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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 9
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 60085)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 9
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text
string | meta
dict |
|---|---|
\section{Introduction}
Some time ago it was realized \cite{BEP,Espriu2013,Espriu2014} that
the presence of a cosmological constant had an effect on the propagation
of gravitational waves (GW) beyond the modification of the effective frequency
due to the redshift induced by the acceleration of the Universe -
the only effect that is usually taken into account. While this effect
is very small, it was found that it could possibly have observational
consequences.
Later on, the analysis was extended in order to include the effect of the various
cosmological parameters \cite{AEG,EGR,Alfaro}, in particular the matter density $\rho_{\text{dust}}$.
Like in the case when only the dark energy density $\rho_\Lambda$ was included,
the effect led to different corrections in the frequency and in the wave number.
The effective frequency agrees with the usual redshifted one, as expected.
This effect that, as mentioned, has been largely unnoticed does not
have any implications for interferometric experiments
such as LIGO or Virgo \cite{LIGOVirgoFirst,LIGO,Virgo} that depend only on the
GW frequency, but it may be relevant and indeed observable in pulsar timing
arrays (PTA) \cite{Romani1989,FosterBecker1990} where the optical path is much longer
and therefore sensitive to modifications in the wave number vector.
Still, the effect being roughly proportional to $H_0 L$ (where $L$ is a
characteristic galactic distance and $H_0$ is the present value of the Hubble
constant) is certainly very small
and it is only through a fortuitous combination of various quantities that it is
potentially observable in PTA for GW originating from binary mergers of
very massive black holes, of the order of $10^6$ solar masses at distances
of a few hundreds of Mpc. Although most such mergers take place at further
distances \cite{Supermassive} it was suspected that the modifications should also be visible
for mergers at distances in the Gpc range.
A definite conclusion could not be reached, however, as we were able to solve
the wave equation describing GW propagation in the cosmological medium
only at order $\sqrt{\Lambda}$, or for that matter at order $H_0$ \cite{EGR}. In fact,
at this order of approximation, the various cosmological densities entered only
in the combination forming $H_0$.
In this article we remove these limitations and we are able to provide a
detailed solution up to order $\Lambda^\frac{3}{2}$. This is enough to ensure
the validity of the conclusions for mergers originating in the Gpc range, in fact
almost up to the confines of the visible universe. Interestingly, going beyond
the leading $\sqrt\Lambda$ order (or, equivalently, $H_0$) removes the degeneracy
of the cosmological parameters and $\rho_\Lambda$ and $\rho_{\text{dust}}$ appears in
various combinations.
Of course this is not to say that a potential observation of the effect
at closer distances is not interesting. On the contrary, as emphasized in
\cite{BEP}, this observation could provide a `local' measurement of the
cosmological constant -something interesting in itself. The issue is of course
also related to the ongoing controversy regarding the value of the Hubble constant \cite{Plank,Wong,Verde}.
In addition, we are also able to quantify the possible influence of the gravitational
mass of the source on the wave propagation. As expected, this turns out to be
minute and irrelevant for cosmological distances, but it might be of interest
in other situations, provided that spherical symmetry is still valid to
describe the physical situation.
Just to set the right frame of mind, it is convenient to state outright
the physical cause of the effect. Let us assume that, as a first approximation,
a GW can be described at large distances (but still close to the source)
by a simple trigonometric function of the form $\frac{1}{r}\cos \omega (t-r)$. In
this formula $t$ and $r$ are the time and radial distance in spherically symmetric
coordinates and for the leading harmonic $\omega$ is equal to twice the orbital frequency
of the collapsing system. The effect arises because in the presence
of the various cosmological components the universe is not asymptotically flat and the coordinates
$(t,r)$ differ from the ones where observations are made, the
Friedmann-Lemaître-Robertson-Walker comoving system of coordinates $(T,R)$. The
relation is non-trivial and this is the explanation of the fact that wave number
and frequency differ.
General Relativity rules that the spacetime metric reacts to the presence of energy and momentum
according to the Einstein field equations
\begin{equation}\label{Eq: Einstein equations}
R_{\mu \nu} - \frac{1}{2} \, R \, g_{\mu \nu} - \Lambda \, g_{\mu \nu} = \kappa \, T_{\mu \nu},
\end{equation}
where $R_{\mu \nu}$ and $R$ are the Ricci tensor and scalar curvature, $T_{\mu \nu}$ is the energy-momentum
tensor and $\kappa = 8 \pi G/c^4$. In this paper we will use the $(+---)$ signature convention and
natural units $c=1$. We have also explicitly included the cosmological constant $\Lambda$ term.
In order to determine the propagation of GW we have to
consider small perturbations around a background metric $\tilde{g}_{\mu \nu}$
\begin{equation}
g_{\mu \nu} = \tilde{g}_{\mu \nu} + h_{\mu \nu}, \qquad |h_{\mu \nu}| \ll 1\
\end{equation}
and the linearized vacuum Einstein equations at first order on the perturbation $h_{\mu \nu}$ then read
\begin{equation}
G_{\mu\nu}(\tilde{g}+h) = G_{\mu\nu}(\tilde{g}) +
\frac{\delta G_{\mu\nu}}{\delta g_{\alpha \beta}}\, \biggl\rvert_{\tilde{g}}\ h_{\alpha\beta} + \ .\ .\ .\
= \Lambda \tilde{g}_{\mu \nu} + \Lambda h_{\mu \nu},
\end{equation}
where $G_{\mu \nu}$ is the Einstein tensor. Clearly, Einstein field equations are satisfied for
the unperturbed metric, $G_{\mu\nu}(\tilde{g}) = \Lambda \tilde{g}_{\mu \nu}$. In
order to avoid redundancies under coordinate transformations, it is mandatory to choose a gauge.
Even though there is freedom in the gauge choice, it is convenient to choose coordinates where the
perturbation is purely spatial $h_{\mu 0} = 0$, transverse and traceless, known as the TT-gauge.
For a wave propagating in the radial direction, transversality implies that the only non-vanishing
components of the purely spatial metric perturbation are the angular ones, i.e.
$h_{\theta \theta}, h_{\theta \phi}, h_{\phi \theta}, h_{\phi \phi}$. Moreover, due to the symmetry of
the metric tensor, $h_{\theta \phi} = h_{\phi \theta}$. Finally, the traceless condition establishes a
relation between $h_{\theta \theta}$ and $h_{\phi \phi}$
\begin{equation}
h = g^{\mu \nu} \, h_{\mu \nu} = 0 \qquad \Rightarrow \qquad h_{\phi \phi} = -\sin^2 \theta \, h_{\theta \theta}.
\end{equation}
The previous considerations apply to any coordinate system displaying rotational symmetry. In
this work two different coordinate systems will be of interest to us: Schwarzschild-de Sitter (SdS) and
Friedman-Lemaitre-Robertson-Walker (FLRW).
\section{Perturbations in Schwarzschild-de Sitter}\label{Section:SdS}
We will first consider the Schwarzschild-de Sitter (SdS) metric
\begin{equation}\label{Eq: SdS metric}
ds^{2}=\left(1-\frac{2 G M}{r}-\frac{\Lambda}{3} r^{2}\right) dt^{2}-
\left(1-\frac{2 G M}{r}-\frac{\Lambda}{3} r^{2}\right)^{-1} dr^{2}-r^{2} d \Omega^{2}.
\end{equation}
This metric possesses spherical symmetry and describes a background consisting of a mass $M$
in a universe endowed with a cosmological constant $\Lambda$. It is the background 'seen' by
a gravitational wave close to its source, but sufficiently far ($r \gg r_S$) from it so that
spherical symmetry can be considered to hold at least approximately.
This background leads to the following equation of motion for the non-zero components of the metric perturbation.
The equation is the same for $h_{\theta \theta}, h_{\theta \phi}, h_{\phi \theta}$ and $h_{\phi \phi}$ in
the TT gauge
\begin{align}\label{Eq: h spherical equation}
-\frac{1}{2} \frac{1}{f} \, \ddot{h}_{\mu \nu}
+ \frac{1}{2} f \, h''_{\mu \nu}
+ \left(\frac{1}{2} \, f' - \frac{f}{r} \right) \, h'_{\mu \nu}
- \left(2 \frac{f'}{r} - \frac{f}{r^2} + \frac{1}{2} \, f'' \right) \, h_{\mu \nu}
+ \mathcal{O}(h^2) = \Lambda h_{\mu \nu}
\end{align}
where $f(r)=1-\frac{2 G M}{r}-\frac{\Lambda}{3} r^{2}$ is the $g_{tt}$ metric component, a dot $\dot{h}_{\mu \nu}$
stands for derivative with respect to time, primes $h'_{\mu \nu} $ represent radial coordinate derivatives
and $\{\mu, \nu\} = \{\theta, \phi\}$.
Since we observe gravitational waves emitted by very distant sources, we are interested in plane wave solutions.
Then a more practical coordinate system is the cartesian set of coordinates $\{x,y,z\}$, where spatial coordinates
can be chosen such as the metric perturbation travels in the z-direction and the source is located in the x-y plane.
Again, in the TT-gauge for a purely spatial metric perturbation, transversality implies that the only non-zero
components are $h_{xx},h_{xy},h_{yx}$ and $h_{yy}$. Furthermore, since gravitational waves sources are at located
at very large distances, we are interested in the small polar angle limit $\theta \approx 0$. Using the
transformation law for a rank 2 tensor and these considerations, we end up with the following relations
\begin{equation}\label{Eq: hxx transformation}
h_{xx} = + \frac{1}{r^2} \cos(2\phi) \, h_{\theta \theta} - \frac{1}{r^2} \frac{\cos \theta}{\sin \theta} \sin(2\phi) \, h_{\theta \phi}
\end{equation}
\begin{equation}\label{Eq: hyy transformation}
h_{yy} = - \frac{1}{r^2} \cos(2\phi) \, h_{\theta \theta} + \frac{1}{r^2} \frac{\cos \theta}{\sin \theta} \sin(2\phi) \, h_{\theta \phi}
\end{equation}
\begin{equation}\label{Eq: hxy transformation}
h_{xy} = + \frac{1}{r^2} \sin(2\phi) \, h_{\theta \theta} + \frac{1}{r^2} \frac{\cos \theta}{\sin \theta} \cos(2\phi) \, h_{\theta \phi}.
\end{equation}
We observe that indeed $h_{xx} = - h_{yy}$, as is required in a traceless gauge. Applying this transformation
in \eqref{Eq: h spherical equation}, the equations of motion for the non-zero cartesian components become
\begin{equation}\label{Eq: h cartesian equation}
\ddot{h}_{ij}-\left(f^2 \, h''_{ij} +\left(f f' + \frac{2}{r}f^2\right) \, h'_{ij}\right) = 0,
\end{equation}
where now $\{i,j\} = \{x,y\}$. Notice that the equations of motion are actually simpler using cartesian components,
where all terms proportional to $h_{ij}$ without derivatives cancel out. As expected, in the minkwoskian limit where
$f \rightarrow 1$, this equation is reduced to the usual spherical wave equation.
In order to solve this equation, we extract a factor $1/r$ from the metric perturbation, since it is
expected that the amplitude of gravitational waves decreases with the distance from the source
\begin{equation}
h_{ij} (t,r) = \frac{p_{ij}(t,r)}{r},
\end{equation}
and equation \eqref{Eq: h cartesian equation} takes the following form
\begin{equation}
\ddot{p}_{ij}-\left(f^2 \, p''_{ij} + f f' p'_{ij}\right) + \frac{ff'}{r} \, p_{ij} = 0.
\end{equation}
Finally, defining the tortoise coordinate \cite{Tortoise} as $dr^*=\frac{1}{f}dr$, the above equation is reduced to
\begin{equation}\label{Eq: p}
\ddot{p}_{ij} - \partial^2_{r^*} \, p_{ij} + V(r) \, p_{ij} = 0,
\end{equation}
which is a wave equation in a potential $V(r)$ defined as
\begin{equation}\label{Eq: Potential}
V(r) = \frac{f \, f'}{r}=-\frac{2\Lambda}{3}+\frac{2}{9} \Lambda^2 r^2+\frac{2GM}{r^3}
+\frac{2GM\Lambda}{3\,r}-\frac{(2G M)^2}{r^4}.
\end{equation}
The tortoise coordinate
in SdS spacetime is usually given in terms of the event
horizon $r_S \approx 2GM$ and cosmological horizon $r_c \approx \sqrt{3/\Lambda}$, which are solutions of $f(r)=0$.
Additionally, the function $f(r)$ has another zero at $r_0 = - (r_S+r_c)$, which is not a physical horizon.
Notice that, while $0<r<r_c$, the tortoise coordinate can take values from $0<r^*<\infty$. The surface gravity
$\kappa_i$ associated with the horizon $r_i$ is also needed, defined as $\kappa_{i}=\frac{1}{2}|d f / d r|_{r=r_{i}}$.
With these quantities, the tortoise coordinate can be expressed as \cite{Tortoise}
\begin{equation}
r^*=\frac{1}{2\kappa_s} \, \log \left(\frac{r}{r_S} -1\right)-\frac{1}{2\kappa_c} \,
\log \left(1-\frac{r}{r_c}\right)+\frac{1}{2\kappa_0} \, \log \left(1-\frac{r}{r_0}\right).
\end{equation}
In order to solve equation \eqref{Eq: p}, the potential in terms of the tortoise coordinate is needed, so we are
interested in inverting the above relation to obtain $r(r^*)$. This is not easy but we recall that we are interested
in GW coming from very distant sources, so $r \gg r_S$. Then, the cosmological horizon is
$r_c \approx \sqrt{3/\Lambda}$, $r_0 \approx - r_c$ and $\kappa_c \approx \kappa_0 = \sqrt{\Lambda/3}$.
Therefore, in this situation
\begin{equation}\label{Eq: Tortoise coord}
r^* \approx \sqrt{\frac{3}{\Lambda}} \text{Arctanh} \left(\sqrt{\frac{\Lambda}{3}} \, r\right),
\end{equation}
which can be inverted as $r\approx \sqrt{\frac{3}{\Lambda}} \text{Tanh} \left(\sqrt{\frac{\Lambda}{3}} \, r^*\right)$.
With this result, it is possible to approximate the wave equation potential in terms of the tortoise coordinate as
\begin{equation}\label{Eq: Potential approx}
V(r^*) \approx -\frac{2\Lambda}{3}+\frac{2\Lambda}{3} \, \text{Tanh}^2 \left(\sqrt{\frac{\Lambda}{3}}r^*\right)
+\frac{4GM\Lambda}{3\,r^*}-\frac{(4GM)^2 \Lambda}{9\,{r^*}^2}+\frac{2GM}{{r^*}^3}-\frac{(2GM)^2}{{r^*}^4},
\end{equation}
where $\mathcal{O}(M \Lambda^2)$ and higher order terms in $M$ and $\Lambda$ have been neglected.
Notice that the potential tends to zero as $r^* \rightarrow +\infty$, as it is expected from its
definition \eqref{Eq: Potential} since $f(r)$ vanishes at $r_c$.
The relative importance of the various terms in the potential depends of course on the masses and distances involved.
Assuming that the cosmological constant has the currently preferred value $\Lambda=10^{-52} \text{ m}^{-2}$ \cite{LambdaValue},
for masses in the range of 10$^6$ solar masses at $r^* = 10^{24}$ m, the dominant term is $-2 \Lambda/3$, which is of
order $10^{-53} \text{ m}^{-2}$. The next one, involving a hyperbolic tangent, is a $\mathcal{O}(\Lambda^2)$ term and
equivalent to $10^{-57} \text{ m}^{-2}$ at this distance. Finally, the mass terms are some orders of magnitude smaller
in this regime, with $2GM/{r^*}^3$ being the leading one contributing with $10^{-63} \text{ m}^{-2}$ to the potential $V(r^*)$.
Making the comparison one should remember that $r$ here
is not the usual comoving coordinate in FLRW and, in fact, the relation is not one-to-one because the
transformation involves the time coordinate (see below).
Extracting a Fourier factor, we can search
for solutions $p(t,r^*)=u(r^*) \, e^{-i \omega t}$ in (\ref{Eq: p})
\begin{equation}\label{Eq: u}
-\omega^2 \, u(r^*)- \partial^2_{r*} \, u(r^*) + V(r^*) \, u(r^*) = 0.
\end{equation}
To begin with, we restrict our problem at distances $r_S \ll r \ll r_c$, where terms in \eqref{Eq: Potential approx}
proportional to $M$ can be neglected. Also, due to the smallness of the cosmological constant value, we are not
interested in $\mathcal{O}(\Lambda^2)$ terms. Therefore, the dominant term in the wave equation potential is
\begin{equation}
V(r^*) \approx -\frac{2\Lambda}{3}.
\end{equation}
With this approximation, equation \eqref{Eq: u} can be solved easily and the desired solution of the metric perturbation
reads at large distances
\begin{equation}\label{Eq: hij solution}
h^{\text{SdS}}_{ij} (t,r) = \frac{\epsilon_{ij}}{r} \, \cos{ (\omega t - k r^*)},
\end{equation}
where $\epsilon_{ij}$ is the polarization tensor and the wave number is defined as
\begin{equation}\label{Eq: k definition}
k^2 = \omega^2+\frac{2\Lambda}{3}.
\end{equation}
Notice that this resembles a dispersion relation corresponding to a massive wave. The reader should not
be alarmed by this. This 'mass-like' term is precisely what is needed for gravitons to have only two polarizations.
This issue is discussed in detail in \cite{Novello}, where the authors analysed the propagation of a massive spin-2 field
in a de Sitter background and showed that the field has only two degrees of freedom when the mass is $m^2_g = -2 \Lambda /3$,
and five degrees of freedom otherwise (including $m_g = 0$).
It may be interesting to remind the reader that had the term proportional to $\Lambda r^2$ be omitted altogether
in the Schwarzschild-de Sitter metric, certainly a good approximation close to the source given the smallness of the
cosmological constant, the solution would have of course been proportional
to $\cos{ (\omega t - k r)}$. After transformation to FLRW coordinates (see next section)
corrections proportional to $\sqrt\Lambda$ would appear.
The dependence of the tortoise coordinate $r^*$ on $\Lambda$ and having $\cos{ (\omega t - k r^*)}$
is therefore instrumental to recover corrections proportional
to higher powers of $\Lambda$.
\subsection{Mass corrections}
Let us take into account the dominant mass term in the wave equation potential \eqref{Eq: Potential approx}.
The following approximation for the potential $V(r^*)$ is considered
\begin{equation}
V(r^*) \approx -\frac{2\Lambda}{3}+\frac{2GM}{{r^*}^3}.
\end{equation}
Taking $u(r^*) =F(r^*) \, e^{ikr^*}$ as an ansatz, where $k$ is given by \eqref{Eq: k definition}, the wave
equation \eqref{Eq: u} can be written as follows
\begin{equation}
F''(r^*) + 2 i k \, F'(r^*) - \frac{2GM}{{r^*}^3} F(r^*) = 0,
\end{equation}
where $F'(r^*) \equiv dF(r^*)/dr^*$. This equation can be solved by defining the variable $x=1/r^*$,
which transforms the above equation into a second-order differential equation that admits an
infinite series solution
\begin{equation}
x^2 F'' + (2x - 2 i k) \, F' - 2GM x \, F = 0
\end{equation}
where now $F'=dF/dx$. Therefore, we search for solutions of the form
\begin{equation}
F = \sum^{\infty}_{n=0} a_n x^{n+s}
\end{equation}
and the following relations between $a_n$ coefficients are found
\begin{align}
&a_0 = a_0, \qquad a_1 = 0 \\
&n(n-1)a_n + 2n a_n - (2ik)(n+1) a_{n+1} + (-2GM) a_{n-1} = 0. \nonumber
\end{align}
Since $u(r^*)$ has a plane wave behaviour in the $r^* \rightarrow \infty$ limit, we take $a_0=1$. Finally,
\begin{equation}\label{Eq: F solution}
F(r^*) = 1 + \frac{-2GM}{2(2ik)} \, \frac{1}{{r^*}^2} + \frac{-2GM}{(2ik)^2} \, \frac{1}{{r^*}^3} + ...
