had its beginning with the formal organization of the state Knights of Columbus Historical Commission in 1924, when the group, with the endorsement of the Texas bishops, resolved to publish a history of the church in Texas. Soon after launching their project, the commission named as chairman Paul J. Foik, librarian at St. Edward's University, Austin, who had established the Catholic Archives of America at Notre Dame University, where he was librarian for twelve years. Carlos E. Castañeda, professor of history at the University of Texas, was selected as historiographer.
Castañeda spent years researching and collecting source materials from depositories in Mexico City and other seats of government such as Saltillo, Matamoros, and Monterrey. His work culminated in the seven-volume Our Catholic Heritage in Texas, 1519–1936, published between 1936 and 1958. Castañeda gave care of the collection to Bishop Lawrence FitzSimon in 1948, and it became the core of the Catholic Archives of Texas. FitzSimon transferred the materials to Amarillo and during several trips to Europe added materials. These documents include correspondence of early French missionaries, biographical information on early priests, and information on Catholic newspapers, of which the Southern Messenger collection is the most valuable.
After FitzSimon died in 1958, the collection was again transferred to Austin, where it was housed in the chancery of the Catholic Diocese of Austin. With the establishment of the Texas Catholic Conference in 1963, the bishops of Texas placed the care of the archives under its direction. Holdings include records of the Texas Catholic Conference, the Texas Knights of Columbus Historical Commission, the Texas Catholic Historical Society, and religious associations, societies, and Catholic clubs in Texas; papers of Paul J. Foik, William H. Oberste, Sam Houston, Charles S. Taylor, Francis Bouchu, and Fred Bomar; diocese and parish collections; copies of governmental and religious documents in other archives; personal papers and biographical files of the bishops and clergy of Texas; documents dealing with the various religious orders formed or stationed in the state; and newspapers, pamphlets, books, maps, photographs, artifacts, and periodicals relating to the history of the Catholic Church in Texas. The depository, owned and supported by the Catholic hierarchy of Texas, solicits records from institutions and organizations in all parts of the state and is open to researchers.
Sister M. Claude Lane, O.P., Catholic Archives of Texas: History and Preliminary Inventory (M.A. thesis, University of Texas, 1961). Karl M. Schmitt, "The History of the Texas Catholic Historical Society," U.S. Catholic Historian 3 (1983).
Organizations, Agencies, etc.
Spanish Texas
Handbook of Texas Online, Sister Dolores Kasner, O.P., "CATHOLIC ARCHIVES OF TEXAS," accessed January 23, 2020, http://www.tshaonline.org/handbook/online/articles/lcc02.