\end{equation}
At large distances from the source $r_S \ll r$, all the terms in the series but the first one are negligible,
so it is fair to approximate $F(r^*) \approx 1$, recovering the result of the previous analysis
\eqref{Eq: hij solution}. Putting some numbers, for a supermassive black hole of $10^{10}$ solar masses, at 100 Mpc
the second coefficient of $F(r^*)$ is approximately $10^{-20} \ll 1$ and totally negligible in that situation.
However, we take note of this correction as it may be relevant in other physical situations.
\section{Perturbations in FLRW}
We now turn to the description of an expanding de Sitter universe in Friedmann–Lemaître–Robertson–Walker metric,
which incorporates the physical principles of isotropy and homogeneity. It is expressed
in comoving coordinates $\{T,R\}$
\begin{equation}
ds^2 = dT^2 - a(T)^2\left(\frac{dR^2}{1-KR^2}+ R^2 d\Omega^2 \right),
\end{equation}
where $a(T)$ is the scale factor. In this work we consider a spatially flat ($K=0$) universe. As discussed
in \cite{AEG}, a comoving cosmological observer will not see the functional form \eqref{Eq: hij solution}
since $T\neq t$ and $R \neq r$. Therefore, our aim is to relate the previous analysis to this coordinate system.
In order to find the corresponding linearized equations of motion for metric perturbations on a background
FLRW metric, we proceed as in the previous section. In the transverse and traceless TT-gauge,
for a wave propagating on the radial direction, the only non-zero metric perturbation components are
$h_{\theta \theta}, h_{\theta \phi}, h_{\phi \theta}$ and $h_{\phi \phi}$. Then, we switch to a cartesian set of
coordinates $\{X,Y,Z\}$, where it is possible to choose the $Z$-axis as the propagation direction of the wave
emitted for a very distant source in the $X$-$Y$ plane. In these coordinates, the non-zero metric perturbation
components are $h_{XX}, h_{XY}, h_{YX}$ and $h_{YY}$, which are related with the angular ones
by \eqref{Eq: hxx transformation}-\eqref{Eq: hxy transformation}, where now $r$ is replaced by
the comoving coordinate $R$.
Finally, the equations of motion for metric perturbations at first order on the FLRW metric in the TT-gauge are
\begin{align}
-\ddot{h}_{i j} + \left( \frac{\dot{a}}{a} \right) \, \dot{h}_{i j} + \frac{1}{a^2} \left(h''_{i j} +
\frac{2}{R} \, h'_{i j} \right) + 6\left( \frac{\ddot{a}}{a} \right) \, h_{i j} +
2\left( \frac{\dot{a}}{a} \right)^2 h_{i j}
= 2 \Lambda \, h_{i j},
\end{align}
where now $\{i,j\}=\{X,Y\}$, and dots and primes stand for derivatives with respect to $T$ and $R$, respectively.
In this section, a vacuum-dominated universe with only a positive cosmological constant is assumed, with scale factor
\begin{equation}\label{Eq2: Scale factor}
a(T) = a_0 \exp{\sqrt{\frac{\Lambda}{3}} \, T},
\end{equation}
where $a_0=a(T_0)=1$ is taken at the current time. In this situation, the equations of motion become
\begin{equation}\label{Eq2: EquationFLRW}
\Box_{\text{FLRW}} \, h_{ij} \equiv \left[\partial^2_T - \sqrt{\frac{\Lambda}{3}} \, \partial_T -
\frac{1}{a^2} \left(\partial^2_R + \frac{2}{R} \, \partial_R \right) - 2 \, \frac{\Lambda}{3} \right] h_{i j}= 0,
\end{equation}
where now the $\partial_i$ notation is used for derivatives for the sake of clarity. This equation again reduces to the Minkwoskian
wave equation in the absence of a cosmological constant, as expected. For a non-vanishing cosmological constant,
it is clear that a harmonic function of the variables $T,R$ is not at all a solution of these equations.
In \cite{EGR} it was found that a solution was
\begin{equation}
h_{ij}^{\text{FLRW}}= \frac{\epsilon'_{ij}}{R} \, \left(1 + \sqrt{\frac{\Lambda}{3}}T \right) \, \cos \left[\omega \, (T-R)
+ \omega \, \sqrt{\frac{\Lambda}{3}} \left(\frac{1}{2}R^2-TR\right) \right].
\end{equation}
An arbitrary superposition of such solutions with various frequencies $\omega$ would of course be a solution too
up to order $\sqrt\Lambda$. This is in fact the form that a harmonic wave in coordinates $(t,r)$ of
Schwarzschild-de Sitter takes when is transformed into FLRW coordinates using the coordinate transformation
up to order $\sqrt\Lambda$ (see below). However, at order $\Lambda$ and beyond this will not work. The reason of course
is that, as we saw in the previous section, a simple harmonic is not a solution already at order $\Lambda$
in Schwarzschild-de Sitter coordinates either. Let us now discuss this point in some more detail.
\subsection{Discussion}
At large distances from the source and assuming \eqref{Eq2: Scale factor} as scale factor, the exact transformation
between SdS and FLRW coordinates is given by \cite{BEP}
\begin{subequations}
\label{Eq2: Canvi coord TR}
\begin{align}
t(T,R) &= T - \frac{1}{2}\sqrt{\frac{3}{\Lambda}} \log \left(1- \frac{\Lambda}{3} \, a^2 \, R^2 \right)\approx T
+ \frac{1}{2} \sqrt{\frac{\Lambda}{3}} \, R^2 + \frac{\Lambda}{3} \, R^2 T + ... \label{Eq2: Canvi coord T}\\
r(T,R) &= a(T) \ R \approx R + \sqrt{\frac{\Lambda}{3}} \, RT + \frac{1}{2}\frac{\Lambda}{3} \, RT^2
+ ...\label{Eq2: Canvi coord R}
\end{align}
\end{subequations}
The transformation omits the presence of the term proportional to $M$. As we have seen in the previous section
this is totally negligible in the present setting.
Gravitational waves produced by two massive objects in orbit around each other
would be approximately described, far enough from the source, by harmonic functions periodic in time $t$
in SdS coordinates
\begin{equation}\label{Eq2: Previous solution}
h_{ij} (t,r) = \frac{\epsilon_{ij}}{r} \, \cos{\omega(t - r)}.
\end{equation}
As shown in \cite{EGR}, transforming this harmonic function into comoving coordinates using \eqref{Eq2: Canvi coord TR}
leads to a solution of the FLRW equation \eqref{Eq2: EquationFLRW} at $\mathcal{O}(\sqrt{\Lambda})$ order
but it is no longer a solution at the next order.
As discussed in the previous section, a perturbation will propagate in the Schwarzschild-de Sitter spacetime
approximately as \eqref{Eq: hij solution} far from the source and neglecting $\mathcal{O}(\Lambda^2)$ terms.
The main differences between \eqref{Eq: hij solution} and \eqref{Eq2: Previous solution} are the appearance of
the tortoise coordinate $r^*$ inside the argument of the cosine and a different wave number, both of which are
$\mathcal{O}(\Lambda)$ corrections. Using the relations \eqref{Eq: Tortoise coord} for $r^* \rightarrow r$
and \eqref{Eq2: Canvi coord TR} for $\{t,r\} \rightarrow \{T.R\}$, the SdS solution can be expressed
in comoving coordinates as
\begin{align}\label{Eq2: Transformation solutions}
h^{\text{SdS}}_{ij} (t,r) = \frac{\epsilon_{ij}}{r} \, \cos{ (\omega t - k r^*)}
\quad \rightarrow \quad h^{\text{FLRW}}_{ij} (T,R) = \frac{\epsilon'_{ij}}{R} \, a(T) \, \cos \Theta(T,R)
\end{align}
where $\epsilon'_{ij}$ is the transformed polarization tensor, $a(T)$ is the scale factor and the
argument $\Theta(T,R)$ of the cosine is given by
\begin{equation}\label{Eq2: Argument in TR}
\begin{split}
\Theta(T,R) = \omega \left[T-\sqrt{\frac{3}{\Lambda}} \, \frac{1}{2}\log \left(1- \frac{\Lambda}{3} \,
a^2 R^2 \right)\right] \\
- \sqrt{\omega^2+\frac{2\Lambda}{3}} \sqrt{\frac{3}{\Lambda}} \text{Arctanh}
\left(\sqrt{\frac{\Lambda}{3}} \, a R\right).
\end{split}
\end{equation}
Now, this functional form of $h^{\text{FLRW}}_{ij}$ should be a solution of the equations of motion obtained
by considering first-order perturbations on a FLRW background metric \eqref{Eq2: EquationFLRW} at
$\mathcal{O}(\Lambda)$ order. And indeed it can be checked that it is.
In fact, it is a solution for the next order also, i.e. $\mathcal{O}(\Lambda^{3/2})$ order. This analysis is
valid at large distances from the GW source $r_S \ll r$ and well inside the cosmological horizon,
where $\Lambda a^2 R^2 \ll 1$ and it is reasonable to neglect the leftover term in \eqref{Eq2: EquationFLRW}
\begin{align}
\Box_{\text{FLRW}} \, h_{ij} = \left(-\frac{2\Lambda}{3}+\frac{2\Lambda}{3} \frac{1}{1-\frac{\Lambda}{3} a^2 R^2 } \right)
\, h_{ij} \sim \mathcal{O}(\Lambda^2 h).
\end{align}
\subsection{Effective frequency and wave number}\label{Section: Effective freq}
In order to have a closer look at the solution written in comoving coordinates \eqref{Eq2: Transformation solutions}
and its trigonometric argument \eqref{Eq2: Argument in TR}, we expand them in powers of $\Lambda$
\begin{equation}\label{Eq2: Expanded solution}
h^{\text{FLRW}}_{ij} (T,R) = \frac{\epsilon'_{ij}}{R} \, \left(1 + \sqrt{\frac{\Lambda}{3}}T
+ \frac{1}{2} \frac{\Lambda}{3} T^2 \right) \, \cos \Theta(T,R)
\end{equation}
\begin{equation}\label{Eq2: Expanded argument}
\begin{split}
\Theta(T,R) = \omega \, (T-R) + \omega \, \sqrt{\frac{\Lambda}{3}} \left(\frac{1}{2}R^2-TR\right) \\
+ \omega \, \frac{\Lambda}{3} \left(-\frac{1}{3}R^3+R^2T-\frac{1}{2}RT^2 - \frac{R}{\omega^2}\right),
\end{split}
\end{equation}
where higher terms in $\Lambda$ have been neglected. With this expansion, the anharmonic
behaviour of the wave as seen by a cosmological observer becomes clear.
For a GW propagating in an expanding universe, physical intuition tells us that its frequency should
be redshifted as
\begin{equation}
\omega_{\text{eff}} = \frac{\omega}{1+z}.
\end{equation}
At distances $\Lambda R^2 \ll 1$ that we are considering, the cosmological redshift can be approximated
by the linear redshift-distance relation $z = H_0 R$, where $H_0$ is the Hubble constant. In fact, this result
is exact for all distances when the Hubble parameter is constant in time \cite{Harrison}, like in the
present situation with $H_0 = \sqrt{\Lambda/3}$. Therefore, the expected redshifted frequency at $\Lambda$ order is
\begin{align}\label{Freq redshift}
\omega_{\text{eff}} = \omega \, \left(1-\sqrt{\frac{\Lambda}{3}} \, R + \frac{\Lambda}{3} \, R^2 \right).
\end{align}
At $\sqrt{\Lambda}$ order, the redshift correction term on the frequency appears naturally in \eqref{Eq2: Expanded argument}.
Imposing the previous relation for the frequency to all orders in $\Lambda$ and rearranging terms
in \eqref{Eq2: Expanded argument}, the following effective wave number is found
\begin{equation}\label{Eq2: Effective k}
k_{\text{eff}} = \omega \left(1- \frac{1}{2} \, \sqrt{\frac{\Lambda}{3}}R+\frac{\Lambda}{3} \,
\left(+\frac{1}{3}R^2+\frac{1}{2}T^2 + \frac{1}{\omega^2}\right)\right)
\end{equation}
and \eqref{Eq2: Expanded solution} can be written as
\begin{equation}
h^{\text{FLRW}}_{ij} (T,R) = \frac{\epsilon'_{ij}}{R} \, \left(1 + \sqrt{\frac{\Lambda}{3}}T
+ \frac{1}{2} \frac{\Lambda}{3} T^2 \right) \, \cos \left(\omega_{\text{eff}} \, T - k_{\text{eff}} \, R\right).
\end{equation}
So much for the discussion concerning the cosmological constant only.
\section{General background}
In the previous sections, we have studied the propagation of gravitational waves in a vacuum-dominated universe,
only filled with a cosmological constant $\Lambda$. Although being the dominant part of the energy and
matter budget of the universe, the $\Lambda$CDM model incorporates also matter (dark or baryonic)
and radiation. In terms of their density parameter $\Omega_i$ defined by
\begin{equation}
\Omega_i \equiv \frac{8\pi G \, \rho_i}{H^2},
\end{equation}
where $\rho_i$ is the density of each species and $H$ the Hubble parameter. The present-day values are
$\Omega_{\Lambda,0} \sim 0.7$, $\Omega_{\text{dust},0} \sim 0.3$, $\Omega_{\text{rad},0} \sim \mathcal{O}(10^{-5})$.
While it is safe to neglect the presence of radiation and relativistic matter, the contribution of dust
is of the same order as the cosmological constant one. The Hubble constant $H_0$ is the current value
of $H$ and is given by
\begin{equation}\label{Eq3: Hubble constant}
H_0 = \sqrt{\frac{\kappa \rho_{\Lambda}}{3}+\frac{\kappa \rho_{\text{dust,0}}}{3}+\frac{\kappa \rho_{\text{rad,0}}}{3}}.
\end{equation}
The different types of matter and energy of the Universe are considered as perfect fluids with equation
of state $p_i=w_i \rho_i$ and included in the energy-momentum tensor as
\begin{equation}
T_{\mu \nu} = (\rho + p)U_{\mu}U_{\nu} - p \, g_{\mu \nu},
\end{equation}
where the fluid four-velocity fulfills the normalization condition $g_{\mu \nu} U^{\mu} U^{\nu} = 1$. It is
important to note that, while the four-velocity in FLRW comoving coordinates is given by $U^{\mu} = (1,0,0,0)$,
we have $U^t \neq 1, U^r \neq 0$ in SdS static coordinates. The angular components do vanish in both coordinates
systems, $U^{\theta} = U^{\phi} = 0$. For a detailed discussion, see Appendix B of \cite{AEG}.
Considering radiation, with $w_{\text{rad}} = 1/3$, pressureless matter, $w_{\text{dust}} = 0$, and vacuum energy
with a negative pressure, $w_{\Lambda} = -1$, and related with the cosmological constant by
$\rho_{\Lambda} = \Lambda/\kappa$ we have
\begin{align}
T^{(\text{rad})}_{\mu \nu} = \frac{4}{3} \, \rho_{\text{rad}} U_{\mu}U_{\nu} - \frac{1}{3} \, \rho_{\text{rad}} \,
g_{\mu \nu} \qquad T^{(\text{dust})}_{\mu \nu} = \rho_{\text{dust}} \, U_{\mu}U_{\nu} \qquad T^{(\Lambda)}_{\mu \nu}
= \rho_{\Lambda} \, g_{\mu \nu}.
\end{align}
With these ingredients, we proceed as in section \ref{Section:SdS} and consider small perturbations $h_{\mu \nu}$
around a background metric. The Einstein equations can be expanded up to first order in the perturbation as
\begin{equation}\label{Eq3: Perturbation theory}
G_{\mu\nu}(\tilde{g}+h) = G_{\mu\nu}(\tilde{g}) +
\frac{\delta G_{\mu\nu}}{\delta g_{\alpha \beta}}\, \biggl\rvert_{\tilde{g}}\ h_{\alpha\beta} + ...
= \kappa \left( T^{(0)}_{\mu \nu} + T^{(1)}_{\mu \nu} + ...\right)
\end{equation}
where $\kappa = 8 \pi G$. Again, these field equations are satisfied for an unperturbed metric,
$G_{\mu\nu}(\tilde{g}) = \kappa T^{(0)}_{\mu \nu}$.
A spherically symmetric coordinate system that describes the de Sitter space and incorporates the presence of
dust and (eventually) radiation is needed. The following linearized metric deduced in \cite{AEG,Luciano} satisfies
these conditions and reduces to the Schwarzschild-de Sitter metric \eqref{Eq: SdS metric} when dust
and radiation are not present
\begin{equation}\label{Eq3: General metric}
\begin{split}
ds^2 = \left(1 - \frac{\kappa}{6} \, ( 2 \rho_{\Lambda} - 2 \rho_{\text{rad}} - \rho_{\text{dust}}) \, r^2 \right) \, dt^2 \\
- \left(1 + \frac{\kappa}{3} \, ( \rho_{\Lambda} + \rho_{\text{rad}} + \rho_{\text{dust}}) \, r^2 \right) \, dr^2 - r^2 \, d\Omega^2.
\end{split}
\end{equation}
In the above expression the mass term $r_S/r$ has been neglected since it is not important at large distances
$r_S \ll r$ as we discussed before. We perturb around this background \eqref{Eq3: Perturbation theory}
and as we have done in section \ref{Section:SdS}, we work in the TT-gauge and with purely spatial components
of the metric perturbation $h_{\mu \nu}$, where the only non-zero ones are
$h_{\theta \theta}$, $h_{\theta \phi}$, $h_{\phi \theta}$ and $h_{\phi \phi}$, related by
$h_{\phi \phi} = - \sin^2 \theta \, h_{\theta \theta}$ and $h_{\phi \theta} = h_{\theta \phi}$. With these ingredients,
the perturbed Einstein equations neglecting $\mathcal{O}(h^2)$ and higher orders are
\begin{align}\label{Eq3: Equations Spherical}
\frac{1}{f}\, \ddot{h}_{\mu \nu}
+\frac{1}{2}\left(\frac{\dot{g}}{f g}-\frac{\dot{f}}{f^2} \right) \, \dot{h}_{\mu \nu}
- \frac{1}{g} \, h''_{\mu \nu}
- \left(- \frac{2}{r} \frac{1}{g}
- \frac{1}{2} \frac{g'}{g^2}
+ \frac{1}{2} \frac{f'}{f g}\right) h'_{\mu \nu}
- \bigg(\frac{2}{r^2} \frac{1}{g} \\
+\frac{\ddot{g}}{f g}
-\frac{1}{2}\frac{(\dot{g})^2}{f g^2}
- \frac{1}{2} \frac{\dot{f}\dot{g}}{f^2 g}
+ \frac{2}{r}\frac{g'}{g^2}
- \frac{2}{r} \frac{f'}{f g}
+ \frac{1}{2} \frac{f' g'}{f g^2}
+ \frac{1}{2} \frac{(f')^2}{f^2 g}
- \frac{f''}{f g} \bigg)\, h_{\mu \nu}
= -2 \kappa \, T^{(1)}_{\mu \nu},
\nonumber
\end{align}
where $\{\mu, \nu\}$ stands for $\{\theta, \phi\}$, dots and primes for time and radial derivatives, respectively,
and we have defined the $\tilde{g}_{tt}$ and $\tilde{g}_{rr}$ components of the background metric as
\begin{align}\label{Eq3: f definition}
f(t,r) &= 1 - \frac{\kappa}{6} \, ( 2 \rho_{\Lambda} - 2 \rho_{\text{rad}} - \rho_{\text{dust}}) \, r^2 \\
g(t,r) &= 1 + \frac{\kappa}{3} \, ( \rho_{\Lambda} + \rho_{\text{rad}} + \rho_{\text{dust}}) \, r^2 \label{Eq3: g definition}.
\end{align}
Before proceeding, an expression for the perturbed energy-momentum tensor is also needed. In this gauge, we are
only interested in the angular components so, at first order in the perturbation, they are given by
\begin{align}
T^{(1)}_{\mu \nu} = (\rho_{\Lambda} -\frac{1}{3} \rho_{\text{rad}}) \, h_{\mu \nu}.
\end{align}
Now, considering a GW travelling in the z-direction, we express the components of the metric perturbation in
cartesian coordinates using the same reasoning as in section \ref{Section:SdS} and the
\eqref{Eq: hxx transformation}-\eqref{Eq: hxy transformation} relations. In this coordinate system, equations
\eqref{Eq3: Equations Spherical} take the following form
\begin{equation}\label{Eq3: Equations Cartesian}
\begin{split}
\ddot{h}_{ij}+\frac{1}{2}\left(\frac{\dot{g}}{g}-\frac{\dot{f}}{f} \right) \, \dot{h}_{ij}-\frac{f}{g} \, h''_{ij}
- \left(\frac{2}{r}\frac{f}{g} + \frac{1}{2} \frac{f'}{g} - \frac{1}{2} \frac{fg'}{g^2} \right) \, h'_{ij}
-\bigg( \frac{\ddot{g}}{g}-\frac{1}{2}\frac{(\dot{g})^2}{g^2} \\
- \frac{1}{2} \frac{\dot{f}\dot{g}}{fg}
- \frac{f'}{g} \frac{1}{r} + \frac{f g'}{g^2} \frac{1}{r} + \frac{(f')^2}{2 f g} + \frac{f' g'}{2 g^2}
- \frac{f''}{g} \bigg) \, h_{ij}
= - 2\kappa f (\rho_{\Lambda} -\frac{1}{3} \rho_{\text{rad}}) \, h_{ij}
\end{split}
\end{equation}
where now $\{i,j\}$ stands for $\{x, y\}$. These equations are not easy to solve, but we recall that our aim
is to find a solution at $\mathcal{O}(\rho)$ order, equivalent to the $\Lambda$ order in the previous sections.
With this purpose in mind, we study the order in the densities $\rho$ of each term in equation
\eqref{Eq3: Equations Cartesian}. For the time-derivative terms, assuming a constant $\rho_{\Lambda}$, we have
\begin{align}
\dot{f} &= \frac{\kappa}{6} \, (2 \partial_t \rho_{\text{rad}} + \partial_t \rho_{\text{dust}}) \, r^2 \\
\dot{g} &= \frac{\kappa}{3} \, (\partial_t \rho_{\text{rad}} + \partial_t \rho_{\text{dust}}) \, r^2,
\end{align}
and from \cite{AEG,Luciano}, the time derivatives of the densities are
\begin{equation}
\partial_t (\kappa \rho_{\text{dust}}) = - \frac{A}{3-\kappa \rho_{\text{dust}} r^2} \,
\frac{(\kappa \rho_{\text{dust}})^{4/3}}{t^{1/3}}
\end{equation}
\begin{equation}
\kappa \, \partial_t \rho_{\text{rad}} = - \frac{B}{3-\kappa \rho_{\text{rad}} r^2} \,
(\kappa \rho_{\text{rad}})^{3/2},
\end{equation}
where $A=3 \sqrt[3]{6}$ and $B=4 \sqrt{3}$. Then, in the limit $\kappa \rho_i r^2 \ll 1$, we can approximate
$\dot{f}$ and $\dot{g}$ as
\begin{equation}
\dot{f}, \dot{g} \sim (\kappa \rho_{\text{rad}})^{1/2} (\kappa \rho_{\text{rad}} \, r^2)
+ \left(\frac{\kappa \rho_{\text{dust}}}{t}\right)^{1/3} (\kappa \rho_{\text{dust}} \, r^2).
\end{equation}
Therefore, these terms are of higher order in the densities in this distance regime and can be neglected.
The same reasoning applies for the $\ddot{g}$ term. Then, the remaining terms proportional to the metric
perturbation $h_{ij}$ at this order are
\begin{equation}
- \left( - \frac{f'}{g} \frac{1}{r} + \frac{f g'}{g^2} \frac{1}{r} + \frac{(f')^2}{2 f g}
+ \frac{f' g'}{2 g^2} - \frac{f''}{g} \right) \, h_{ij} + 2 \kappa f \left(\rho_{\Lambda}
-\frac{1}{3} \rho_{\text{rad}} \right) \, h_{ij}.
\end{equation}
Using the $f$ and $g$ definitions \eqref{Eq3: f definition}-\eqref{Eq3: g definition}, we can study these
terms at linear order with the densities and observe that they vanish, so this combination of terms proportional
to the metric perturbation is of order $\rho^2$ or higher
\begin{equation}
\begin{split}
\bigg[ - 4 \, \frac{\kappa}{6} \, ( 2 \rho_{\Lambda} - 2 \rho_{\text{rad}} - \rho_{\text{dust}})
- 2 \, \frac{\kappa}{3} \, ( \rho_{\Lambda} + \rho_{\text{rad}} + \rho_{\text{dust}}) \\
+ 2 \kappa \left(\rho_{\Lambda} -\frac{1}{3} \rho_{\text{rad}} \right) + \mathcal{O}(\rho^2) \bigg] \, h_{ij}
\sim \mathcal{O}(\rho^2) \, h_{ij}.
\end{split}
\end{equation}
Consequently, the equations of motion for the cartesian components \eqref{Eq3: Equations Cartesian} at this
order are simplified as
\begin{equation}\label{Eq3: Final equations}
\ddot{h}_{ij}-\frac{f}{g} \, h''_{ij} - \left(\frac{2}{r}\frac{f}{g} + \frac{1}{2} \frac{f'}{g}
- \frac{1}{2} \frac{fg'}{g^2} \right) \, h'_{ij}
= 0.
\end{equation}
These equations reduce to the analogous ones on the Schwarzschild-de Sitter metric when $g=1/f$, as expected.
Using the same strategy to solve these equations than in section \ref{Section:SdS}, we factor out the expected
$1/r$ behaviour of propagating gravitational waves as $h(t,r)=p(t,r)/r$, so
\begin{equation}\label{Eq3: p equation}
\ddot{p}_{ij}-\frac{f}{g} \, p''_{ij} - \left( \frac{1}{2} \frac{f'}{g} - \frac{1}{2} \frac{fg'}{g^2} \right) \left(p'_{ij}
- \frac{p}{r}\right) = 0
\end{equation}
We would like now to introduce a generalized tortoise coordinate that simplifies this differential equation,
analogous to the SdS case. In fact, this coordinate should recover the SdS form, $dr^* = \frac{1}{f}dr$,
when $g=1/f$. The desired tortoise coordinate is
\begin{equation}\label{Eq3: Tortoise coord general}
dr^* = \sqrt{\frac{g}{f}} \, dr
\end{equation}
under which \eqref{Eq3: p equation} takes the following form
\begin{equation}
\ddot{p}_{ij}- \partial^2_{r^*} \, p_{ij}+ V(r) \, p_{ij} = 0,
\end{equation}
where we have defined the potential $V(r)$ with the following expression and, at first order in the densities
and in the $\rho r^2 \ll 1$ regime, is given by
\begin{equation}
V(r) = \frac{1}{2} \frac{1}{r}\left( \frac{f'}{g} - \frac{fg'}{g^2} \right) \approx - \frac{\kappa}{3} \,
( 2 \rho_{\Lambda} + \frac{1}{2} \rho_{\text{dust}}).
\end{equation}
Although integrating the definition \eqref{Eq3: Tortoise coord general} is not as straightforward as in
the de Sitter space with only a cosmological constant, a solution can be found in terms of elliptic integrals,
which can be approximated at first order in the densities as
\begin{equation}\label{Eq3: Tortoise approximation}
r^* \approx r + \frac{1}{6} \, \frac{\kappa}{3} \, ( 2 \rho_{\Lambda} + \frac{1}{2} \rho_{\text{dust}}) \, r^3.
\end{equation}
Therefore, combining the above results it is possible to find the following solution of equations
\eqref{Eq3: Final equations} at the considered order
\begin{equation}\label{Eq3: hij general solution}
h_{ij}(t,r) = \frac{\epsilon_{ij}}{r} \cos (\omega t - kr^*)
\end{equation}
with a wave number
\begin{equation}
k^2 = \omega^2 + \frac{\kappa}{3} \, ( 2 \rho_{\Lambda} + \frac{1}{2} \rho_{\text{dust}}).
\end{equation}
This solution is a generalization of expression \eqref{Eq: hij solution} for a universe filled with pressureless
matter and radiation. Notice that $\rho_{\text{rad}}$ does not appear at first order in
the expression of the potential $V(r)$, so it does not appear in the wave number $k$ one either. This is so because
the radiation coefficients in the $f$ and $g$ definitions \eqref{Eq3: f definition}-\eqref{Eq3: g definition}
cancel out when added, a behaviour that affects the studied first order. At higher order in the densities some
radiation density contributions would eventually emerge. For the same reason, $\rho_{\text{rad}}$ does not appear in the
tortoise coordinate approximation \eqref{Eq3: Tortoise approximation} either.
\section{Comoving coordinates}
We are interested in how GW are seen by a cosmological observer, so our aim is to express
the above solution \eqref{Eq3: hij general solution} in comoving coordinates. First, the inclusion of other
cosmological parameters in \eqref{Eq2: Canvi coord T}-\eqref{Eq2: Canvi coord R} is needed, where only a
cosmological constant was considered.
For simplicity, in the following analysis the radiation density $\rho_{\text{rad}}$ will be neglected, since
it does not appear at leading order in the above expressions and its contribution in the current observed
universe budget seems to be some orders of magnitude lower than $\rho_{\Lambda}$ and $\rho_{\text{dust}}$. Then, it
is possible to obtain an expression for the scale factor by solving the first Friedmann equation, coming from the
Einstein field equations
\begin{equation}
\left(\frac{\dot{a}}{a}\right)^2 = \frac{\kappa \, \rho_{\Lambda}}{3} + \frac{\kappa \,
\rho_{\text{d0}}}{3}\left(\frac{a_0}{a} \right)^3,
\end{equation}
where $\rho_{\text{d0}}$ stands for the current value of the non-relativistic matter density.
The scale factor is given by \cite{AEG}
\begin{equation}
a(T)=a_{0}\left[\sqrt{1+\frac{\rho_{\text{d0}}}{\rho_{\Lambda}}} \sinh \left(\sqrt{3 \kappa \rho_{\Lambda}} \frac{\Delta T}{2}\right)
+\cosh \left(\sqrt{3 \kappa \rho_{\Lambda}} \frac{\Delta T}{2}\right)\right]^{2/3},
\end{equation}
where $\Delta T = T-T_0$ and $a(T_0)=a_0$, taken as 1 as in the previous section. The scale factor for a
$\Lambda$-dominated universe \eqref{Eq2: Scale factor} is recovered when $\rho_{\text{d0}} \rightarrow 0$.
Using this result, it is possible to find the transformation between static SdS coordinates and FLRW comoving ones
that also preserves spherical symmetric, so the angular element $r^2 d\Omega^2$ becomes $a(T)^2R^2 d\Omega^2$.
Linearization in terms of the densities leads to the results in \cite{AEG}.
\begin{subequations}
\label{Eq3: Canvi general TR}
\begin{align}
t(T,R) &\approx T+\frac{1}{2} \sqrt{\frac{\kappa}{3}( \rho_{\Lambda}+ \rho_{\text{d0}})} \,
R^{2}+\left(\frac{\kappa \rho_{\Lambda}}{3}+\frac{\kappa \rho_{\text{d0}}}{12}\right) \Delta T R^{2}
+\ldots \label{Eq3: Canvi general T}\\
r(T,R) &\approx R+\sqrt{\frac{\kappa}{3}( \rho_{\Lambda}+ \rho_{\text{d0}})} \,
\Delta T R+ \frac{\kappa}{12}\left(2 \rho_{\Lambda}- \rho_{\text{d0}}\right)\Delta T^2 R+\ldots \label{Eq3: Canvi general R}
\end{align}
\end{subequations}
In the limit $\rho_{\text{d0}} \rightarrow 0$ we recover the relations \eqref{Eq2: Canvi coord T}-\eqref{Eq2: Canvi coord R},
recalling that $\kappa \rho_{\Lambda}=\Lambda$. Moreover, at order $\sqrt{\rho_0}$ all cosmological densities appear in
such a combination that reproduces the Hubble constant $H_0$ \eqref{Eq3: Hubble constant}, but this is not so when
the next order is considered, linear with the densities. Consequently, it is possible {\em a priori} to
distinguish the contribution from the various densities.
By using these relations along with \eqref{Eq3: Tortoise approximation}, we can transform \eqref{Eq3: hij general solution}
into FLRW comoving coordinates, keeping $\rho_0$ order terms
\begin{equation}\label{Eq3: General solution comoving}
h_{ij} (T,R) = \frac{\epsilon'_{ij}}{R} \left(1+\sqrt{\frac{\kappa}{3}( \rho_{\Lambda}+ \rho_{\text{d0}})} \,
T + \frac{\kappa}{12}\left(2 \rho_{\Lambda}-\rho_{\text{d0}}\right)T^2 \right) \cos \Theta(T,R)
\end{equation}
where $\epsilon'_{ij}$ is the transformed polarization tensor and this time the trigonometric argument is given by
\begin{gather}\label{Eq3: General argument}
\Theta(T,R) = \omega \, (T-R) + \omega \, \sqrt{\frac{\kappa}{3}( \rho_{\Lambda}
+ \rho_{\text{d0}})} \left(\frac{1}{2}R^2-TR\right) \nonumber\\
+ \omega \, \left[-\left(\frac{\kappa \rho_{\Lambda}}{9} + \frac{\kappa \rho_{\text{d0}}}{36}\right)R^3
+\left(\frac{\kappa \rho_{\Lambda}}{3} + \frac{\kappa \rho_{\text{d0}}}{12}\right)R^2T
-\left(\frac{\kappa \rho_{\Lambda}}{6} - \frac{\kappa \rho_{\text{d0}}}{36}\right)RT^2\right] \\
- \frac{1}{\omega} \left(\frac{\kappa \rho_{\Lambda}}{3} + \frac{\kappa \rho_{\text{d0}}}{12}\right)R \nonumber
\end{gather}
These expressions are a generalization of \eqref{Eq2: Expanded solution}-\eqref{Eq2: Expanded argument},
where only a cosmological constant was considered. Also, the different combination of coefficients proportional
to the densities $\rho_0$ makes clear that it is not possible to write them in terms of $H^2_0$ only, as stated before.
It is also interesting to express the trigonometric argument as $\omega_{\text{eff}} \, T - k_{\text{eff}} \, R$,
where the effective frequency $\omega_{\text{eff}}$ satisfies the expected cosmological redshift.
Analogously to what has been done in section \ref{Section: Effective freq}, the GW frequency will be
redshifted as $z=H_0 R$
\begin{equation}
\omega_{\text{eff}} = \omega \left(1 - \sqrt{\frac{\kappa}{3}( \rho_{\Lambda}+ \rho_{\text{d0}})} \,
R + \frac{\kappa}{3}( \rho_{\Lambda}+ \rho_{\text{d0}}) R^2\right)
\end{equation}
and the remaining terms form the following effective wave number
\begin{gather}
k_{\text{eff}} = \omega \bigg[1 - \frac{1}{2} \sqrt{\frac{\kappa}{3}( \rho_{\Lambda}+ \rho_{\text{d0}})} \, R
+ \left(\frac{\kappa \rho_{\Lambda}}{9}+\frac{\kappa \rho_{\text{d0}}}{36} \right) R^2
+ \left(\frac{\kappa \rho_{\Lambda}}{6}-\frac{\kappa \rho_{\text{d0}}}{12} \right) T^2 + \\
+ \frac{\kappa \rho_{\text{d0}}}{4} TR
+ \frac{1}{\omega^2} \left(\frac{\kappa \rho_{\Lambda}}{3}+\frac{\kappa \rho_{\text{d0}}}{12} \right) \bigg], \nonumber
\end{gather}
which agrees with \eqref{Eq2: Effective k}.
\section{Observational consequences}
Let us summarize briefly our findings. In the previous sections we have been able to derive and solve
the differential equation governing the propagation of GW in a universe endowed with a cosmological
constant and matter density up to order $\rho_\Lambda^\frac{3}{2}$
for a vacuum dominated universe and up to order $\rho_\Lambda, \rho_{\text{dust}}$ in the general case.
A limitation of the previous results obtained in \cite{EGR}, where these results had been obtained up to order
$\rho_\Lambda^\frac12$ only, was that going beyond a few Mpc was questionable. The new terms obtained allow us to
explore sources in the Gpc range reliably. In this section we will explore the consequences of the
new corrections in the context of PTA.
\begin{figure}[t]
\centering
\includegraphics[width=0.33\linewidth]{PTAdiagram.pdf}
\caption{Diagram of the relative position of a GW source at a distance $Z_E$ from the Earth and a pulsar
located at $\vec{P}$ from the source. The angles $\alpha$ and $\beta$ are the polar and azimuthal angles
of the pulsar with respect to the Earth-source axis.}
\label{Fig: PTA diagram}
\end{figure}
Consider the configuration described in Fig. \ref{Fig: PTA diagram}, where the relative position of a GW source,
a nearby pulsar and the Earth is shown. The pulsar emits electromagnetic pulses with a time-dependent
phase $\phi_0 (T)$, which measured from the Earth reads \cite{Deng}
\begin{equation}
\phi (T) = \phi_0 \left[T - \frac{L}{c} - \tau_0 (T) - \tau_{\text{GW}} (T) \right],
\end{equation}
where we have recovered the speed of light factor $c$, $\tau_0 (T)$ takes into account some corrections on the
motion of the Earth and the Solar
System and $\tau_{\text{GW}} (T)$ is a timing correction due to the effect of GW. Since a non-zero value modifies
the pulse arrival time, $\tau_{\text{GW}} (T)$ is known as gravitational wave timing residual. It is given by \cite{Deng}
\begin{equation}
\tau_{\text{GW}} (T) = - \frac{1}{2} \, \hat{n}^i \hat{n}^j \, \mathcal{H}_{ij} (T)
\end{equation}
where $\hat{n}$ is a unit vector in the Earth-pulsar direction and $\mathcal{H}_{ij} (T)$ is the integral of the
metric perturbation along the null geodesic from the pulsar to the Earth. The pulsar-Earth path can be parametrized
as $\vec{R}(x) = \vec{P} + L \, (1 +x) \, \hat{n}$, with $x \in [-1, 0]$, so the integral is given by
\begin{equation}\label{Eq4: H integral}
\mathcal{H}_{ij} (T) = \frac{L}{c} \int_{-1}^{0} h_{ij}^{\text{FLRW}} \left(T_E + \frac{x L}{c}, \vec{R} (x) \right) dx.
\end{equation}
Before proceeding, we make some reasonable approximations. For neighbor pulsars, which are inside our Galaxy, and
GW sources, such as galaxy mergers, we have $L \ll Z_E$. Then, the parametrized path that light follows from the pulsar
to the Earth, in modulus, is $R (x) \approx Z_E + x \, L \cos \alpha$.