You all have known me long enough to know I love made in the USA. But what I love even more is things handmade that are actually one of a kind and very reasonable.
I happened upon this web site I believe through Livingsocial.com . I fell in love with these pieces of jewelry and what a great way to support an artist living in the USA. Birmingham, AL to be exact. Necklaces, earrings, bracelets.. ALL very nice.
Ok you shopping for your favorite Alabama fan or you know elephants are a symbol of good luck and we all can use that.. Here is an elephant necklace.
I being an Auburn fan and and old hippie..I loved this. So if you need to get that special someone in your life a gift, here it is! You probably could get your team on one of these.
Everybody knows somebody that is having a bundle of joy arriving . This would be something special they could keep forever and even pass down to their child when they are awaiting for their own bundle of joy to arrive. You could make this a family tradition. Get one for you and pass it to your child and they pass it to their child on down the line.
What the world needs is more love and you could spread it around with a pair of these earrings.
This would be an adorable necklace for a mother or grandmother or you could even have just your and your spouse name put on there. It would be cool to spell out your own name.
Look I could go on and on about what Rececca has on the site but why do you just not go to www.jennarosityllc.com and see and shop for yourself. I am sure you and she could work something out.
You know when I find MADE IN THE USA..I am going to pass it on to you. We need to support the artists in the USA!
Go there and look at all the pretties. Let me know in a comment what you think.
When I contacted Rececca and asked for permission to blog about her company and asked her about it, she said she was located in Birmingham, AL and that she hand stamps every piece. Now how cool is that??
Made in the USA and her prices are very reasonable. That special gift you need for Christmas or whatever occassion could be personalized and just what you are looking for. Something no one else has. Christmas is coming very soon!
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Now that you're officially in the sweepstakes as a partner, you'll want to make sure you've reviewed the Official Rules and the landing page.
Official rules are the terms and conditions clearly communicated to your entrants and agreed to as a condition for entering the giveaway.
A good set of official rules protect your brand from liability and educated entrants about the limitations to and guidelines for entering to win.
By entering the sweepstakes, your entrants are agreeing to these rules.
Before agreeing to participate in a campaign, you'll want to review the campaign's Official Rules and make sure they work for your brand.
What if I see an issue with the Official Rules language?
Only the Host brand can edit the actual Official Rules, so follow up with them directly to discuss any edits.
Follow these steps to review the Official Rules and landing page.
Preview the campaign by heading to My Partnerships > Current > and then opening the sweepstakes you want to Preview. Click the Preview button.
While the Host ultimately controls the design of the page, be sure to examine the landing page to make sure the imagery and copy match your expectations.
Ensure that the logo on the bottom of the landing page is the correct one. Need to change your logo? Upload the one you'd like to use to your Brand Profile and it will appear on the landing page.
4. Open the Official Rules.
Find the link to the Official Rules, next to the Enter button. The rules will open in a new window. Read them carefully - these will be used to govern the sweepstakes.
Only the Host can edit the Official Rules and Disclaimer.


Stephen E. Arnold: Amazon Police State Stuff….
I have been following the Amazon response to employee demands to cut off the US government. Put that facial recognition technology on "ice." The issue is an intriguing one; for example, Rekognition plugs into DeepLens. DeepLens connects with Sagemaker. The construct allows some interesting policeware functions.
Ah, you didn't know that? Some info is available if you view the October 30 and November 6, 2018, DarkCyber. Want more info? Write benkent2020 at yahoo dot com.


GitHub's collaborative approach to development depends on publishing commits from your local repository for other people to view, fetch, and update.
A remote URL is Git's fancy way of saying "the place where your code is stored." That URL could be your repository on GitHub, or another user's fork, or even on a completely different server.
Git associates a remote URL with a name, and your default remote is usually called origin.
For information on the differences between these URLs, see "Which remote URL should I use?"
This associates the name origin with the REMOTE_URL.
You can use the command git remote set-url to change a remote's URL.