Moreover, for a GW propagating in the $Z$ direction, the only non-zero components of the metric perturbation in
the TT-gauge are the spatial $X,Y$ ones. Therefore, we assume for simplicity that the non-zero components of the
transformed polarization tensor are $\epsilon \sim |\epsilon_{ij}|$ for $i, j = X, Y$. Furthermore, we can always
choose a reference plane defined by the position of the Earth, the pulsar and the GW source, so the azimuthal angle
in Fig. \ref{Fig: PTA diagram} can be set to $\beta = 0$. This angle is not important in the timing residual behaviour,
but this is not the case for the polar angle $\alpha$. Finally, the timing residual of the arrival time due to the
passing of gravitational waves reads
\begin{equation}\label{Eq5: Timing residual}
\tau_{\text{GW}} (T_E, Z_E, L, \alpha, \epsilon, \omega, \rho_i) =
- \frac{L \epsilon}{2c} \sin^2 \alpha \int_{-1}^{0} h_{ij}^{\text{FLRW}} \left(T_E + \frac{x L}{c}, Z_E + xL \cos \alpha \right) dx.
\end{equation}
where $h_{ij}^{\text{FLRW}}$ is given by \eqref{Eq2: Expanded solution} for a vacuum-dominated universe and
\eqref{Eq3: General solution comoving} for the $\rho_\Lambda+\rho_\text{dust}$ case.
Our purpose here is simply to assess the relevance of the $\mathcal{O}(\rho)$ corrections with respect to
the $\mathcal{O}(\sqrt\rho)$ corrections previously known.
In order to perform a numerical analysis, we take reasonable values for some parameters appearing in
\eqref{Eq5: Timing residual}. We choose $\epsilon= 1.2 \times 10^9 \text{ m}$, so that $|h| \sim \epsilon / Z \sim 10^{-15}$,
and $\omega = 10^{-8} \text{ rad/s}$, corresponding to ultra-low frequency GW signals, which are values within
the sensibility of PTA projects \cite{Supermassive,Jenet}. We consider one pulsar located at $L = 0.3 \text{ kpc}$ from the Earth.
In \cite{Espriu2013,Espriu2014,AEG,EGR,Alfaro} it was shown that the presence of a non-zero cosmological constant and
other cosmological fluids
could affect the timing residuals. In Fig. \ref{Fig: Sqrt vs Rho} we compare the resulting timing residual
for the already known solution $h_{ij}$ at $\sqrt{\rho}$ order with the inclusion of $\rho$ order corrections,
given by \eqref{Eq3: General solution comoving}, for a universe filled with dark energy and dark matter.
A remarkable feature of these plots is an important enhancement of the signal for a particular value of the
angle $\alpha_m$, where the timing residual reaches its maximum. The position of this peak depends on the distance
to the source $Z_E$, which occurs at larger angles for further sources. While the peak appears at similar
angular positions for both cases, the corrections linear with the densities $\rho$ allow us to safely
explore remoter sources in the Gpc region where most mergers are expected to occur. The
peak $\alpha_m$ is at slightly lower values when the $\rho$ corrections are included, particularly for
very distant sources. The main conclusion
of the present study is that it is actually viable to seek for the effect of very massive black hole
mergers at any distance.
\begin{figure}[t]
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\linewidth]{PlotRho100Mpc.pdf}
\label{Fig: 100Mpc}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\linewidth]{PlotRho500Mpc.pdf}
\label{Fig: 500Mpc}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\linewidth]{PlotRho1Gpc.pdf}
\label{Fig: 1Gpc}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\linewidth]{PlotRho2Gpc.pdf}
\label{Fig: 2Gp}
\end{subfigure}
\caption{Comparison of the absolute timing residual $|\tau_{\text{GW}}|$ between the $\sqrt{\rho}$ (blue) and $\rho$ (green) order solutions for sources located at 100 Mpc, 500 Mpc, 1 Gpc and 2 Gpc. The figures are symmetrical for $\pi \leq \alpha \leq 2\pi$.}
\label{Fig: Sqrt vs Rho}
\end{figure}
\newpage
\section{Conclusions}
It was already known that a harmonic function like \eqref{Eq2: Previous solution} in $\{t,r\}$ coordinates,
which describes well gravitational waves far away from their source, is a solution of the equations of motion for
perturbations on a FLRW background metric \eqref{Eq2: EquationFLRW} only up to $\sqrt{\Lambda}$ order when
transformed into comoving coordinates $\{T,R\}$. In order to go to the next order, we have studied metric
perturbations on the SdS metric in section \ref{Section:SdS} and obtained $h^{\text{SdS}}(t,r)$ \eqref{Eq: hij solution},
which includes $\mathcal{O}(\Lambda)$ corrections inside the argument of the cosine. This functional form,
transformed into comoving coordinates, does satisfy the FLRW perturbation equations in the TT-gauge up to $\Lambda^2$ terms.
The previous discussion is valid well inside the cosmological horizon.
In addition, we have extended the analysis to include all other cosmological fluids up to order $\rho$.
We provide explicit formulae
for the effective wave number. This result is non-trivial. Furthermore, it is found that beyond the leading order
the densities appear in combinations other than $H_0$. This potentially removes degeneracies in what concerns the
propagation of gravitational waves in a cosmological background. These results support the conclusions put forward
in previous works \cite{Espriu2013,Espriu2014,AEG,EGR,Alfaro} concerning the possible measurement of the cosmological
parameters in PTA observations. In fact, as emphasized
e.g. in \cite{AEG}, this effect may {\em facilitate} a positive detection of GW in PTA.
In passing, we have derived a number of interesting results, such as the dependence of the propagation equation on the
final mass of the merger producing the gravitational waves, which is minute but possibly conceptually relevant.
\section*{Acknowledgements}
We would like to thank our collaborators J. Alfaro, J. Bernabeu, L. Gabbanelli and D. Puigdom\`enech.
This research is partly supported by the Ministerio de Ciencia e Inovaci\'on under research grants
PID2019-105614GB-C21, CEX2019-000918-M of ICCUB (Unidad de Excelencia Mar\'\i a de Maeztu),
and by grant 2017SGR0929 (Generalitat de Catalunya).
|
{
"timestamp": "2021-08-24T02:14:28",
"yymm": "2108",
"arxiv_id": "2108.09567",
"language": "en",
"url": "https://arxiv.org/abs/2108.09567"
}
|
\section{\label{sec:intro}Introduction\protect\\}
It is a common practice to investigate planetary fluid dynamics assuming the flow to be incompressible and inviscid.
Such flows are typically characterized by two-dimensional
(2D) turbulence wherein there is direct cascade of enstrophy to smaller
scales and inverse cascade of energy to larger scales. Based on a
2D turbulence model, Lorenz\cite{doi:10.3402/tellusa.v21i3.10086}
predicted the atmospheric flow as a function of spatial scales. Leith\cite{1971JAtS...28..145L}, and Leith \& Kraichnan\cite{Leith1972}
refined Lorenz\textquoteright s predictability estimates to further
illustrate the application of a turbulence theory to important meteorological
problems. In fact, the topic of 2D turbulence dates back to the theoretical
studies of Kraichnan\cite{doi:10.1063/1.1762301}, Kraichnan \& Montgomery\cite{Kraichnan_1980},
and Leith\cite{doi:10.1063/1.1691968} on an infinite plane. They
predicted the classical power laws of $k^{-3}$ in the forward enstrophy-cascade
range and of $k^{-5/3}$ in the inverse energy-cascade range for
the energy spectrum. Lilly\cite{doi:10.1063/1.1692444} numerically
integrated the 2D incompressible Navier-Stokes equations in order
to test the validity of Kraichnan's predictions on the structure of
2D turbulence. Batchelor\cite{doi:10.1063/1.1692443} computed the
energy spectrum in homogeneous 2D turbulence. Newell\cite{newell_1969} proposed a mechanism comprising of the
resonant interaction of Rossby wave packets to generate planetary
zonal flows. However, these studies do not examine the reasons for the existence of Rossby waves.
Furthermore, these classical studies
did not consider the effect of planetary rotation ("beta"), and
the presence of long-lived coherent vortices in 2D turbulence was
not widely known \citep{Huang1998}.
\subsection{2D turbulence on a $\beta$-plane}
Rhines\cite{rhines_1975} presented the first
geophysical application of the 2D turbulence theory. The effect of
planetary rotation on 2D turbulence was investigated with
a numerical model on a $\beta$-plane. The inverse energy cascade
was shown to cease roughly at a characteristic wave number $k_{\beta}=\sqrt{\beta/2U}$
(also known as the Rhines scale $k_{R}$), where $k$ is the total wave
number, $U$ is the RMS velocity and $\beta$ is the meridional gradient
of the Coriolis parameter (later denoted by $f$). The turbulence transforms into Rossby
waves around the wave number $k_{\beta}$. The $\beta$-effect (the
meridional variation of the Coriolis parameter $\beta$) causes the flow field to be anisotropic, and a zonal band structure consisting
of alternating easterly and westerly jets emerges. The stronger
$\beta$-effect on eddies that are meridionally elongated compared with zonally elongated
eddies makes the Rhines scale anisotropic.
Holloway \& Hendershott\cite{holloway_hendershott_1977} investigated decaying 2D turbulence
on a $\beta$-plane and observed zonal anisotropy for wavenumber
$k\leq k_{\beta}^{H}\equiv\beta/Z$, where $Z$ is the RMS vorticity.
Shepherd\cite{shepherd_1987} extended the theory of homogeneous
barotropic $\beta$-plane turbulence to include effects arising from spatial
inhomogeneity in the form of a zonal shear flow and demonstrated profound
effects of the background shear flow on the inhomogeneous turbulence.
The shear straining process induced a downscale enstrophy transfer
similar to the traditional downscale enstrophy cascade due to local
eddy-eddy interaction in spectral space. The shear was shown to induce
transfer of disturbance energy into the range of $k\leq k_{\beta}$
and make the disturbance flow field to become meridionally anisotropic
in the low-wave number range. The observed atmospheric energy spectrum
was explained with these concepts \citep{1987JAtS...44.1166S}. Yamada \& Yoneda\cite{YAMADA20131} proved, with a mathematically
rigorous theorem, that at a high $\beta$, the resonant interactions
of Rossby waves (which are expected to dominate the $\beta$-plane
dynamics) govern the flow dynamics for an incompressible 2D flow on
a $\beta$-plane.
Several studies\cite{10.1175/1520-0485,maltrud_vallis_1991,doi:10.1063/1.1327594,doi:10.1063/1.1327594,CHEKHLOV1996321,Basdevant1981,holloway_hendershott_1977,doi:10.1080/03091928008241180,bartello_holloway_1991,10.1175/1520-0485,doi:10.2514/5.9781600866340.0108.0120,Panetta1993,doi:10.1063/1.166007,doi:10.1063/1.166011,10.1175/JAS4013.1} present the dynamics of forced 2D turbulence (such as ``wave-turbulence boundary", zonostrophic turbulence, etc.) on a $\beta$-plane.
\subsection{2D turbulence on a rotating sphere\label{sec:decaying_2d_turbulence}}
Yoden \& Yamada\cite{Yoden1993}
investigated the effects of rotation and sphericity on decaying
2D turbulence on a rotating sphere. Large rotation rates, under the
existence of Rossby waves, revealed an easterly jet at high latitudes.
Huang \& Robinson\cite{Huang1998}
examined the dynamics of 2D turbulence on a rotating sphere, and
derived and verified the anisotropic Rhines scale in decaying turbulence
simulations. The inverse energy cascade along the zonal axis (zonal
wavenumber $m=0$) was not directly arrested by beta in their simulations.
Coherent polar vortices emerge in decaying 2D turbulence simulations
on a rotating sphere, although multiple zonal jets are difficult to
obtain \citep{Yoden1993}. However,
rotating shallow-water turbulence with a finite radius of deformation
does generate multiple zonal jets \citep{doi:10.1063/1.868929}. For
a variety of initial conditions and a sufficiently large rotation rate,
a band structure at mid-latitudes and westward circumpolar jets
in the polar regions have been observed \citep{YODEN1999,Hayashi2000,ishioka:hal-00301974}.
Hayashi {\it et al.}\cite{10.1175/2007JAS2209.1} reviewed jet formation
/ zonal mean flow generation in decaying 2D turbulence
on a rotating sphere from the view-point of Rossby waves with
two parameter space of the rotation rate and Froude number Fr. For
a nondivergent flow (low Fr) and large rotation rate, intense easterly
circumpolar jets in addition to a banded structure of zonal mean flows
with alternating flow directions were found. However, for divergent
flows (with increasing Fr), circumpolar jets disappear and an equatorial
easterly jet emerges. Takehiro {\it et al.}\cite{Takehiro_2007a} investigated
the strength and width of the easterly circumpolar jets and discovered
asymptotic behaviors in rapidly rotating cases. Extremely inhomogeneous
banded structure of zonal flows and accumulation of most of the kinetic
energy inside the easterly circumpolar jets were revealed. Takehiro {\it et al.}\cite{Takehiro_2007b} confirmed these revelations
by performing 2D decaying turbulence simulations for a barotropic fluid on a rotating
sphere. Establishment of the banded structure of zonal flows was observed
relatively early in the simulations.
At late times, only the
circumpolar jets were intensified gradually, while no further evolution
in the banded structure in the low and midlatitudes was observed.
The easterly momentum transport from the low and midlatitudes associated
with Rossby waves contributes to the maintenance of the circumpolar
easterly jets \citep{Hayashi2000,10.1175/2007JAS2209.1}. Yoden {\it et al.}\cite{10.1007/978-4-431-67002-5_22} investigated 2D
decaying turbulence for a nondivergent barotropic fluid on a rotating
sphere to survey the nature of pattern formation from random initial
fields. Isolated coherent vortices emerged in non-rotational cases
as in the planar 2D turbulence. However, a westward circumpolar vortex
in high-latitudes and zonal band structures in mid and low-latitudes
appeared with increasing rotation rate. They investigated dependence
of these features on the initial energy spectrum and discussed the
dynamics of such pattern formations with a weakly nonlinear Rossby
wave-zonal flow interaction theory. Sasaki {\it et al.}\cite{sasaki_takehiro_yamada_2012}
examined the stability of inviscid zonal jet flows on a rotating sphere. Saito \& Ishioka\cite{Izumi_SAITO20162016-002}
obtained a quasi-invariant associated with the emergence of
zonally elongated structures in 2D turbulence on a rotating sphere
by a minimization process. They explained the anisotropic energy transfer
that favors zonally elongated structures depicting airfoil-shaped contours
for the weighting coefficient distribution. The quasi-invariant (defined as a weighted sum of the energy
density in the wavenumber space)
was shown to conserve well if nonlinearity
of the system is sufficiently weak. Obuse \& Yamada\cite{PhysRevFluids.4.024601} investigated
three-wave resonant interactions of Rossby-Haurwitz waves in 2D turbulence
on a rotating sphere. According to them, the zonal waves of the form
$Y_{l}^{m=0}\exp\left(i\omega t\right)$ with odd $l$ should be considered
for inclusion in the resonant wave set to ensure that the dynamics
of the resonant wave set determine the overall dynamics of the turbulence
on a rapidly rotating sphere. Here, $Y_{l}^{m}$ are the spherical
harmonics and $\omega=-2\Omega m/\left[l\left(l+1\right)\right]$
is the frequency of a Rossby-Haurwitz wave.
\subsubsection{Shallow-water turbulence}
The shallow-water equations are the simplest model of planetary flows
considering the effects of divergence \citep{farge_sadourny_1989,yuan_hamilton_1994}.
Cho \& Polvani\cite{doi:10.1063/1.868929} found remarkably different
characteristic flow patterns in shallow-water turbulence from that
in 2D non-divergent turbulence. Their investigations revealed a retrograde
equatorial jet instead of a polar jet (which is dominant in 2D turbulence)
because of the effects of planetary rotation. The asymmetry between
a cyclone and an anticyclone is attributed to the predominance of a retrograde
jet in the shallow-water system \citep{doi:10.1063/1.869898}. Kitamura \& Ishioka\cite{10.1175/JAS4015.1} performed ensemble experiments
of decaying shallow-water turbulence on a rotating sphere to confirm
the robustness of the emergence of an equatorial jet. Predominance
of a prograde jet, although less likely in shallow-water turbulence,
was also noted. From the examination of a zonal-mean flow induced by wave-wave
interactions, using a weak nonlinear model, they found that Rossby
and mixed Rossby-gravity waves induce second-order acceleration.
Yoden {\it et al.}\cite{10.1007/978-94-007-0360-5_21}
reviewed jet formation in decaying 2D turbulence on a rotating sphere
considering wave mean-flow interaction for both shallow-water case
and non-divergent case (in the limit Fr $\rightarrow$ zero). They investigated
the behavior of mean zonal flow generation in the two dimensional ($\Omega$, Fr) parameter space
where $\Omega$ is the non-dimensional rotation rate. For the non-divergent
flow and large $\Omega$, an intense retrograde circumpolar jet and a
banded structure of mean zonal flows with alternating flow directions
in middle and low latitudes emerged. With increasing Fr, the circumpolar
jets disappeared and a retrograde jet emerged in the equatorial region.
The appearance of the intense retrograde jets was attributed to the
angular momentum transport associated with the propagation and absorption
of Rossby waves. In non-divergent flows, long Rossby waves tend to
be absorbed around the poles. However, for large Fr, Rossby waves
hardly propagate towards the poles and are absorbed near the equator.
The equatorial jet was not always retrograde, the emergence of a less
likely prograde jet was also found.
Although investigating the dynamics of {\em forced} 2D turbulence on a rotating sphere is beyond the scope of the present work, we, nonetheless, note several interesting studies\cite{Williams1978,Basdevant1981,doi:10.1063/1.869518,doi:10.1063/1.869327,Huang1998,doi:10.1063/1.34273,doi:10.1063/1.1327594,PhysRevLett.89.124501,10.1175/JAS4003.1,doi:10.1063/1.3407652,10.1175/1520-0485,doi:10.1063/1.1373684,doi:10.1029/2004GL019691,galperin:hal-00302701,Galperin_2008,vallis_2017,CHEKHLOV1996321,doi:10.1029/2004GL020106,galperin:hal-00302701,10.1175/2007JAS2219.1,PhysRevLett.101.178501}.
\subsection{Discrete exterior calculus (DEC)}
Before delving into some conclusions drawn from the review, we digress to present a brief introduction of DEC. Exterior calculus deals with the calculus of differential geometry,
hence with the calculus of differential forms, and provides an alternative
to the vector calculus. DEC is a numerical exterior calculus and deals
with the discrete differential forms. A discreet differential form
is an integral quantity on a mesh object, e.g., integral of a vector
along a mesh edge $\int\mathbf{v}\cdot\mathbf{dl}$ represents a discrete
1-form. DEC retains at the discrete level many of the identities of
its continuous counterpart. It is coordinate independent, therefore
suitable for solving flows over curved surfaces. For the further reading, a few representative DEC references include the references \onlinecite{flanders1963differential, abraham2012manifolds, hirani2003discrete, desbrun2003discrete, desbrun2005discrete, grinspun2006discrete, hirani2008numerical, desbrun2008discrete, crane2013digital, perot2014differential, mohamed2016discrete, de2016subdivision}.
\subsection{Scope of the present study}
The literature suggests that there is no uniform agreement about the arrest of the cascade at the Rhines scale. Moreover, the focus of the previous investigations was to analyze the effect of rotation on the inverse energy cascade, and the effect on the direct enstrophy cascade was implied. A comprehensive analysis investigating the effect of rotation on the vorticity dynamics on a sphere is warranted. We investigate the effect of rotation on vorticity dynamics on a unit
sphere (which falls in the category of decaying 2D turbulence as reviewed in section \ref{sec:decaying_2d_turbulence}) using a DEC method \citep{jagad2020primitive}. The evolution of the
vorticity field from an arbitrary initial vorticity distribution,
is examined with an emphasis on investigating the effect of rotation. Presently, we vary the rate of rotation
of the sphere $\left(\Omega\right)$ from zero (Ro = $\infty$) to 320 (Ro = $1.30 \times 10^{-3}$). Here, the Rossby number $\mathrm{Ro}=U/2\Omega L $, where $U$ is the characteristic velocity scale (which is assumed to be equal to square root of total kinetic energy), $L$ is the characteristic length scale (assumed as the radius of sphere). The sphere undergoes $\Omega/\left(2\pi\right)$ rotations per unit time. Considering $U\approx 10$ m/s for the atmosphere, and $L\approx 6 \times 10^{6}$ m, then $\Omega = 36.47$ rad/s for the unit sphere represents the rotating earth. Moreover, we choose the eddy turnover time ( = $4\pi / \left | \omega_{max} \right|$ with $\left | \omega_{max} \right |$ denoting the maximum vorticity magnitude) as the characteristic time scale for the representation of non-dimensional time $t$. In addition, we vary the range of spherical harmonics wave numbers (with total initial kinetic energy held approximately constant) constituting the initial vorticity field, and investigate the effect of differential initial spectral representations. Table \ref{tab:Simulation-parameters} shows the parameters employed in the present study. We vary the initial condition from case A to F, and for each case, we vary Ro as indicated in table \ref{tab:Simulation-parameters}.