text:
Beyond 'necropolitics'
"I have reason to believe this would be his wish for us, too."
The first thing I did when word was getting around that former President Benigno Simeon Cojuangco Aquino II (PNoy for short) was rushed to the hospital was pray that he survives and gets better after another procedure or so.
I thought this was just one of those things as reports that he had been having bouts of such intermittent hospital visits for treatments have been swirling a year or so after he stepped down in 2016. Thankfully, he has survived most of those earlier attacks. Lately, however, some of his close friends advised that things got worse during the pandemic.
Nonetheless, I was hopeful he would make it yet again if only to lend his voice in tempering the passions that usually attend the incredible suffering or even the death of a member of the storied Aquino-Cojuangco clan.
But that was not meant to be. PNoy passed away and is now reunited with his parents in a place of peace and quiet, away from the passions engendered by their deaths and the battles of the past which, sadly, their followers continue to inject in the life of a grieving nation. It is this narrow passage to what people now describe as "necropolitics" which, hopefully, we will now be able to overcome as a people as we prepare for the 2022 presidential polls.
I am not optimistic though that such will be the case. Even before PNoy was rushed to the hospital last Thursday, we had been getting reports that the Liberal Party (LP) where he remained Chairman Emeritus was already angling for the lead role in the opposition table. Incumbent LP Chairman VP Leni Robredo, the likely 1Sambayan (yes, that aggrupation touting itself as the fount of good governance and the country's savior from the Duterte maladministration) nominee for President demurred, giving LP President Kiko Pangilinan the opportunity to organize a complete line up for the upcoming polls in coalition with other like minded groups hooked on their never ending mantra of "anti-corruption, good governance and decency in public service."
With PNoy's demise, that initiative will now surely be on overdrive — and how!!
Even as PNoy's remains were still being transported to the Church of the Gesu at the Ateneo de Manila University Loyola campus, whisperings about the possibility of another Aquino taking over the mantle of the former President was already getting around. It did not take long before a "Kris 2022 in bright yellow sheets" surfaced in social media.
Not to be outdone, scores of Pnoy loyalists moved to center stage bandying the need for "sacrifice" to get the country out of the "Dark Ages" under this administration and back to the "Golden Years" of Aquino magic. Ever faithful and seemingly seething crowds of Aquino loyalists also began purveying all kinds of posts including fighting anti-administration ones and urging the hanging of yellow ribbons – all meant to get people to get into the groove all over again.
A Jesuit priest, Fr. Alejo, said it all when he harangued those present during the first day of the wake to come around together and bring in a "new and better batch of politicians" as it is their duty to "reclaim politics as a lofty vocation with integrity and competence like PNoy." That call was reiterated by no less than Archbishop Socrates Villegas, the main celebrant at the burial mass, when he advised those present and beyond to rise up and raise the level of "good governance and integrity in public life" as he bewailed the "vulgarity, fakery and persecution" being practiced by the current regime.
It was well that Ballsy Aquino-Cruz, the oldest of the siblings, skewed that route preferring instead to deliver a tempered and humanizing eulogy to his brother as she lovingly recalled the counsels of their parents as they together suffered through the "slings and arrows of outrageous fortune." She could have easily given in to the whisperings of the crowd, as it were, and lead them to battle, so to speak.
But she chose not to. Which was, of course, largely well received especially by those who had long hoped to see the day when their family would finally come around to still the polarizing tendencies in our political life using their clan's life journey in settling scores (imaginary or otherwise) and reliving the battles of the past.
She must have read the statement of her Teresiana — or was it Povedan? — classmate and also the eldest of their family's political nemesis, Senator Imee Marcos, who tweeted that despite their political differences she had very good memories of PNoy as a colleague in Congress as she prayed for the eternal repose of his soul. Or words to that effect.
Hopefully, President Duterte's ordering ten days of national mourning, flying the flag at half mast in all government buildings and openly condoling with the family as he thanked PNoy on behalf of a grateful nation for his service to the country and that Imee tweet would all add up to stilling any efforts to use Pnoy's death to stir another round of 'necropolitics" which, if we go by Ballsy's eulogy, seems to be the Aquino family's wish as well.
I am, of course, crossing my fingers that such a passage beyond necropolitics and into reality and the need to come around together as a people to tackle the many challenges in building a nation we can all be proud of will finally come to fruition in our lifetime. Like all others who have gone through decades of acrimony and rounds of back and forth between the political families which have somehow deterred our passage from a middling economy to a mid-size one.
We need to harness the best in us, not unleash the worst. We need to forego the stirrings of partisan politics and recover our humane and compassionate selves. We need to get back to the values and virtues of a people intent on building a nation, not encourage those bent on tearing it apart. That should be our singular initiative, something which, I have every reason to believe, PNoy would have wanted to preach himself had he survived that attack last Thursday.
Tags: 1Sambayan2022 presidential pollsJ.A. Dela CruznecropoliticsPNoyPresident Benigno Simeon Cojuangco Aquino III


In transporting goods, parcels, and items – whether in bulk or individual packaging –there are various modes of transport: land, air, and sea. Each has its own advantages and benefits; you just have to familiarize yourself with how this industry works.
When you are planning to ship items and parcels, you can choose whether to ask the help of land, air or sea freight companies in the UAE – depending on your preference. However, at Logtrans, you won't need to worry much about the quality of work because we give land, sea, and air freight services in Dubai, as well as relocating and warehouse services. We are considered to be one of the leading and most prominent freight companies in the industry because of our unsurpassed services and high-quality of work.
Being included in the pioneers of our field has made us distinctive and remarkably unique – our clients trust us and our future customers can do the same.
If you compare land transport to the other modes of transport, you can immediately say that this is the cheapest. But, having the lowest packing expenses doesn't mean you'll receive inadequate service in this mode of transport – if you hire a credible and reliable provider, guarantee a high-quality service.
Being on land makes it easier for companies to have safe transit – hence, as customers, you can assure that your goods will be delivered safely with lesser damage and breakage.
If the distance is just near, the best option to choose is land transport. It's fast and perfect for short distances – delays might be a bit impossible and the goods can be delivered from door to door.
It offers a wide coverage – national and local destinations can be reached through this mode of transportation. If the customer is just within the vicinity, state, or country, land transport is a perfect choice.
And as one of the leading land transportation companies, we, Logtrans, can give the four aforementioned benefits to you. If you ever need any of our services, we are always ready to take your inquiries and expect that we'll give our best to meet and even exceed your expectations. You can come and visit us anytime. Not only do we give freight services but warehouse storage and relocation services in Dubai as well.


text:
class CreateEveWars < ActiveRecord::Migration[6.0]
  def change
    create_table :eve_wars do |t|
      t.bigint :war_id
      t.datetime :declared
      t.datetime :finished
      t.boolean :mutual
      t.boolean :open_for_allies
      t.datetime :retracted
      t.datetime :started

      t.timestamps
    end
  end
end

redpajama_set_name: RedPajamaGithub


text:
Q: How to deal with a force close I am currently developing a game application for Android. My problem is that it sometimes crashes and displays the "Unfortunately...has stopped" dialog. But running the application again does NOT force close the app when getting to the same point, it will just appear every so often.
What are some potential reasons that this might happen? I've done some research but couldn't really find anything

A: You should review your logcat for possible Exceptions.
If you have no Exceptions causing your app to crash -meaning your code is aparently "ok"- there's the chance the application is consuming lots of memory. If your application is making your Garbage Collector to overwork, Android may shut down your application.
You must take care you are not overcreating objects, or leaving a load behind to be cleaned by the Garbage Collector. This means, if you can reuse objects, that's a good idea.
There's an <application> element in the Android Manifest. You can set there an attribute named android:largeHeap="true". In this way Android will assign more memory for your application. But do do this as a last resort. Try to first improve the object management in your app. This ain't easy task, so good luck.

redpajama_set_name: RedPajamaStackExchange


It's not unusual for people to be anxious about visiting the dentist, or to fail to properly care for their teeth. But people who have psychiatric disorders struggle with those problems to an even greater degree than the average patient. They may be phobic or distrustful, so a visit to the dentist becomes an even bigger issue for them than for most people.
That means it's incumbent on dentists to be aware of the special needs of these patients and take steps to provide them the proper care. For example, in some cases the dentist might not be able to accomplish everything they would like in one appointment as they would with another patient. We need to be flexible and willing to adapt.
Each of these concerns – and others – can be addressed if dentists demonstrate patience and familiarize themselves with some of the problems surrounding people with mental illnesses. Sometimes it may seem like a patient is making progress, but then they have a relapse and regress to poor oral hygiene again. That's not unexpected, and can be handled if the dentist continues to show empathy and tolerance for what may seem like eccentric behavior.
Dr. Scott Shamblott is a general dentist and the founder of Shamblott Family Dentistry (www.shamblottfamilydentistry.com) in Hopkins, Minn. He is the author of two books, Help! My Tooth Hurts: Your Guide to Feeling Better Fast, and Fear-Free Dental Care: Finding a Dentist You Can Love. Dr. Shamblott earned a B.S. in Finance at the University of Arizona and a Doctor of Dental Surgery degree with high distinction at the University of Minnesota School of Dentistry. He completed a general practice residency at the University of Tennessee Hospital. Dr. Shamblott earned a fellowship from the Academy of General Dentistry and from the Dental Organization for Conscious Sedation. Also a product evaluator, he is one of 450 dentists worldwide selected to test dental products and techniques.