As discussed in detail later, the
time evolution of the vorticity field at different Ro indicates that
increasing rotation diminishes the forward cascade of enstrophy and zonalizes
the vortical structures. However, the zonalization of the structures
does not continue monotonically with ever decreasing Rossby numbers,
and the structures tend to a non-zonal state below a certain
Rossby number for the initial vorticity field comprising
of intermediate-wavenumber spherical harmonics (for test cases A, B, and C). Whereas, for the initial vorticity field comprising also of large-wavenumber spherical harmonics (for test cases D, E, and F), the tendency to zonalization is monotonic. Furthermore, we express the vorticity distribution in terms of spherical harmonics, and determine the spherical harmonic modes. From the spectral content, we determine the vorticity power spectrum and the effect of rotation on it. Moreover, we investigate the effect of rotation on the spectral distribution of the vorticity power. We also determine the effect of rotation on the vorticity probability density, which is a measure of the area occupied by each vorticity level. Additionally, we identify individual vortical structures using a segmentation procedure, and determine the effect of rotation on the probability density of aspect
ratio (ratio of major to minor axis length of the vortical structures),
and that of the cosine of the angle between the major axis and the
azimuthal direction (which is a measure of orientation of the vortical
structures). The analysis further confirms the non-monotonic and monotonic nature of zonalization
with decreasing Ro observed qualitatively in the time evolution of
the vorticity field, respectively for intermediate-wavenumbers and intermediate to large-wavenumbers constituting the initial vorticity field. Moreover, it reveals that although, rotation
diminishes the forward cascade of enstrophy, it does not completely
cease/arrest the cascade.
The outline of the paper
is as follows. The simulations results are
presented along with detailed quantification of the important
dynamical quantities first. The paper closes by emphasizing the key
conclusions deduced from the conducted simulations and analysis of the results. A brief description of the physical setup and numerical procedure is presented in appendix \ref{sec:num_proc}, followed by a case for the verification of the numerical procedure in appendix \ref{appB0}. The segmentation algorithm employed for identifying individual vortical
structures is presented in appendix \ref{appB}.
\begin{table*}
\centering{}%
\begin{tabular}{|ccccccc|}
\hline
&\multicolumn{6}{c|}{Test case} \tabularnewline
&\multirow{1}{*}{A} & \multirow{1}{*}{B} & \multirow{1}{*}{C} & \multirow{1}{*}{D} & \multirow{1}{*}{E} & \multirow{1}{*}{F}\tabularnewline
\hline
Spherical harmonics wavenumber $\left(l\right)$ range & \multirow{2}{*}{4 - 10} & \multirow{2}{*}{4 - 10 } & \multirow{2}{*}{4 - 10} & \multirow{2}{*}{4 - 20} & \multirow{2}{*}{4 - 40} & \multirow{2}{*}{4 - 80}\tabularnewline
constituting the initial vorticity field & & & & & & \tabularnewline
\hline
\multirow{8}{*}{Rossby number (Ro)} & \multicolumn{6}{c|}{$\infty$}\tabularnewline
& \multicolumn{6}{c|}{$8.34\times10^{-2}$}\tabularnewline
& \multicolumn{6}{c|}{$4.17\times10^{-2}$}\tabularnewline
& \multicolumn{6}{c|}{$2.08\times10^{-2}$}\tabularnewline
& \multicolumn{6}{c|}{$1.04\times10^{-2}$}\tabularnewline
& \multicolumn{6}{c|}{$5.20\times10^{-3}$}\tabularnewline
& \multicolumn{6}{c|}{$2.60\times10^{-3}$}\tabularnewline
& \multicolumn{6}{c|}{$1.30\times10^{-3}$}\tabularnewline
\hline
\end{tabular}
\begin{centering}
\caption{Simulation parameters. \label{tab:Simulation-parameters}. Note that spherical harmonics wavenumber $\left(l\right)$ range constituting the initial vorticity field is the same for the cases A, B, and C, but the amplitude is different.}
\par\end{centering}
\end{table*}
\section{Results and Discussion}
In this section, we first present the results for one of the cases (case A) with arbitrary initial vorticity field comprising of spherical harmonics wavenumbers $l = 4 - 10$ (see table \ref{tab:Simulation-parameters}) . This is followed by a discussion of other cases emphasizing the effect of wavenumber range comprising the initial vorticity field.
We compute the initial stream function distribution at the primal mesh nodes as
\begin{equation}
\psi\left(\theta,\phi\right)=\sum_{l=4}^{10}\sum_{m=-l}^{l}\psi_{lm}Y_{l}^{m}\left(\theta,\phi\right),
\label{eq:shexpansion}
\end{equation}
\noindent where $\theta$ is the colatitude, $\phi$ is the longitude, $Y_{l}^{m}$ is the spherical harmonic function of degree $l$ and order $m$, and $\psi_{lm}$ are the expansion coefficients (or spherical harmonic modes) with $\psi_{l-m}=\left(-1\right)^{m}\psi_{lm}^{\ast}$. Here, $\psi_{lm}^{\ast}$ is the complex conjugate of $\psi_{lm}$. The coefficients $\psi_{lm}$ are assigned values as given in Dritschel {\it et al.}\cite{dritschel2015late}. The degree $l$ characterizes the total wavenumber and order $m$ characterizes the azimuthal/zonal wavenumber of the spherical harmonics. The initial mass flux 1-form for a primal mesh edge is now computed as $u^{\ast}=d_{0}\psi$. The vorticity distribution (that of the component of vorticity vector normal to the surface) at the primal mesh nodes is computed as $\omega=\ast_{0}^{-1}\left[-d_{0}^{T}\right]\ast_{1}u^{\ast}$. Figure \ref{fig:Initial-conditions} shows the initial vorticity distribution.
\subsection{Vorticity field evolution}
Figure \ref{fig:Vorticity-distribution-at} shows the vorticity distribution with varying Rossby numbers at four time instants (early, mid, and late-to-very late). For the non-rotating case $\mathrm{Ro}=\infty$ (top panel in Figure \ref{fig:Vorticity-distribution-at}) the vortices rapidly develop thin filaments due to the forward enstrophy cascade by time $t=18.461$ (eddy turnover time).
Smaller scale vortices merge to form larger ones due to the inverse energy cascade.
At late times ($t=92.304$, and $t=292.60$) much of the enstrophy has cascaded beyond the smallest scale resolved in the simulation. Due to the biharmonic term, the grid level small scale enstrophy dissipates numerically. An oscillating quadrupolar vortical structure, similar to that in Dritschel {\it et al.}\cite{dritschel2015late}, is the late time outcome.
On the other hand, for the rotating cases (finite $\mathrm{Ro}$ number) we note that the evolution into the thin filaments in inhibited and the absence of the quadrupolar vortical structure at late times. Thus, rotation diminishes the forward cascade of enstrophy (and also the inverse cascade of energy). Moreover, as the Rossby number decreases, up to Ro $=2.08 \times 10^{-2}$, the vortices tend to become elongated and aligned along the azimuthal direction (tend to become zonal - a process dubbed as ``zonalization"). With increased rotation, i.e., further decrease in Ro, these structures tend towards smaller elongation and non-zonal in character. Thus, the zonalization of the structures with decreasing Ro is non-monotonic (for the present case) and this is apparent at late times. We also note the presence of circumpolar vortices for the rotating cases as previously reported \citep{Yoden1993,doi:10.1063/1.868929}.
\begin{figure}
\centering{}\includegraphics[scale=0.18]{figures/initial_vorticity_distribution}\caption{Arbitrary initial vorticity field computed from the spherical harmonic modes. The bottom panel shows the spherical surface projected using the standard orthographic projection.
\label{fig:Initial-conditions}}
\end{figure}
\begin{figure*}
\centering{}\includegraphics[scale=0.2]{figures/omega_vs_Ro_and_t_turnover}\caption{Time evolution of vorticity field as a function of Ro, showing diminishing enstrophy cascade and zonalization of vortical structures with decreasing Ro.\label{fig:Vorticity-distribution-at}}
\end{figure*}
In order to quantify the effect of rotation, vorticity power spectra, spectral distribution of vorticity power, vorticity probability density, probability density of aspect ratio (ratio of major to minor axis length of vortical structures) and the cosine of the angle between the major axis and the azimuthal direction (a measure of orientation of the vortical structures) are determined. These quantifications are discussed subsequently.
\subsection{Vorticity power spectra}
Spherical harmonics satisfying the following orthogonality relation \citep{Wieczorek2018}
\begin{equation}
\int_{\Omega}Y_{lm}\left(\theta,\phi\right)Y_{l'm'}\left(\theta,\phi\right)\mathrm{d}\Omega=4\pi\delta_{ll'}\delta_{mm'},
\end{equation}
\noindent where $\Omega$ is the surface area of the sphere, and $\delta_{ij}$ is the Kronecker delta function. With this, the spherical harmonic expansion coefficients are expressed as
\begin{equation}
\omega_{lm}=\frac{1}{4\pi}\int_{\Omega}\omega\left(\theta,\phi\right)Y_{lm}\left(\theta,\phi\right)\mathrm{d}\Omega.
\end{equation}
The total vorticity power in the physical and spectral domains is related as
\begin{equation}
\frac{1}{4\pi}\int_{\Omega}\omega^{2}\left(\theta,\phi\right)\mathrm{d}\Omega=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\omega_{lm}^{2}=\sum_{l=0}^{\infty}S_{ff}\left(l\right), \label{eq:sffl} \end{equation}
\noindent where $S_{ff}\left(l\right)$ is the power spectrum of vorticity. We use Eq. (\ref{eq:sffl}) to compute the vorticity power spectrum $S_{ff}\left(l\right)$.
Figure \ref{fig:Vorticity-power-spectra} shows vorticity power spectra for three time instants and for different rotation rates. The initial (at t = 0) vorticity power is confined to a wavenumber range of $l=4-10$. This power cascades to the larger waver numbers as the flow evolves with time. As the Rossby number decreases (increase in rotation rate), the slope of the vorticity power spectra increases. This implies that the forward cascade of enstrophy diminishes with decreasing Rossby number. With decreasing Ro, the proportion of the spectral power contained in the wave numbers $l=4-10$ increases, even at a late time t = 92.304, further supporting the observation that the enstrophy cascade diminishes with decreasing Ro. However, there is still significant power contained in the larger wave numbers, even at a very small Ro = $1.30 \times 10^{-3}$, showing that the forward enstrophy cascade is not suppressed/arrested completely which is somewhat contrary to the previously held notions of the enstrophy cascade \citep{rhines_1975,Yoden1993,doi:10.1063/1.1327594}.
\begin{figure*}
\centering{}
\subfloat[]{\begin{centering}
\includegraphics[scale=0.35]{figures/Power_spectra_t4}
\par\end{centering}
}\subfloat[]{
\centering{}\includegraphics[scale=0.35]{figures/Power_spectra_t40}} \\
\subfloat[]{
\centering{} \includegraphics[scale=0.35]{figures/Power_spectra_t200}}
\caption{Vorticity power spectra as a function of wavenumber for varying Ro, depicting diminishing enstrophy cascade with decreasing Ro. Time instants shown are (a) $t=1.846$, (b) $t=18.461$, (c) $t=92.304$. \label{fig:Vorticity-power-spectra}}
\end{figure*}
\subsection{Spectral distribution of vorticity power}
It is interesting to examine the effect of rotation on the distribution of the vorticity power in spectral space (i.e. using the $m-l$ space of spherical harmonics). The results are plotted in figure \ref{fig:Vorticity-power-distribution} where the spectral distribution is shown using right angle triangles: the vertical (resp. horizontal) edge of each triangle is the total wavenumber $l$ (azimuthal wave number $m$, resp.).
In figure \ref{fig:Vorticity-power-distribution}(a) we plot the
initial vorticity power, comprising
of the total modes $l=4-10$ and all of the initial zonal modes $m=4-10$,
and is isotropic
with respect to the zonal modes.
The vorticity power cascades to smaller
scales (larger $l$) as the flow evolves with time. At later times,
for the non-rotating case ($\mathrm{Ro}=\infty$), the vorticity power
still remain nearly uniformly distributed over all of the comprising
zonal modes, and isotropic with respect
to the zonal modes. With the Rossby number decreasing from infinity
to $2.08 \times 10^{-3}$, the vorticity power distribution tends to become
concentrated
in the smaller zonal modes and the distribution of the vorticity power
becomes anisotropic with respect to the zonal modes. This represents
the zonalization of the vortical structures and transition to Rossby
wave like motions from turbulent flow with decreasing Ro. With further
decrease in Ro, however, all of the comprising zonal modes tend to
become equally significant, i.e., the vorticity power returns to
isotropy in the zonal modes, and this is more clearly observable at late
times. Thus, the zonalization of the vortical structures with decreasing
Ro is non monotonic for the present case.
\begin{figure*}
\centering{}
\subfloat[]{
\centering{}\includegraphics[scale=0.22]{figures/t0}}
\hspace{3mm}\subfloat[]{\centering{}\includegraphics[scale=0.15]{figures/t4}}
\caption{}
\end{figure*}
\begin{figure*}
\ContinuedFloat
\centering{}
\subfloat[]{
\centering{}\includegraphics[scale=0.15]{figures/t_40}}\hspace{3mm}\subfloat[\label{fig:t-=00003D-92.304}]{
\begin{centering}
\includegraphics[scale=0.15]{figures/t_200}
\par\end{centering}
}\caption{Vorticity power distribution in spectral space ($l,m$) for different Ro, showing diminishing enstrophy cascade with decreasing Ro and zonalization of the power. Time instants shown are (a) $t = 0$, (b) $t = 1.846$, (c) $t = 18.461$, (d) $t = 92.304$ \label{fig:Vorticity-power-distribution}}
\end{figure*}
Table \ref{tab:Rhynes-scale-for} shows the Rhines scales \citep{rhines_1975} for the present case. We choose the reference latitude to be 45{\textdegree} for the computation of $\beta$ here. For Ro$\leq 1.04 \times 10^{-2}$, the wavenumbers ($l=4-10$) comprising the initial vorticity field are smaller than the Rhines scale (as a wavenumber, $k_{\beta}$), i.e., the scales comprising the initial vorticity field are larger than the Rhines scale (as a length scale, 1/$k_{\beta}$). Hence, according to the Rhines theory \citep{rhines_1975}, for the present case for Ro$\leq 1.04 \times 10^{-2}$, the eddies in the initial vorticity field should not merge and initiate the inverse energy cascade (and therefore the forward enstrophy cascade) because the initial scales are already larger than the Rhines scale. However, the spectral analysis (see figures \ref{fig:Vorticity-power-spectra}, \ref{fig:Vorticity-power-distribution}) does show forward enstrophy cascade even for Ro$\leq 1.04 \times 10^{-2}$, indicating that the cascade does not cease completely at the Rhines scale.
\begin{center}
\begin{table}
\begin{centering}
\begin{tabular}{|ccccc|}
\hline
Ro & $k_{\beta}$ & & Ro & $k_{\beta}$\tabularnewline
\hline
8.34$\times10^{-2}$ & 3.878 & & 5.20$\times10^{-3}$ & 15.514 \tabularnewline
4.17$\times10^{-2}$ & 5.485 & & 2.60$\times10^{-3}$ & 21.940\tabularnewline
2.08$\times10^{-2}$ & 7.757 & & 1.30$\times10^{-3}$ & 31.027\tabularnewline
1.04$\times10^{-2}$ & 10.970 & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c|}{}\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{Rhines scale for case A\label{tab:Rhynes-scale-for}}
\end{table}
\par\end{center}
\subsection{Vorticity probability density}
The vorticity probability density/vorticity measure is defined as
the area occupied by each vorticity level and expressed as
\begin{equation}
p\left(\omega\right)=\frac{\textrm{the area occupied by vorticity in the range \ensuremath{\left[\omega+\Delta\omega\right]}}}{4\pi \Delta\omega}. \label{eq:pd_vorticity}
\end{equation}
The conservation of all Casimirs/inviscid invariants is equivalent
to the conservation of $p\left(\omega\right)$ $\left ( p\left(\omega + f \right) \right )$ for the non-rotating (for the rotating) inviscid flow.
However, there is a lack of conservation in simulations at finite
resolution because of the inevitable cascade of vorticity to small
scales \citep{dritschel2015late}. Here, we investigate the probability density of vorticity in the same spirit as Dritschel {\it et al.} \citep{dritschel2015late}. Figure \ref{fig:Vorticity-probability-density}
shows the vorticity probability density for different Rossby numbers
at $t=1.846,\,18.461$ and $92.304$. The initial (at $t=0)$ vorticity
probability density is broadly distributed between $\omega_{min}=-5.1187$
and $\omega_{max}=5.7996$. For the non-rotating case, $p\left(\omega\right)$
narrows and diminishes everywhere except for values of $\omega$ near
zero at late times. However, for the rotating cases, the distribution
of $p\left(\omega\right)$ remains broad even at late times, and smaller
the Rossby number, broader is the distribution of $p\left(\omega\right)$
at late times. This shows that as the Rossby number decreases, the
forward cascade of vorticity to smaller scales decreases, consistent
with the inferences made from the vorticity and power spectra plots.
\begin{figure*}
\begin{centering}
\subfloat[]{
\begin{centering}
\includegraphics[scale=0.35]{figures/pd_omega_t4}
\par\end{centering}
}\subfloat[]{
\centering{}\includegraphics[scale=0.35]{figures/pd_omega_t40}}
\par\end{centering}
\centering{}\subfloat[]{
\begin{centering}
\includegraphics[scale=0.35]{figures/pd_omega_t200}
\par\end{centering}
}\caption{Vorticity probability density as a function of time and Ro. The distribution of probability density becomes broader with decreasing Ro. (a) $t=1.846$, (b) $t=18.461$, (c) $t=92.304$. \label{fig:Vorticity-probability-density}}
\end{figure*}
\subsection{Aspect ratio and orientation of the vortical
structures}
We identify individual vortical structures using an algorithm (see
appendix \ref{appB}). We determine the centroid of each individual structure
from the computation of first moment of area. Hence, we have
\begin{equation}
\bar{\theta}_{i}=\frac{\int_{\Omega}\theta_{i}\left|\omega\right|\textrm{d}\Omega}{\int_{\Omega}\left|\omega\right|\textrm{d}\Omega}=\frac{\sum_{n=1}^{N}\left(\theta_{i}\right)_{n}\left|\omega\right|_{n}\left(\Delta\Omega\right)_{n}}{\sum_{n=1}^{N}\left|\omega\right|_{n}\left(\Delta\Omega\right)_{n}};i=1,2.