text:
Q: Swift - Decode/encode an array of generics with different types How can I decode/encode an array of different generic types?
I have a data structure, which has properties, that conform to a protocol Connection, thus I use generics:
// Data structure which saves two objects, which conform to the Connection protocol
struct Configuration<F: Connection, T: Connection>: Codable {
    var from: F
    var to: T
    private var id: String = UUID.init().uuidString

    enum CodingKeys: String, CodingKey {
        case from, to, id
    }

    init(from decoder: Decoder) throws {
        let container = try decoder.container(keyedBy: CodingKeys.self)
        self.from = try container.decode(F.self, forKey: .from)
        self.to = try container.decode(T.self, forKey: .to)
        self.id = try container.decode(String.self, forKey: .id)
    }

    func encode(to encoder: Encoder) throws {
        var container = encoder.container(keyedBy: CodingKeys.self)
        try container.encode(from, forKey: .from)
        try container.encode(to, forKey: .to)
        try container.encode(id, forKey: .id)
    }
}

protocol Connection: Codable {
    var path: String { get set }
}

// Two implementations of the Connection protocol
struct SFTPConnection: Connection, Codable {
    var path: String
    var user: String
    var sshKey: String
}

struct FTPConnection: Connection, Codable {
    var path: String
    var user: String
    var password: String
}

This works fine when I know of what type the connections F and T are. But I have cases, where I want to load a configuration, not knowing which type F and T are.
public static func load<F: Connection, T: Connection>(for key: String) throws -> Configuration<F, T>? {
    // Load from UserDefaults
    guard let configurationData = defaults.object(forKey: key) as? Data else {
        return nil
    }

    // Decode
    guard let configuration = try? PropertyListDecoder().decode(Configuration<F, T>.self, from: configurationData) else {
        return nil
    }

    return configuration
}

// OR

func loadAll<F:Connection, T: Connection>() -> [String: Configuration<F, T>]? {
    return UserDefaults.standard.dictionaryRepresentation() as? [String: Configuration<F, T>]
}

In the above cases F and T could be of any unknown type, that conforms to the Connection protocol. So the above functions wouldn't work, since I would need to specify a specific type for F and T when calling the function, which I don't know.
In the second function, F alone could actually be of different types. That's where it gets difficult. I figured I need to somehow store the types of F and T in the User Defaults as well and then use them in the decode and encode function (thus discarding the generics). But I have no idea how I would elegantly do that.
I would appreciate any ideas on how to solve this problem!

A: The following solutions resolves all the issues, that I had with generics and not knowing the specific type of Connection. The key to the solution was

*

*saving the type of a Connection implementation in the implementation itself and

*Using superEncoder and superDecoder to encode/decode the from and to properties.

This is the solution:
import Foundation

protocol Connection: Codable {
    var type: ConnectionType { get }
    var path: String { get set }
}

struct LocalConnection: Connection {
    let type: ConnectionType = ConnectionType.local

    var path: String
}

struct SFTPConnection : Connection {
    let type: ConnectionType = ConnectionType.sftp

    var path: String
    var user: String
    var sshKey: String

    init(path: String, user: String, sshKey: String) {
        self.path = path
        self.user = user
        self.sshKey = sshKey
    }
}

struct FTPConnection: Connection {
    let type: ConnectionType = ConnectionType.ftp

    var path: String
    var user: String
    var password: String
}

struct TFTPConnection: Connection {
    let type: ConnectionType = ConnectionType.tftp

    var path: String
}

enum ConnectionType : Int, Codable {
    case local
    case sftp
    case ftp
    case tftp

    func getType() -> Connection.Type {
        switch self {
        case .local: return LocalConnection.self
        case .sftp: return SFTPConnection.self
        case .ftp: return FTPConnection.self
        case .tftp: return TFTPConnection.self
        }
    }
}