\end{equation}
Here, $\theta_{1}$ and $\theta_{2}$ are the coordinates longitude
and latitude respectively, and the overbar stands for the centroid
coordinates, $\left|\omega\right|$ is the vorticity magnitude, $\Omega$
is the total surface area of the sphere, $\Delta\Omega$ is the dual
area surrounding a primal node, and $N$ is the total number of primal
nodes. We approximate each individual structure by an ellipse, and
determine the major and minor axes from the computation of the second
moment of area as follows. The expression of second moment of area
is as follows.
\begin{eqnarray}
& \mu_{ij}=\frac{\int_{\Omega}\left(\theta_{i}-\bar{\theta}_{i}\right)\left(\theta_{j}-\bar{\theta}_{j}\right)\left|\omega\right|\textrm{d}\Omega}{\int_{\Omega}\left|\omega\right|\textrm{d}\Omega} \nonumber \\ &=\frac{\sum_{n=1}^{N}\left[\left(\theta_{i}\right)_{n}-\bar{\theta}_{i}\right]\left[\left(\theta_{j}\right)_{n}-\bar{\theta}_{j}\right]\left|\omega\right|_{n}\left(\Delta\Omega\right)_{n}}{\sum_{n=1}^{N}\left|\omega\right|_{n}\left(\Delta\Omega\right)_{n}};i=1,2,j=1,2.
\end{eqnarray}
Then, we compute a covariant matrix as
\begin{equation}
M=\left[\begin{array}{cc} \mu_{11} & \mu_{12}\\ \mu_{21} & \mu_{22} \end{array}\right].
\end{equation}
Now, we determine the lengths of the major and minor axes of the vortical structure
from the larger ($\lambda_{larger}$) and smaller ($\lambda_{smaller}$) eigenvalues of the covariant matrix, respectively. The aspect ratio, a measure of the elongation,
of the vortical structure is computed as the ratio of the major axis length to
the minor axis length. The orientation of the vortical structure is
determined from the angle ($\alpha$, see the schematic in figure \ref{fig:vortical_structures}) between
the major axis and the azimuthal direction. We determine this angle from the dot product of
the eigenvector corresponding to the larger eigenvalue with the
basis vector in the azimuthal direction. A representative plot showing individual structures, the major and minor axes of these structures, and the definition of the angle $\alpha$ is given in figure \ref{fig:vortical_structures}. We compute the probability density of the aspect ratio or $\cos\alpha$ as
\begin{equation}
p\left(f\right)=\frac{\textrm{The number of \ensuremath{f} values in the range \ensuremath{\left[f+\Delta f\right]}}}{\textrm{Total number of \ensuremath{f} values \ensuremath{\times}}\Delta f}.
\end{equation}
\begin{figure}
\centering{}\includegraphics[scale=0.35]{figures/visit0041_coord_axes_added}\caption{A representative plot showing individual vortical structures, major and minor axes, and the definition of angle $\alpha$.
\label{fig:vortical_structures}}
\end{figure}
Here, the function $f$ represents the aspect ratio or $\cos\alpha$ distribution. Figure \ref{fig:Probability-density-function-aspect-ratio} shows
the probability density of the aspect ratio of the vortical structures
as a function of Ro at late times (over the turnover time from 69.228
- 115.379). Here, the structures with aspect ratio > 20 are excluded for
the sake of clarity. The lower limit of the aspect ratio (= 1) represents circular structures, and the higher values represent the elongated structures.
For the non-rotating case (Ro$=\infty$), the
peak in the value of probability density at aspect ratio $\approx1$
corresponds to the presence of quadrupolar vorticity field. The decreasing
probability density at larger aspect ratios corresponds to the presence
of small scale vortices due to the forward enstrophy cascade. With
the Rossby number decreasing from infinity to $2.08 \times 10^{-2}$, the probability
density peak moves away from aspect ratio $\approx1$, showing that
the vortical structures tend to elongate with decreasing Ro. In fact,
these elongated structures tend to be zonal (see figure \ref{fig:Probability-density-function-orientation}).
With further decrease in Ro, the probability density peak tends back to
move towards aspect ratio $\approx1$. This shows that the vortical
structures tend back to become circular and non-zonal (see figure \ref{fig:Probability-density-function-orientation})
with further decrease in Ro. This is consistent with the aforementioned
claim that the zonalization of the vortical structure is non-monotonic
with decreasing Ro (for the present case).
\begin{figure}
\centering{}\includegraphics[scale=0.4]{figures/pdf_aratio}\caption{Probability density function of the aspect ratio of individual vortices
as a function of Ro over the turnover time from 69.228 - 115.379.
\label{fig:Probability-density-function-aspect-ratio}}
\end{figure}
Similarly, figure \ref{fig:Probability-density-function-orientation}
shows the probability density of the orientation of the vortical structures
(that of the cosine of the angle between the major axis and the azimuthal
direction) as a function of Ro at late times (over the turnover time
from 69.228 - 115.379). The upper limit $\cos\alpha=1$ represents the structures aligned in the zonal direction, and the lower values represent the non-zonal structures. For the non-rotating case (Ro$=\infty$), the relatively higher values of $p\left(\cos\alpha\right)$ at larger $\cos\alpha$
values correspond to the presence of small scale vortices due to the
forward enstrophy cascade. For the rotating cases,
with Ro decreasing up to
$2.08 \times 10^{-2}$ from infinity, $p\left(\cos\alpha\right)$ tends to increase
at $\cos\alpha\approx1$, and to vanish at smaller values of $\cos\alpha$.
This indicates the zonalization of the structures with decreasing
Ro. Moreover, these zonal structures have larger aspect ratio / are
elongated (see figure \ref{fig:Probability-density-function-aspect-ratio}).
With further decrease in Ro, $p\left(\cos\alpha\right)$ tends to decrease
at $\cos\alpha\approx1$, and to augment at smaller values of $\cos\alpha$.
Thus, the structures tend to become non-zonal and circular (see figure
\ref{fig:Probability-density-function-aspect-ratio}, the aspect ratio
of these structures tend to be smaller) with further decrease in Ro.
This further confirms the aforementioned claim of non-monotonic nature
of the zonalization of the vortical structure with decreasing Ro (for the present case).
\begin{figure}
\centering{}\includegraphics[scale=0.4]{figures/pdf_cos_angle}\caption{Probability density function of the cosine of the angle between the
major axis and the azimuthal direction as a function of Ro over the
turnover time from 69.228 - 115.379.\label{fig:Probability-density-function-orientation}}
\end{figure}
\subsection{Effect of wavenumber rage constituting the (arbitrary) initial vorticity field}
We consider two different arbitrary initial vorticity fields comprising of wavenumbers $l=4-10$ (test cases B, and C), and three arbitrary initial vorticity fields comprising of wavenumbers $l=4-20$, $4-40$ and $4-80$ (test cases D, E, and F). We investigate the effect of these initial conditions on the evolution of the vorticity field. The results are presented in the following sub-sections.
\subsubsection{Influence of initial vorticity distribution}
So far we have examined the details of vorticity evolution for case A wherein the initial vorticity was confined to the wavenumber range $l=4-10$.
Figure \ref{fig:Effect-of-wavenumber-range} shows the evolution of the vorticity field for different initial wavenumber distributions. For each distribution we ensure that the total kinetic energy is approximately constant. Although, the initial wavenumber range for the cases B and C is the same (also, it is the same for case A), the initial amplitudes are different for these cases.
For the initial vorticity field comprising of intermediate wavenumbers ($l=4-10$, test cases B, and C), at a later time, the emerging vortices from the merger of smaller scales (due to inverse cascade) tend to be zonal due to the rotation. However, the zonalization is non-monotonic with decreasing Ro. Whereas, for the initial vorticity field comprising of intermediate to large wavenumbers ($l=4-20$, $l=4-40$, and $l=4-80$; test cases D, E, and F),
the zonalization tends to be monotonic with decreasing Ro. Moreover, the enstrophy cascade diminishes but does not completely cease with decreasing Ro even for the initial vorticity spectrum comprising of the scales larger than the Rhines scale. This diminishing effect tends to be weaker from test case B to F. As we move from the test case B to F, the range of scales smaller than the Rhines scale present in the initial vorticity field increases. Therefore, the cascade becomes stronger from test case B to F, and therefore the effect of rotation becomes weaker from B to F. Moreover, the scales formed from the merger of smaller scales during the inverse cascade tend to be zonal. Hence, the stronger cascade from B to F
is a probable cause for the zonalization to become non-monotonic to monotonic.
\begin{figure*}
\centering{}\includegraphics[scale=0.125]{figures/omega_vs_Ro_and_l_range_scientific_notation}\caption{Vorticity distribution showing the effect of wavenumber rage comprising the (arbitrary) initial vorticity field.\label{fig:Effect-of-wavenumber-range}}
\end{figure*}
\subsubsection{Spectral distribution of vorticity power}
Figure \ref{fig:Vorticity-power-Effect-of-wavenumber-range} shows the distribution of vorticity power in the spectral $\left( m - l \right)$ space as a function of the wavenumber range constituting the initial vorticity field (test cases B - F, see table \ref{tab:Simulation-parameters}) and Ro. For cases B and C, at a later time ($t\approx18$), as Ro decreases the vorticity power tends to confine to smaller $m$, i.e., there is zonalization of the vortical structures. However, for Ro smaller than $5.20 \times 10^{-3}$, the vorticity power tends also to be equally significant for larger $m$, showing non-monotonic nature of the zonalization. However, the confinement of the vorticity power to smaller $m$ with decreasing Ro tends to be monotonic for the cases D, E, and F, i.e., the zonalization tends to be monotonic with decreasing Ro for these cases.
\begin{figure*}
\centering{}\includegraphics[scale=0.12]{figures/spectral_power_vs_Ro_and_l_range_scientific_notation}\caption{Spectral distribution of vorticity power showing the effect of wavenumber rage comprising the (arbitrary) initial vorticity field.\label{fig:Vorticity-power-Effect-of-wavenumber-range}}
\end{figure*}
\subsubsection{Probability density of orientation of vortical structures}
Figure \ref{fig:pdf-cos-alpha-Effect-of-wavenumber-range} shows the probability density of cosine of the angle between the major axis of the vortical structure and the azimuthal direction (that of the orientation of the vortical structures) as a function of the wavenumber range constituting the initial vorticity field and Ro. For the intermediate wavenumber range comprising the initial vorticity field (for cases B, and C), as the Ro decreases, the probability density diminishes for all of the cos$\alpha$ values except near one up to Ro of about $5.20 \times 10^{-3}$. This shows the zonalization of the vortical structures. With further reduction in Ro, the probability density for cos$\alpha$ values smaller than one tends back to augment, showing nonmonotonic nature of the zonalization with decreasing Ro. However, for the initial vorticity field comprising of intermediate to large wavenumbers (for cases D, E, and F), the diminishing of the probability density for all of the cos$\alpha$ values except near unity is almost monotonic with decreasing Ro (there is insignificant augmentation of probability density values at very low Ro), revealing monotonic zonalization.
\begin{figure*}
\centering{}\includegraphics[scale=0.25]{figures/pdf_cos_angle_Ro_ics}\caption{Probability density of cos$\alpha$ showing the effect of wavenumber rage comprising the (arbitrary) initial vorticity field.\label{fig:pdf-cos-alpha-Effect-of-wavenumber-range}}
\end{figure*}
\section{Conclusions}
The present study examines the effect of rotation on the vorticity dynamics on a unit sphere using discrete exterior calculus.
In addition to examining the effect of rotation, we also investigate different initial spectra constituting the initial vorticity field and the differences in the late time evolution of
the vorticity field.
The visualization of the evolving vorticity field, reveals diminishing of the forward enstrophy cascade and the non-monotonic nature of the zonalization of the vortical structures for the initial vorticity field comprising of intermediate-wavenumbers. On the other hand, the zonalization is monotonic for the initial vorticity field comprising of intermediate -to- large wavenumbers. We analyze the vorticity field on the sphere by
computing the vorticity power spectrum,
distribution of spectral power, vorticity probability density, probability
density of aspect ratio and that of the orientation of the vortical
structures. The analyses further confirms the phenomenon of zonalization although we note the tendency to zonalization reverses for high rotation rates where the initial vorticity is confined to a wavenumber range $l=4-10$. Moreover, we observe a forward cascade of enstrophy for the cases with initial vorticity field comprising of larger scales than the Rhines scale. Thus, while the forward
enstrophy cascade diminishes with decreasing Rossby number,
the cascade does not cease completely. The cases with initial vorticity field comprising of the scales much smaller (larger wavenumbers) than the Rhines scale have a higher potential for the cascade, and therefore, the diminishing effect of rotation is weaker for these cases. Moreover, the scales emerging from the merger of smaller scales during the inverse cascade tend to be zonal. Hence, the zonalization tend to be monotonic for these cases as the diminishing effect of rotation is weaker. Our investigation suggests that in addition to the dependence of the Rhines scale on $\beta$ and RMS velocity, the dependence on the spectrum constituting the initial vorticity field should be included in order for it to represent the scale arresting the cascade more effectively.
\begin{acknowledgments}
This research was supported by the KAUST Office of Sponsored Research under Award URF/1/3723-01-01.
\end{acknowledgments}
|
{
"timestamp": "2021-08-24T02:14:56",
"yymm": "2108",
"arxiv_id": "2108.09583",
"language": "en",
"url": "https://arxiv.org/abs/2108.09583"
}
|
\section{Introduction}
X-ray imagery is a widely used modality for non-destructive testing \cite{Tang2020TII}, especially for screening illegal and smuggled items at airports, cargoes, and malls. Manual baggage inspection is a tiring task and susceptible to errors caused due to exhausting work routines and less experienced personnel. Initial systems proposed to address these problems employed conventional machine learning \cite{bastan2013BMVC}. Driven by hand-engineered features, these methods are only applicable to limited data and confined environmental settings \cite{turcsany2013improving}. Recently, attention has turned to deep learning methods, which gave a neat boost in accuracy and generalization capacity towards screening prohibited baggage items \cite{Hu2020ACCV, akccay2016transfer}. However, deep learning methods are also prone to clutter, and occlusion \cite{akcay2018using}. This limitation emanates from the proposal generation strategies which have been designed for the color images \cite{gaus2019evaluation}. Unlike RGB scans, X-ray imagery lack texture and exhibit low-intensity variations between cluttered objects. This intrinsic difference makes the region-based or anchor-based proposal generation methods such as Mask R-CNN \cite{maskrcnn}, Faster R-CNN \cite{fasterrcnn}, RetinaNet \cite{retinanet}, and YOLO \cite{yolov3} less robust for detecting the cluttered contraband data \cite{akcay2018using}. Moreover, the problem is further accentuated by the class imbalance nature of the contraband items in the real-world \cite{gaus2019evaluation}.
Despite the considerate strategies proposed to alleviate the occlusion and the imbalance nature \cite{opixray, miao2019sixray}, recognizing threatening objects in highly cluttered and concealed scenarios is still an open problem \cite{ackay2020}.
\RV{
\subsection{Contributions}
\noindent In this paper, we propose a novel multi-scale contour instance segmentation framework for identifying suspicious items using X-ray scans. Unlike standard models that employ region-based or keypoint-based techniques to generate multiple boxes around objects \cite{akccay2016transfer, akcay2018using, gaus2019evaluating}, we propose to derive proposals according to the hierarchy of the regions defined by the contours.
The insight driving this approach is that contours are the most reliable cue in the X-ray scans due to the lack of surface texture. For example, the occluded items exhibit different transitional patterns based upon their orientation, contrast, and intensity. We try to amplify and exploit this information through the multi-scale scan decomposition, which boosts the proposed framework's capacity for detecting the underlying contraband data in the presence of clutter. Furthermore, we are also motivated by the fact that organic material's suspicious items show only their outlines in the X-ray scans \cite{hassan2019}.
To summarize, the main features of this paper are:
\begin{itemize}[leftmargin=*]
\item Detection of overlapping suspicious items by analyzing their predominant orientations across multiple scales within the candidate scan. Unlike \cite{hassan2019, Hassan2020ACCV, hassan2020Sensors}, we propose a novel tensor pooling strategy to decompose the scan across various scales and fuses them via a single multi-scale tensor. This scheme results in more salient contour maps (see Figure \ref{fig:fig1}), boosting our framework's capacity for handling dulled, concealed, and overlapping items.
\item A thorough validation on three publicly available large-scale baggage X-ray datasets, including the OPIXray \cite{opixray}, which is the only dataset allowing a quantitative measure of the level of occlusion.
\item Unlike state-of-the-art methods such as CST \cite{hassan2019}, TST \cite{Hassan2020ACCV}, and DTS \cite{hassan2020Sensors}, the performance of the proposed framework to detect occluded items has been quantitatively evaluated on \RV{OPIXray \cite{opixray} dataset}. Please see Table \ref{tab:tab5} for more details.
\end{itemize}
}
\section{Related Work} \label{sec:related}
\noindent
Many researchers have developed computer-aided screening systems to identify potential baggage threats \cite{mery2016}. While a majority of these frameworks are based on conventional machine learning \cite{bastan2013object}, the recent works also employ supervised \cite{akccay2016transfer}, and unsupervised \cite{akccay2019skip} deep learning, and these methods outperform conventional approaches both in terms of performance, and efficiency \cite{akcay2018using}. In this section, we discuss some of the major baggage threat detection works. We refer the readers to \cite{Mery2017TMSC,ackay2020} for an exhaustive survey.
\subsection{Traditional Methods}
\noindent The early baggage screening systems were driven via classification \cite{turcsany2013improving}, segmentation \cite{heitz2010} and detection \cite{bastan2015} approaches to identify potential threats and smuggled items. Here, the work of Bastan et al. \cite{bastan2013BMVC} is appreciable, which identifies the suspicious and illegal items within the multi-view X-ray imagery through fused SIFT and SPIN driven SVM model. Similarly, SURF \cite{heitz2010}, and FAST-SURF \cite{kundegorski2016} have also been used with the Bag of Words \cite{bastan2011} to identify threatening items from the security X-ray imagery. Moreover, approaches like adapted implicit shape model \cite{riffo2015automated} and adaptive sparse representation \cite{mery2016} were also commendable for screening suspicious objects from the X-ray scans.
\subsection{Deep Learning Frameworks}
\noindent The deep learning-based baggage screening frameworks have been broadly categorized into supervised and unsupervised \RV{learning schemes.}
\subsubsection{Supervised Methods}
The initial deep learning approaches involved scan-level classification to identify the suspicious baggage content \cite{akccay2016transfer}. However, with the recent advancements in object detection, researchers also employed sophisticated detectors like RetinaNet \cite{retinanet}, YOLO \cite{yolo, yolov2}, and Faster R-CNN \cite{fasterrcnn} to not only recognize the contraband items from the baggage X-ray scans but also to localize them via bounding boxes \cite{akcay2018using}. Moreover, researchers also proposed semantic segmentation \cite{an2019} and instance segmentation \cite{Hassan2020ACCV} models to recognize threatening and smuggled items from the grayscale and colored X-ray imagery. Apart from this, Xiao et al. \cite{_45} presented an efficient implementation of Faster R-CNN \cite{fasterrcnn} to detect suspicious data from the TeraHertz imagery. Dhiraj et al. \cite{_42} used Faster R-CNN \cite{fasterrcnn}, YOLOv2 \cite{yolov2}, and Tiny YOLO \cite{yolov2} to screen baggage threats contained within the scans of a publicly available GDXray dataset \cite{mery2015gdxray}. Gaus et al. \cite{gaus2019evaluation} utilized RetinaNet \cite{retinanet}, Faster R-CNN \cite{fasterrcnn}, Mask R-CNN \cite{maskrcnn} (driven through ResNets \cite{he2016deep}, VGG-16 \cite{vgg16}, and SqueezeNet \cite{i2016squeezenet}) to detect prohibited baggage items. In another approach \cite{gaus2019evaluating}, they analyzed the transferability of these models on a similarly styled X-ray imagery contained within their local dataset as well as the SIXray10 subset of the publicly available SIXray dataset \cite{miao2019sixray}. Similarly, Ak\c{c}ay et al. \cite{akcay2018using} compared Faster R-CNN \cite{fasterrcnn}, YOLOv2 \cite{yolov2}, R-FCN \cite{rfcn}, and sliding-window CNN with the AlexNet \cite{alexnet} driven SVM model to recognize occluded contraband items from the X-ray imagery. Miao et al. \cite{miao2019sixray} explored the imbalanced nature of the contraband items in the real-world by developing a class-balanced hierarchical refinement (CHR) framework. Furthermore, they extensively tested their framework (backboned through different classification models) on their publicly released SIXray \cite{miao2019sixray} dataset. Wei et al. \cite{opixray} presented a plug-and-play module dubbed De-occlusion Attention Module (DOAM) that can be coupled with any object detector to enhance its capacity towards screening occluded contraband items. DOAM was validated on the publicly available OPIXray \cite{opixray} dataset, which is the first of its kind in providing quantitative assessments of baggage screening frameworks under low, partial, and full occlusion \cite{opixray}. \RV{
Apart from this, Hassan et al. \cite{hassan2019} also addressed the imbalanced nature of the contraband data by developing the cascaded structure tensors (CST) based baggage threat detector. CST \cite{hassan2019} generates a balanced set of contour-based proposals, which are then utilized in training the backbone model to screen the normal and abnormal baggage items within the candidate scan \cite{hassan2019}. Similarly, to overcome the need to train the threat detection systems on large-scale and well-annotated data, Hassan et al. \cite{hassan2020Sensors} introduced meta-transfer learning-based dual tensor-shot (DTS) detector. DTS \cite{hassan2020Sensors} analyzes the scan's saliency to produce low and high-density contour maps from which the suspicious contraband items are identified effectively with few-shot training \cite{hassan2020Sensors}. In another approach, Hassan et al. \cite{Hassan2020ACCV} developed an instance segmentation-based threat detection framework that filters the contours of the suspicious items from the regular content via trainable structure tensors (TST) \cite{Hassan2020ACCV} to identify them accurately within the security X-ray imagery.
}
\subsubsection{Unsupervised Methods}
\noindent While most baggage screening frameworks involved supervised learning, researchers have also explored adversarial learning to screen contraband data as anomalies. Ak\c{c}ay et al. \cite{akcay2018ganomaly}, among others, laid the foundation of unsupervised baggage threat detection by proposing GANomaly \cite{akcay2018ganomaly}, an encoder-decoder-encoder network trained in an adversarial manner to recognize prohibited items within baggage X-ray scans. In another work, they proposed Skip-GANomaly \cite{akccay2019skip} which employs skip-connections in an encoder-decoder topology that not only gives better latent representations for detecting baggage threats but also reduces the overall computational complexity of GANomaly \cite{akcay2018ganomaly}.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{Fig1.png}
\caption{(A) An exemplar X-ray scan from the GDXray dataset \cite{mery2015gdxray}, (B) contour map obtained through the modified structure tensors in \cite{hassan2019} and \cite{Hassan2020ACCV}, (C) contour map obtained through proposed tensor pooling strategy.}
\label{fig:fig1}
\end{figure}
\noindent The rest of the paper is organized as follows: Section \ref{sec:proposed} presents the proposed framework. Section \ref{sec:exp} describes the experimental setup. Section \ref{sec:results} discusses the results obtained with three public baggage X-ray datasets. Section \ref{sec:discussion} concludes the paper and enlists future directions.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{Picture212.png}
\caption{Block diagram of the proposed framework. The input scan is passed to the tensor pooling module to extract the tensor representations encoding the baggage items' contours at different orientations. These representations are fused into a single multi-scale tensor and passed afterward to an asymmetric encoder-decoder backbone that segments and recognizes the contraband item's contours while suppressing the rest of the baggage content. For each detected contour, the corresponding bounding box and mask is generated to localize the detected contraband items. The abbreviations are CV: Convolution, BN: Batch Normalization, SPB: Shape Preserving Block, IB: Identity Block, MP: Max Pooling, AP: Average Pooling, ZP: Zero Padding, SM: Softmax.}
\label{fig:fig2}
\end{figure*}
\section{Proposed Approach } \label{sec:proposed}
\noindent The block diagram of the proposed framework is depicted in Figure \ref{fig:fig2}. The input scan is fed to the tensor pooling module (block A) to generate a multi-scale tensor representation, revealing the baggage content's transitional patterns at multiple predominant orientations and across various scales. Afterward, the multi-scale tensor is passed to the encoder-decoder backbone (block B), implementing the newly proposed contour maps-based instance segmentation. This block extracts the contours of the prohibited data while eliminating the irrelevant scan content. In the third stage (block C), each extracted contour, reflecting the contraband item instance, is utilized in generating the respective mask and the bounding box for localization. In the subsequent sections, we present a detailed description of each module within the proposed framework.
\subsection{Tensor Pooling Module}
\noindent The tensor pooling module decomposes the input scan \RV{into $n$ levels of a pyramid. From each level of the pyramid,} the baggage content's transitional patterns are generated by analyzing their distribution of orientations within the associated image gradients. In the proposed tensor pooling scheme, we highlight the transitional patterns in $N$ image gradients (corresponding to $N$ directions) by computing the following $N \times N$ block-structured symmetric matrix \cite{hassan2019, Hassan2020ACCV}:
\begin{equation}
\begin{bmatrix}
\phi * (\nabla^0 . \nabla^0) & \phi * (\nabla^1 . \nabla^0) & \cdots & \phi * (\nabla^{N-1} . \nabla^0) \\
\phi * (\nabla^0 . \nabla^1) & \phi * (\nabla^1 . \nabla^1) & \cdots & \phi * (\nabla^{N-1} . \nabla^1) \\
\vdots & \vdots & \ddots & \vdots \\
\phi * (\nabla^0 . \nabla^{N-1}) & \phi * (\nabla^1 . \nabla^{N-1}) & \cdots & \phi * (\nabla^{N-1} . \nabla^{N-1}) \\
\end{bmatrix}
\label{eq:eq3},
\end{equation}
\noindent
Each tensor ($\phi * (\nabla^k . \nabla^m)$) in the above block-matrix is an outer product of two image gradients and a smoothing filter $\phi$. Moreover, the orientation ($\theta$), of the image gradient $\nabla^j$, is computed through: $\theta = \frac{2 \pi j}{N}$, where $j$ ranges from $0$ to $N-1$. Since the block-structured matrix in Eq. \ref{eq:eq3} is symmetric, we obtain $\frac{N(N-1)}{2}$ unique tensors. From this group, we derive the \RV{coherent tensor, reflecting the baggage items' predominant orientations. The coherent tensor is a single tensor representation generated by adding the most useful tensors out of the $\frac{N(N+1)}{2}$ unique tensor set. Here, it should be noted that these useful tensors are selected by ranking all the $\frac{N(N+1)}{2}$ unique tensors according to their norm.}
\noindent Moreover, the coherent tensor also reveals the variations in the intensity of the cluttered baggage items, aiding in generating individual contours for each item. However, this scheme analyzes only the intensity variations of the baggage items at a single scale, limiting the extraction of the objects having lower transitions with the background \cite{hassan2019, Hassan2020ACCV}. To address this limitation, we propose a multi-scale tensor fusing the X-ray scan transitions from coarsest to finest levels so that each item, even having a low-intensity difference with the background, can be adequately highlighted.
For example, see the boundaries of: the \textit{razor} in a multi-scale tensor representation in Figure \ref{fig:fig1} (C), the \textit{straight knife} in Figure
\ref{fig:multiscale} (G), the two \textit{knives} and a \textit{gun} in Figure \ref{fig:multiscale} (H), and the two \textit{guns} and a \textit{knife} in Figure \ref{fig:multiscale} (I).
\begin{algorithm}[t]
\SetAlgoLined
\DontPrintSemicolon
\textbf{Input: } X-ray scan ($I$), Scaling Factor ($n$), Number of Orientations ($N$)
\textbf{Output: } Multi-scale Tensor ($M_t$)
[$r$, $c$] = size($I$)
Initialize $M_t$ (of size $r \times c$) with zeros
Set $\eta=2$ // pyramid pooling factor
\For{$i=0$ to $n-1$}
{
\eIf{$i$ is 0}
{
$\Im$ = ComputeTensors($I$, $N$) // $\Im$: Tensors
$\Im_c$ = GetCoherentTensor($\Im$)
$M_t$ = $M_t$ + $\Im_c$
}
{
[$s$, $t$] = size($I$)
\If{(min($s$, $t$) \% $\eta) \neq 0$ or min($s$, $t$) $< \eta$}
{
break
}
$I$ = Pool($I$, $\eta$)
$\Im_c$ = Pool($\Im_c$, $\eta$)
$I$ = $I \times \Im_c$
$\Im$ = ComputeTensors($I$, $N$)
$\Im_c$ = GetCoherentTensor($\Im$)
$M_t$ = $M_t$ + Unpool($\Im_c$, $\eta^i$)
}
}
\caption{Tensor Pooling Module}
\label{algo:algo1}
\end{algorithm}
\begin{figure}[htb]
\centering
\includegraphics[width=1\linewidth]{Picture3.jpg}
\caption{Difference between conventional structure tensors (used in \cite{hassan2019, Hassan2020ACCV}), and proposed multi-scale tensor approach. First row shows the original scans from OPIXray \cite{opixray}, GDXray \cite{mery2015gdxray}, and SIXray \cite{miao2019sixray} dataset. The second row shows the output for the conventional structure tensors \cite{hassan2019, Hassan2020ACCV}. The third row shows the output for the proposed tensor pooling module.}
\label{fig:multiscale}
\end{figure}
\noindent As mentioned earlier, the multi-scale tensors are computed through pyramid pooling (up to $n^{th}$ level). At any $l^{th}$ level, (such that $2 \leq l \leq n$), we multiply, pixel-wise, the decomposed image with the transitions obtained at the previous ($l-1$) levels. In so doing, we ensure that the edges of the contraband items (procured earlier) are retained across each scale. The full procedure of the proposed tensor pooling module is depicted in Algorithm \ref{algo:algo1} and also shown in Figure \ref{fig:fig2}.
\begin{figure}[htb]
\centering
\includegraphics[width=1\linewidth]{Picture7.jpg}
\caption{Contour instance segmentation from multi-scale tensors. The first column shows the original scans, the second column shows the multi-scale tensor representations, the third column shows the ground truths, and the fourth column shows the extracted contours of the contraband items.}
\label{fig:contours}
\end{figure}
\noindent The multi-scale tensor is then passed to the proposed encoder-decoder model to extract the contours of the individual suspicious items. A detailed discussion about contour instance segmentation is presented in the subsequent section.
\subsection {Contour Instance Segmentation}
\noindent The contour instance segmentation is performed through the proposed asymmetric encoder-decoder network, which assigns the pixels in the multi-scale tensors to one of the following categories $\mathcal{C}_{k=1:\mathcal{M}+1}$ where $\mathcal{M}$ denotes the number of prohibited items' instances
to which we add the class \textit{background} which include background and irrelevant pixels (i.e., pixels belonging to a non-suspicious baggage content).
\noindent Furthermore, to differentiate between the contours of the normal and suspicious items, the custom shape-preserving (SPB) and identity blocks (IB) have been added within the encoder topology. The SPB, as depicted in Figures \ref{fig:fig2} and \ref{fig:fig3} (A), integrates the multi-scale tensor map (after scaling) in the feature map extraction to enforce further the attention on prohibited items' outlines. The IB (Figure \ref{fig:fig3}-A), inspired by ResNet architecture \cite{he2016deep}, acts as a residual block to emphasize the feature maps of the previous layer.
\noindent Apart from this, the whole network encompasses one input, one zero-padding, 22 convolution, 20 batch normalization, 12 activation, four pooling, two multiply, six addition, three lambda (that implements the custom functions), and one reshape layer. Moreover, we use skip-connections (via addition) within the encoder-decoder to refine the extracted items' boundaries. The number of parameters within the network is 1,308,160, from which around 6,912 parameters are non-trainable. The detailed summary of the proposed model (including the architectural details of the SPB and IB blocks) is available in the source code repository\footnote{\label{note1}The source code of the proposed framework along with its complete documentation is available at \url{https://github.com/taimurhassan/tensorpooling}.}.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{Picture33.png}
\caption{(A) Shape Preserving Block (SPB), (B) Identity Block (IB).}
\label{fig:fig3}
\end{figure}
\subsection{Bounding Box and Mask Generation}
\noindent
After segmenting the contours, we perform morphological post-processing to remove tiny and isolated fragments. The obtained outlines contain both open and closed contours of the underlying suspicious items. The closed contours can directly lead towards generating the corresponding item's mask. For open contours, we join their endpoints and then derive their masks through morphological reconstruction. Afterward, we generate the items' bounding boxes from the masks as shown in Figure \ref{fig:fig2} (C).
\section{Experimental Setup} \label{sec:exp}
\noindent This section presents the details about the experimental protocols, datasets, and evaluation metrics which were in order to assess the proposed system's performance and compare it with state-of-the-art methods.
\subsection{Datasets}
We validated the proposed framework on three different publicly available baggage X-ray datasets, namely, GDXray \cite{mery2015gdxray}, SIXray \cite{miao2019sixray}, and OPIXray \cite{opixray}. The detailed description of these datasets \RV{are presented below.}
\subsubsection{GDXray}
GDXray \cite{mery2015gdxray} \RV{was} first introduced in 2015 and it contains 19,407 high-resolution grayscale X-ray scans. The dataset is primarily designed for the non-destructive testing purposes and the scans within GDXray \cite{mery2015gdxray} are arranged into five categories, i.e., \textit{welds}, \textit{baggage}, \textit{casting}, \textit{settings} and \textit{nature}. But \textit{baggage} is the only relevant group for this study and it contains 8,150 grayscale X-ray scans. Moreover, the dataset also presents the detailed annotations for the prohibited items such as \textit{shuriken}, \textit{knives}, \textit{guns}, and \textit{razors}. As per the dataset standard, 400 scans from GDXray \cite{mery2015gdxray} were used for training purposes, while the remaining scans were used for testing purposes.
\subsubsection{SIXray}
SIXray \cite{miao2019sixray} is a recently introduced large-scale security inspection X-ray dataset. It contains a total of 1,059,231 colored X-ray scans from which 8,929 scans are positive (containing prohibited items such as \textit{knives}, \textit{wrenches}, \textit{guns}, \textit{pliers}, \textit{hammer} and \textit{scissors} along with their ground truths), and 1,050,302 are negative (containing only the normal items). To validate the performance against class imbalance, the authors of the dataset presented three subset schemes of the dataset, namely, SIXray10, SIXray100, and SIXray1000 \cite{miao2019sixray}. Moreover, SIXray \cite{miao2019sixray} is also the largest and most challenging dataset (to date) designed to assess threat detection frameworks' performance towards screening extremely cluttered and highly imbalanced contraband data \cite{miao2019sixray, hassan2020Sensors}. As per the SIXray \cite{miao2019sixray} dataset standard, we used 80\% scans for the training and the rest of 20\% for testing.
\subsubsection{OPIXray}
OPIXray \cite{opixray} is the most recent baggage X-ray dataset (released publicly for the research community in 2020). It contains 8,885 colored X-ray scans. As per the dataset standard, out of these 8,885 scans, 7,109 are to be utilized for the training purposes, while the remaining 1,776 are to be used for testing purposes, to detect \textit{scissor}, \textit{straight knife}, \textit{multi-tool knife}, \textit{folding knife}, and \textit{utility knife}. Moreover, the dataset authors also quantified occlusion within the test scans into three levels, i.e., OP1, OP2, and OP3. OP1 indicates that the contraband items within the candidate scan contain no or slight occlusion, OP2 depicts a partial occlusion, while OP3 represents severe or full occlusion cases.
\RV{
\noindent We also want to highlight here that the resolution of the scans within each dataset varies significantly (except for OPIXray \cite{opixray}). For example, on GDXray \cite{mery2015gdxray}, the scan resolution varies as $2688 \times 2208$, $900 \times 1430$, $850 \times 850$ and $601 \times 1241$, etc. Similarly, on SIXray \cite{miao2019sixray}, the scan resolution varies as $681 \times 549 \times 3$, $801 \times 482 \times 3$, $649 \times 571 \times 3$, $1024 \times 640 \times 3$, and $675 \times 382 \times 3$, etc. But on OPIXray \cite{opixray}, the resolution of all the scans is 1225x954x3.
In order to process all the scans with the proposed framework, we have re-sized them to the common resolution of $576 \times 768 \times 3$, which is extensively used in the recently published frameworks \cite{hassan2019, Hassan2020ACCV, hassan2020Sensors}.
}
\subsection{Training and Implementation Details}
\noindent
The proposed framework \RV{was} developed using Python 3.7.4 with TensorFlow 2.2.0 and Keras APIs on a machine havingIntel Core i9-10940X@3.30 GHz CPU, 128 GB RAM and an NVIDIA Quadro RTX 6000 with cuDNN v7.5, and a CUDA Toolkit 10.1.243. Some utility functions are also implemented using MATLAB R2021a. Apart from this, the training on each dataset was conducted for a maximum of 50 epochs using ADADELTA \cite{Zeiler2012ADADELTA} as an optimizer (with the default learning and decay rate configurations) and a batch size of 4. Moreover, 10\% of the training samples from each dataset were used for the validation (after each epoch).
For the loss function, we used the focal loss \cite{retinanet} expressed below:
\begin{equation}
l_f = -\frac{1}{b_s}\sum\limits_{i=0}^{b_s-1}\sum\limits_{j=0}^{c-1} \alpha(1-p(l_{i,j}))^\gamma t_{i,j}\log(p(l_{i,j}))
\label{eq:eq4}
\end{equation}
where $c$ represents the total number of classes, and $b_s$ denotes the batch size. $p(l_{i,j})$ denotes the predicted probability of the logit $l_{i,j}$ generated from $i^{th}$ training sample for the $j^{th}$ class, $t_{i,j}$ tells if the $i^{th}$ training sample actually belongs to the $j^{th}$ class or not, the term $\alpha(1-p(l_{i,j}))^\gamma$ represents the scaling factor that gives more weight to the imbalanced classes (in other words, it penalizes the network to give emphasize to the classes for which the network obtain low prediction scores). Through rigorous experiments, we empirically selected the optimal value of $\alpha$ and $\gamma$ as 0.25 and 2, respectively, as they result in faster learning for each dataset while simultaneously showing good resistance to the imbalanced data. \RV{Apart from this, architecturally, the kernel sizes within the proposed encoder-decoder backbone vary as 3x3 and 7x7, whereas the number of kernels varies as 64, 128, 256, 512, 1024, and 2048. Moreover, the pooling size within the network remained 2x2 across various network depths to perform the feature decomposition (at each depth) by the factor of 2. For more architectural and implementation details of the proposed framework, we refer the reader to the source code, which we have released publicly for the research community on GitHub\textsuperscript{\ref{note1}}.}
\subsection{Evaluation Metrics}
In order to assess the proposed approach and compare it with the existing works, we used the following evaluation metrics:
\subsubsection{Intersection-over-Union}
Intersection-over-Union (IoU) tells how accurately the suspicious items have been extracted, and it is measured by checking the pixel-level overlap between the predictions and the ground truths. Mathematically, IoU is defined as:
\begin{equation}
IoU = \frac{T_p}{T_p+F_p+F_n}
\label{eq:eq5},
\end{equation}
where $T_p$ are true positives (indicating that the pixels of the contraband items are correctly predicted w.r.t the ground truth), $F_p$ represents false positives (indicating that the background pixels are incorrectly classified as positives), and $F_n$ represents false negatives (meaning that the pixels of the contraband items are misclassified as background). Furthermore, we also calculated the mean IoU ($\mu$IoU) by taking an average of the IoU score for each contraband item class.
\subsubsection{Dice Coefficient}
Apart from IoU scores, we also computed the dice coefficient (DC) scores to assess the proposed system's performance for extracting the contraband items. DC is calculated through:
\begin{equation}
DC = \frac{2T_p}{2T_p+F_p+F_n}
\label{eq:eq6},
\end{equation}
\noindent Compared to IoU, DC gives more weightage to the true positives (as evident from Eq. \ref{eq:eq6}). Moreover, the mean DC ($\mu$DC) is calculated by averaging DC scores for each category.