struct Configuration {
    var from : Connection
    var to : Connection
    private var id = UUID.init().uuidString

    var fromType : ConnectionType { return from.type }
    var toType : ConnectionType { return to.type }

    init(from: Connection, to: Connection) {
        self.from = from
        self.to = to
    }
}

extension Configuration : Codable {

    enum CodingKeys: String, CodingKey {
        case id, from, to, fromType, toType
    }

    init(from decoder: Decoder) throws {
        let container = try decoder.container(keyedBy: CodingKeys.self)

        self.id = try container.decode(String.self, forKey: .id)

        var type : ConnectionType

        type = try container.decode(ConnectionType.self, forKey: .fromType)
        let fromDecoder = try container.superDecoder(forKey: .from)
        self.from = try type.getType().init(from: fromDecoder)

        type = try container.decode(ConnectionType.self, forKey: .toType)
        let toDecoder = try container.superDecoder(forKey: .to)
        self.to = try type.getType().init(from: toDecoder)
    }

    func encode(to encoder: Encoder) throws {
        var container = encoder.container(keyedBy: CodingKeys.self)

        try container.encode(self.id, forKey: .id)

        try container.encode(self.fromType, forKey: .fromType)
        let fromContainer = container.superEncoder(forKey: .from)
        try from.encode(to: fromContainer)

        try container.encode(self.toType, forKey: .toType)
        let toContainer = container.superEncoder(forKey: .to)
        try to.encode(to: toContainer)
    }
}


The work of this "eminent, still-wild spirit of Central Europe" (Publishers Weekly) continues to electrify. In The Blue Tower, language is remade with tenderness and abandon: "Rommel was kissing heaven's dainty hands and yet / from his airplane above the Sahara my uncle / Rafko Perhauc still blew him to bits." There is an effervescence and a sense of freedom to Tomaž Salamun's poetry that has made him an inspiration to successive generations of American poets, "a poetic bridge between old European roots and the American adventure" (Associated Press). Trivial and monumental, beautiful and grotesque, healing, ferocious, mad: The Blue Tower is an essential volume.
What you hear through the walls is panic coming here.
In Morocco he whipped slaves. First I open the chest.
T-shirt. I'll show your hand, my hand is your hand.
of snails. Snow is beautiful. The moon calms his lips.
You flash him signals for cricket, eat chickens at midnight.
Isn't the wood for bramblebees rowing the river?
They think nothing of closing the eyebrows of someone like you.
To chop up cotton and read through a cookbook.
To be running behind and hang from your lower jaw.
pathways. Do you see how I lopped off their heads?
Do you see how I step on his scalp as an officer?
mustache. They peeled the enamel off Cassandra's tooth.
a bell has sand and rain pouring from his hat.
made off with Srebrna while she drank at a well.
like Vikings. Zebras have scraped eyes.
Ibrahim, Drago and Miklavž are great guys.
pigeon on the plate. Mother of pearl. Gray head.
Every ball is a bloody, beautiful mask of powerful people.
I always did like chickens.
and steam, but for Cendrars no longer matters.
too," the way Marci and Hudi said it to me.


text:
(function(){
  exports = module.exports = plus = function (one, two) {
    return one + two;
  };
}).call(this);


You're so vain I'll bet you think this song is about you Don't you? Don't you? …given that the song is about him, doesn't that just make him insightful?
Prof. S.: No French speakers in the class? Smartarse Student: German. It'll get me through France!


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Information you provide or that is obtained by us will be used by us to enable us to review, develop and improve the services which we offer and provide you and other customers (via mail, email telephone or otherwise) with information about new products and services and special offers we think you will find valuable. We may also inform you about new products and services and special offers of selected third parties.