\subsubsection{Mean Average Precision}
The mean average precision (mAP) (in the proposed study) is computed by taking the mean of average precision (AP) score calculated for each contraband item class for the IoU threshold $\geq$ 0.5. Mathematically, mAP is expressed below:
\begin{equation}
mAP = \sum_{i=0}^{n_c-1} AP(i)
\label{eq:eq7},
\end{equation}
\noindent where $n_c$ denotes the number of contraband items in each dataset. Here, we want to highlight that to achieve fair comparison with the state-of-the-art, we have used the original bounding box ground truths of each dataset for measuring the proposed framework's performance towards extracting the suspicious and illegal items.
\section{Results} \label{sec:results}
\noindent In this section, we present the detailed results obtained with GDXray \cite{mery2015gdxray}, SIXray \cite{miao2019sixray}, and OPIXray \cite{opixray} datasets.
Before going into the experimental results, we present detailed ablation studies to determine the proposed framework's hyper-parameters. We also report a detailed comparison of the proposed encoder-decoder network with the popular segmentation models.
\subsection{Ablation Studies}
\noindent The ablation studies in this paper aim to determine the optimal values for 1) the number of orientations and scaling levels within the tensor pooling module and 2) the choice of the backbone model for performing the contour instance segmentation.
\begin{figure}[t]
\centering
\textbf{(A)} \includegraphics[width=0.4\linewidth,height=2.8cm]{gdX.png}
\textbf{(B)} \includegraphics[width=0.4\linewidth,height=2.8cm]{SIX.png}
\textbf{(C)} \includegraphics[width=0.4\linewidth,height=2.8cm]{opi1_.png}
\textbf{(D)} \includegraphics[width=0.4\linewidth,height=2.8cm]{gd2_.png}
\textbf{(E)} \includegraphics[width=0.4\linewidth,height=2.8cm]{six2_.png}
\textbf{(F)} \includegraphics[width=0.4\linewidth,height=2.8cm]{opi2_.png}
\caption{Detection performance of the proposed system in terms of mAP (A, B, C), and computational time in terms of seconds (D, E, F) obtained for GDXray \cite{mery2015gdxray}, SIXray \cite{miao2019sixray}, and OPIXray \cite{opixray} datasets, respectively.}
\label{fig:fig33}
\end{figure}
\subsubsection{Number of Orientations and the Scaling Levels}
\noindent
The tensor pooling module highlights the baggage content transitions in the image gradients oriented in $N$ directions and up to $n$ scaling levels.
Increasing these parameters helps generate the best contour representation leading towards a more robust detection, but also incurs additional computational cost. As depicted in Figure \ref{fig:fig33} (A), we can see that for GDXray dataset \cite{mery2015gdxray} with $N=2$, $n=2$, we obtain an mAP score of 0.82. With the combination $N=5$, $n=5$, we get \RV{16.54\%} improvements in the detection performance but at the expense of a \RV{97.71\%} increase in computational time (see Figure \ref{fig:fig33}-D). Similarly, on the SIXray dataset \cite{miao2019sixray}, we obtain \RV{18.36\%} improvements in the detection performance (by increasing $N$ and $n$) at the expense of \RV{95.88\%} in the computational time (see Figure \ref{fig:fig33} (B, E)). The same behavior is also noticed for OPIXray dataset \cite{opixray} in Figure \ref{fig:fig33} (C, F). Considering all the combinations depicted in Figure \ref{fig:fig33}, we found that $N=4$ and $n=3$ provide the best trade-off between the detection and run-time performance across all three datasets.
\subsubsection{Choice of a Backbone Model}
\noindent
The proposed backbone model has been specifically designed to segment the suspicious items' contours while discarding the normal baggage content. In this series of experiments, we compared the proposed asymmetric encoder-decoder model's performance with popular encoder-decoder, scene parsing, and fully convolutional networks. In terms of $\mu$DC and $\mu$IoU, we report the performance results in Table \ref{tab:tab2}.
We can observe that the proposed framework achieves the best extraction performance on OPIXray \cite{opixray} and SIXray \cite{miao2019sixray} dataset, leading the second-best UNet \cite{ronneberger2015unet} by \RV{2.34\%} and \RV{3.72\%}. On the GDXray \cite{mery2015gdxray}, however, it lags from the FCN-8 \cite{fcn8} and PSPNet \cite{zhao2017pyramid} by \RV{6.54\%} and \RV{5.91\%}, respectively. But as our model outperforms all the other architectures on the large-scale SIXray \cite{miao2019sixray} and OPIXray \cite{opixray} datasets, we chose it as a backbone for the rest of the experimentation.
\begin{table}[t]
\centering
\caption{Performance comparison of the proposed backbone network with PSPNet \cite{zhao2017pyramid}, UNet \cite{ronneberger2015unet} and FCN-8 \cite{fcn8} for recognizing the boundaries of the contraband items. The best and second-best performances are in bold and underline, respectively. Moreover, the abbreviations are: Met: Metric, Data: Dataset, GDX: GDXray \cite{mery2015gdxray}, SIX: SIXray \cite{miao2019sixray}, OPI: OPIXray \cite{opixray}.}
\begin{tabular}{cccccc}
\toprule
Met & Data & Proposed & PSPNet & UNet & FCN-8 \\\hline
$\mu$IoU & GDX & 0.4994 & \underline{0.5585} & 0.4921 & \textbf{0.5648} \\
& SIX & \textbf{0.7072} & 0.5659 & \underline{0.6700} & 0.6613 \\
& OPI & \textbf{0.7393} & 0.5645 & \underline{0.7159} & 0.5543 \\\hline
$\mu$DC & GDX & 0.6661 & \underline{0.7167} & 0.6596 & \textbf{0.7219} \\
& SIX & \textbf{0.8285} & 0.7227 & \underline{0.8024} & 0.7961 \\
& OPI & \textbf{0.8501} & 0.7217 & \underline{0.8344} & 0.7132 \\
\bottomrule
\end{tabular}
\label{tab:tab2}
\end{table}
\begin{table}[b]
\centering
\caption{Performance comparison between state-of-the-art baggage threat detection frameworks on GDXray (GDX), SIXray (SIX), and OPIXray (OPI) dataset in terms of mAP scores. '-' indicates that the respective score is not computed. Moreover, the abbreviations are: Data: Dataset, GDX: GDXray \cite{mery2015gdxray}, SIX: SIXray \cite{miao2019sixray}, OPI: OPIXray \cite{opixray}, PF: Proposed Framework, and FD: FCOS \cite{fcos} + DOAM \cite{opixray}.}
\begin{tabular}{cccccc}
\toprule
Data & Items & PF & CST & TST & FD \\\hline
GDX & Gun & \textbf{0.9872} & 0.9101 & \underline{0.9761} & - \\
& Razor & \textbf{0.9691} & 0.8826 & \underline{0.9453} & - \\
& Shuriken & 0.9735 & \textbf{0.9917} & \underline{0.9847} & - \\
& Knife & \underline{0.9820} & \textbf{0.9945} & 0.9632 & - \\
& \color{blue}{mAP} & \textbf{0.9779} & 0.9281 & \underline{0.9672} & - \\\hline
SIX & Gun & \underline{0.9863} & \textbf{0.9911} & 0.9734 & - \\
& Knife & \textbf{0.9811} & 0.9347 & \underline{0.9681} & - \\
& Wrench & \underline{0.9882} & \textbf{0.9915} & 0.9421 & - \\
& Scissor & 0.9341 & \textbf{0.9938} & \underline{0.9348} & - \\
& Pliers & \underline{0.9619} & 0.9267 & \textbf{0.9573} & - \\
& Hammer & 0.9172 & \underline{0.9189} & \textbf{0.9342} & - \\
& \color{blue}{mAP} & \textbf{0.9614} & \underline{0.9595} & 0.9516 & - \\\hline
OPI & Folding & \underline{0.8528} & - & 0.8024 & \textbf{0.8671} \\
& Straight & \textbf{0.7649} & - & 0.5613 & \underline{0.6858} \\
& Scissor & 0.8803 & - & \underline{0.8934} & \textbf{0.9023} \\
& Multi & \textbf{0.8941} & - & 0.7802 & \underline{0.8767} \\
& Utility & \textbf{0.8062} & - & 0.7289 & \underline{0.7884} \\
& \color{blue}{mAP} & \textbf{0.8396} & - & 0.7532 & \underline{0.8241} \\
\bottomrule
\end{tabular}
\label{tab:tab3}
\end{table}
\subsection{Evaluation on GDXray Dataset}
\noindent The performance of the proposed \RV{framework and of the state-of-the-art methods on the GDXray \cite{mery2015gdxray} dataset are reported} in Table \ref{tab:tab3}. We can observe here that the proposed framework outperforms the CST \cite{hassan2019} and the TST framework \cite{Hassan2020ACCV} by \RV{4.98\%} and \RV{1.07\%}, respectively.
Furthermore, we wanted to highlight the fact that CST \cite{hassan2019} is only an object detection scheme, i.e., it can only localize the detected items but cannot generate their masks. Masks are very important for the human observers in cross-verifying the baggage screening results (and identifying the false positives), especially from the cluttered and challenging grayscale scans.
In Figure \ref{fig:fig4}, we report some of the cluttered and challenging cases showcasing the effectiveness of the proposed framework
in extracting the overlapping contraband items. For example, see the extraction of merged \textit{knife} instances in (H), and the cluttered \textit{shuriken} in (J, L). We can also appreciate how accurately the \textit{razors} have been extracted in (J, L). Extracting such low contrast objects in the competitive CST framework requires suppressing first all the sharp transitions in an iterative fashion \cite{hassan2019}.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{Picture4.jpg}
\caption{Qualitative evaluations of the proposed framework on GDXray \cite{mery2015gdxray} dataset. Please zoom-in for best visualization.}
\label{fig:fig4}
\end{figure}
\subsection{Evaluations on SIXray Dataset}
\noindent
The proposed framework has been evaluated on the whole SIXray dataset \cite{miao2019sixray} (containing 1,050,302 negative scans and 8,929 positive scans) and also on each of its subsets \cite{miao2019sixray}. In Table \ref{tab:tab3}, we can observe that the proposed framework achieves an overall performance gain of \RV{0.190\%} and \RV{0.980\%} over CST \cite{hassan2019} and TST \cite{Hassan2020ACCV} framework, respectively.
In Table \ref{tab:tab4}, we report the results obtained with each subset of the SIXray dataset \cite{miao2019sixray}, reflecting different imbalanced normal and prohibited item categories. The results further confirm the superiority of the proposed framework against other state-of-the-art solutions, especially w.r.t the CHR \cite{miao2019sixray}, and \cite{gaus2019evaluating}. In addition to this, in an extremely challenging SIXray1000 subset, we notice that the proposed framework leads the second-best TST framework \cite{Hassan2020ACCV} by \RV{3.22\%}, and CHR \cite{miao2019sixray} by \RV{44.36\%}.
\noindent Apart from this, Figure \ref{fig:fig5} depicts the qualitative evaluations of the proposed framework on the SIXray \cite{miao2019sixray} dataset. In this figure, the first row shows examples containing one instance of the suspicious item, whereas the second and third rows show scans containing two or more instances of the suspicious items.
Here, we can appreciate how accurately the proposed scheme has picked the cluttered \textit{knife} in (B). Moreover, we can also observe the extracted \textit{chopper} (\textit{knife}) in (D) despite having similar contrast with the background. More examples such as (F, H, and J) demonstrate the proposed framework's capacity in picking the cluttered items from the SIXray dataset \cite{miao2019sixray}.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{Picture5.jpg}
\caption{Qualitative evaluations of the proposed framework on SIXray \cite{miao2019sixray} dataset. Please zoom-in for a best visualization.}
\label{fig:fig5}
\end{figure}
\begin{table}[t]
\centering
\caption{Performance comparison of proposed framework with state-of-the-art solutions on SIXray subsets. For fair comparison, all models are evaluated using ResNet-50 \cite{he2016deep} as a backbone. Moreover, the abbreviations are: SIX-10: SIXray10 \cite{miao2019sixray}, SIX-100: SIXray100 \cite{miao2019sixray}, SIX-1k: SIXray1000 \cite{miao2019sixray}, and PF: Proposed Framework}
\begin{tabular}{cccccc}
\toprule
Subset & PF & DTS & CHR & \cite{gaus2019evaluating} & TST\\\hline
SIX-10 & \textbf{0.9793} & 0.8053 & 0.7794 & 0.8600 & \underline{0.9601}\\
SIX-100 & \textbf{0.8951} & 0.6791 & 0.5787 & - & \underline{0.8749}\\
SIX-1k & \textbf{0.8136} & 0.4527 & 0.3700 & - & \underline{0.7814}\\
\bottomrule
\end{tabular}
\label{tab:tab4}
\end{table}
\subsection{Evaluations on OPIXray Dataset}
\noindent
The performance evaluation of the proposed framework on OPIXray dataset \cite{opixray} is reported in Table \ref{tab:tab3}. We can observe here that the proposed system achieves an overall mAP score of 0.8396, outperforming the second-best DOAM framework \cite{opixray} (driven via FCOS \cite{fcos}) by \RV{1.55\%}. Here, although the performance of both frameworks is identical, we still achieve a significant lead of \RV{7.91\%} over the DOAM \cite{opixray} for extracting the \textit{straight knives}.
\noindent Concerning the level of occlusion (as aforementioned, OPIXray \cite{opixray} splits the test data into three subsets, OP1, OP2, OP3, according to the level of occlusion), we can see in Table \ref{tab:tab5} that the proposed framework achieves the best performance at each occlusion level as compared to the second-best DOAM \cite{opixray} framework driven by the single-shot detector (SSD) \cite{Liu2016SSD}.
\noindent Figure \ref{fig:fig6} reports some qualitative evaluation, where we can appreciate the recognition of the cluttered \textit{scissor} (e.g. see B and F), and overlapping \textit{straight knife} (in H). We can also notice the detection of the partially occluded \textit{folding and straight knife} in (D) and (J).
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{Picture6.jpg}
\caption{Qualitative evaluations of the proposed framework on OPIXray \cite{opixray} dataset. Please zoom-in for a best visualization.}
\label{fig:fig6}
\end{figure}
\subsection{Failure Cases}
\noindent In Figure \ref{fig:fig8}, we report examples of failure cases encountered during the testing. In cases (B, H, N, and P), we can see that the proposed framework could not pick-up the whole regions of the contraband items, even though the items were detected correctly. However, such cases are observed in highly occluded scans such as (A and G), where it is difficult, even for a human observer, to distinguish the items' regions properly.
The second type of failure corresponds to the pixels misclassification as shown in (D) where some of the \textit{gun}'s pixels have been misclassified as \textit{knife}. We can address these scenarios through post-processing steps like blob removal and region filling.
The third failure case relates to the proposed framework's inability to generate a single bounding box for the same item. Such a case is depicted in (F), where two bounding boxes were generated for the single orange \textit{knife} item. One possible remedy here is to generate the bounding boxes based upon the minimum and maximum mask value in both image dimensions for each label. Another type of failure is shown in (J) and (L). Here, the scans contain only normal baggage content, but some pixels occupying tiny regions have been misclassified as false positive (i.e., \textit{knife}). We can also address this kind of failure through blob removal scheme.
\noindent Examining the failure cases' statistical distributions, we found a majority of 86.09\% cases belonging to the curable categories (i.e., second, third, and fourth), meaning that the proposed framework's performance can be further improved using the post-processing techniques mentioned above.
\begin{table}[t]
\centering
\caption{Performance comparison of proposed framework with DOAM \cite{opixray} (backboned through SSD \cite{Liu2016SSD}) on different occlusion levels of OPIXray \cite{opixray} dataset.}
\begin{tabular}{cccc}
\toprule
Method & OP1 & OP2 & OP3 \\ \hline
Proposed & \textbf{0.7946} & \textbf{0.7382} & \textbf{0.7291} \\
DOAM + SSD \cite{opixray} & \underline{0.7787} & \underline{0.7245} & \underline{0.7078} \\
SSD \cite{Liu2016SSD} & 0.7545 & 0.6954 & 0.6630\\
\bottomrule
\end{tabular}
\label{tab:tab5}
\end{table}
\section{Conclusion} \label{sec:discussion}
\noindent
In this work, we proposed a novel contour-driven approach for detecting cluttered and occluded contraband items (and their instances) within the baggage X-ray scans, hypothesizing that contours are the most robust cues given the lack of texture in the X-ray imagery. We concretized this original approach through a tensor pooling module, producing multi-scale tensor maps highlighting the items' outlines within the X-ray scans and an instance segmentation model acting on this representation.
We validated our approach on three publicly available datasets encompassing gray-level and colored scans and showcased its overall superiority over competitive frameworks in various aspects. For instance, the proposed framework outperforms the state-of-the-art methods \cite{opixray,hassan2019,Hassan2020ACCV,hassan2020Sensors} by \RV{1.07\%}, \RV{0.190\%}, and \RV{1.55\%} on GDXray \cite{mery2015gdxray}, SIXray \cite{miao2019sixray}, and OPIXray \cite{opixray} dataset, respectively. Furthermore, on each SIXray subsets (i.e., SIXray10, SIXray100, SIXray1000) \cite{miao2019sixray}, the proposed framework leads the state-of-the-art by \RV{1.92\%}, \RV{2.02\%}, and \RV{3.22\%}, respectively.
\noindent In future, we aim to apply the proposed framework to recognize 3D printed contraband items from the X-ray scans. Such items exhibit poor visibility in the X-ray scans because of their organic material, making them an enticing and challenging case to investigate and address.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{Picture8.jpg}
\caption{Failure cases from GDXray \cite{mery2015gdxray}, SIXray \cite{miao2019sixray}, and OPIXray \cite{opixray} dataset.}
\label{fig:fig8}
\end{figure}
\section*{Conflicts of Interest Declarations}
\noindent \textbf{Funding}: This work is supported by a research fund from ADEK (Grant Number: AARE19-156) and Khalifa University (Grant Number: CIRA-2019-047).
\noindent \textbf{Conflict of Interest}: The authors have no conflicts of interest to declare that are relevant to this article.
\noindent \textbf{Financial and Non-Financial interests}: All the authors declare that they have no financial or non-financial interests to disclose for this article.
\noindent \textbf{Employment}: The authors conducted this research during their employment in the following institutes:
\begin{itemize}
\item T. Hassan (Khalifa University, UAE),
\item S. Ak\c{c}ay (Durham University, UK),
\item M. Bennamoun (The University of Western Australia, Australia),
\item S. Khan (Mohamed bin Zayed University of Artificial Intelligence, UAE), and
\item N. Werghi (Khalifa University, UAE).
\end{itemize}
\noindent \textbf{Ethics Approval}: All the authors declare that no prior ethical approval was required from their institutes to conduct this research.
\noindent \textbf{Consent for Participate and Publication}: All the authors declare that no prior consent was needed to disseminate this article as there were no human (or animal) participants involved in this research.
\noindent \textbf{Availability of Data and Material}: All the datasets that have been used in this article are publicly available.
\noindent \textbf{Code Availability}: The source code of the proposed framework is released publicly on GitHub\textsuperscript{\ref{note1}}.
\noindent \textbf{Authors' Contributions}:
T. Hassan formulated the idea, wrote the manuscript, and performed the experiments.
S. Ak\c{c}ay improved the initial design of the framework and contributed to manuscript writing.
M. Bennamoun co-supervised the whole research, reviewed the manuscript and experiments.
S. Khan reviewed the manuscript, experiments and improved the manuscript writing.
N. Werghi supervised the whole research, contributed to manuscript writing, and reviewed the experimentation.
\small
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