Since 2012, Life, Listened has been producing successful, conversational and topical podcasts in the parenting and women's lifestyle space. Our shows have been downloaded millions of times and continue to attract and engage audiences. Whether you're an individual planning to host your show out of a home studio, or a brand who wants to reach rapt audiences through the magic of audio, we'll work with you to bring your idea from concept to a fully-realized, marketable podcast that audiences can easily find.
Podcast listeners are incredibly engaged, and at Life, Listened, we know how to reach them. We make your brand part of the conversation by developing creative, effective podcast advertising campaigns from the ground up. For brands who want to integrate their message more deeply into a podcast episode we also offer "sponsored conversations" that bring your product or service to life with a lively, in-depth interview or discussion featuring your campaign messaging and/or experts.


text:
Ranking the top 5 moments from "Younger" season 5 episode 8: "Can We Stay In Our Bubble Just For Tonight?"
Happy Wednesday TV fans! After watching TV Land's Younger season 5 episode 8 last night, I found myself thinking of Liza's advice when she says, "The good things only happen when we risk letting the bad stuff happen too." We may want to keep our guard up or make decisions with our head rather than our heart, but sometimes we can miss out on something truly great if we do so. Please note spoilers are ahead if you have not gotten a chance to watch the latest episode yet. In last night's episode, "The Bubble," Liza and Charles finally get a chance to explore their feelings for one another. Meanwhile, Kelsey tries to make the right decision, Lauren has a dinner party, and Diana struggles with keeping up with her cleanse. My top 5 moments from Younger season 5 episode 8 are:
"I need to press pause."
Kelsey has clearly been stressed out in her personal and professional life and it doesn't help that she has been stuck in the middle of a love triangle between her author and her co-worker. Kelsey finally makes the decision to choose herself, or at least a time out, when she tells Jake that she wants to cool things at least until after the book comes out. Kelsey is just being honest and tries to explain that she has made this mistake before and it seems like Jake is willing to accept this. However, Jake demonstrates his sneaky side once again and goes behind Kelsey's back to approach Charles and suggest that his book be published by Empirical Press rather than Millennial. Jake's actions prove that he is absolutely not the right match for Kelsey, but will the shadiness stop here? Something tells me it's only the beginning. Watch out Kelsey!
"We all have issues."
Lauren is excited to host a dinner party for her friends and plans on inviting a woman she recently met that she is very much interested in. Josh and Maggie show up and Maggie hilariously brings some food even though Lauren is cooking and reminds her, "You want to impress this girl right?" When the doorbell finally rings, Maggie is beyond shocked to learn that Lauren's date is her ex-girlfriend, Malkie. Lauren's cooking is not the only thing stirring in the pot, as the drama and tension between these two escalates. Malkie and Maggie clearly have unresolved feelings for one another, but Maggie leaves before they can completely rehash everything. After speaking to Liza, Maggie realizes that maybe she does need to let her wall down and give Malkie a chance. With the help of Josh, she paints a beautiful mural with Malkie's portrait to surprise her and it looks like they are both ready to give their relationship another go and put in some work.
"This one is hers."
In the beginning of the episode, Zane is still angry with Kelsey after being taken off of Jake's book. Kelsey tries to talk it out with him, but Zane harshly tells her, "You screwed me and then you screwed me over." Later on in the episode, Charles approaches Zane about shifting Jake's book over to Empirical and Zane proves to be a knight in shining armor. He could easily take the credit here and grab this opportunity, but he tells Charles that Kelsey worked very hard on the book and Charles agrees to keep it at Millennial. Zane may be hurt and frustrated by Kelsey's actions, but he refuses to stoop to the underhanded level that Jake has, proving once again to a good and honest man. Now let's hope our girl Kelsey can realize this too. Men like Zane are hard to come by and I would hate to see these two miss out on an opportunity to be together.
"She might want revenge."
Liza and Diana are meeting with Reese Witherspoon's representative to discuss "Marriage Vacation," and Liza is shocked when Pauline arrives. Pauline clearly knows what is happening between Liza and Charles, as she starts coming up with different premises for her sequel, suggesting that perhaps a younger assistant steals her husband. Liza panics and plays along when she says, "Maybe the assistant was actually trying to help," but Pauline is not buying it and starts talking about revenge ideas. While it is great to see Liza and Charles finally together, there are still many outside forces that can bring consequences. Are they finally ready to face them together?
"Can we stay in our bubble just for tonight?"
The beginning of the episode opens up with us seeing bits and pieces of Liza and Charles's night together which is very sexy and passionate. Charles leaves for ten days and when he returns, he is sporting a new beard. While Liza is not necessarily a fan of the beard, she is so excited to see Charles and the two continue their secret relationship by meeting up in the stairwell and sending texts to one another. When Charles invites Liza out, he takes her to Battery Park to a sea glass carousel and I was absolutely swooning over this date. All of this talk about staying in a bubble and Charles literally takes her to their very own bubble away from the rest of the world. After Liza tells Charles about Pauline, he explains that he has already taken care of it and agrees to let her take her sequel to another publishing company. It's clear that Charles's main priority is Liza, but he does recognize that the challenges are going to keep coming their way. While we know that they cannot stay hidden in their bubble forever, I have to say that it was so wonderful to finally see these two enjoy some quality time together. But what happens when they step out of the bubble and face reality? Something tells me that as long as they stick together, they can make it through any obstacle thrown their way. That doesn't mean it will be smooth sailing; however, as we know these two are in for a bumpy ride. Stay tuned!
I hope you enjoyed the list and feel free to post your own theories and thoughts on Younger below in the comments section. Thanks so much for reading and happy viewing!
@TV_Fanatic_Girl
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Posted on August 1, 2018 August 1, 2018 Author Amy Koto, TV Fanatic GirlCategories Uncategorized, YoungerTags Battery Park, comedy, drama, Empirical Press, episode 8, fans, Hilary Duff, love triangle, Marriage Vacation, Millennial Press, New York, Ranking, Reese Witherspoon, revenge, romance, romcom, Season 5, secret relationship, sitcom, spoilers, Sutton Foster, Team Charles, television, The Bubble, theories, top 5 list, Top 5 moments, TV, TV Land, Younger, Youngertv
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text:
Такмичење у петобоју за жене на Светском првенству у атлетици у дворани 2014.  одржано је 7. марта у Ерго Арени у Сопоту (Пољска).

Титулу освојену у Истанбулу 2012, није бранила Наталија Добринска из Украјине.

Земље учеснице
Учествовало је 8 такмичарки из 7 земаља.

Освајачи медаља

Рекорди пре почетка Светског првенства 2014.
Стање 6. март 2014.

Рекорди после завршетка Светског првенства 2014.

Најбољи резултати у 2014. години
Десет најбољих атлетичарки године у петобоју у дворани пре првенства (6. марта 2014), имале су следећи пласман.

Такмичарке чија су имена подебљана учествују на СП 2014.

Квалификациони стандард
ИААФ је одредио да осам петобојки позове на такмичење по овом критеријуму:
 Победница -{IAAF World Challenge}- у вишебојима 2013.
 Три најбоље пласиране петобојке на светским листама за 2013. (на дан 31. децембра 2013), ограничено на максимално једна по земљи
 Три најбоље петобојке на листи 2014. закључно са 17. фебруаром 2014.
 Једна петобојка може бити позвана по дискреционом праву ИААФ.
Позване могу бити укупно две такмичарке из исте земље. Након одбијања или отказивања, позив ће бити упућен следећој  рангираној у истим листама поштујући горенаведене услове.

Сатница

Резултати

Трка 60 метара са препонама
Такмичење је почело у 12:00.

Скок увис
Такмичење у скоку увис почело је у 12:35.

Бацање кугле
Такмичење у бацању кугле почело је у 18:45.

Скок удаљ
Такмичење у скоку удаљ почело је  у 18:00.

Трка на 800 метара
Трка на 800 метара је почела у 20:10 по локалном времену

Коначан пласман

Референце

Спољашње везе
 Комплетни резултати са СП 2014. на сајту ИААФ

Светско првенство у атлетици у дворани 2014.
Петобој на Светском првенству у атлетици у дворани

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