Dual energy X-ray Computed Tomography (DECT) enables to automatically decompose materials in clinical images without the manual segmentation using the dependency of the X-ray linear attenuation with energy. In this work we propose a deep learning procedure called End-to-End Material Decomposition (E2E-DEcomp) for quantitative material decomposition which directly convert the CT projection data into material images. The algorithm is based on incorporating the knowledge of the spectral model DECT system into the deep learning training loss and combining a data-learned prior in the material image domain. Furthermore, the training does not require any energy-based images in the dataset but rather only sinogram and material images. We show the effectiveness of the proposed direct E2E-DEcomp method on the AAPM spectral CT dataset Sidky and Pan, 2023 compared with state of the art supervised deep learning networks.

§ INTRODUCTION

Dual-energy computed tomography (DECT) is one spectral CT technology which is based on the deployment of two X-ray sources at different energies which can potentially allow to discriminate different materials in a specimen or to reconstruct virtual mono-energetic images which is of utmost interest in clinical imaging applications and industrial non-destructive testing Mendonça et al., 2013. The dependency of the attenuation coefficient of different materials respect to the X-ray energy can be leveraged in the DECT material decomposition procedure whose aim is to estimate each pixel's value as a linear combination of two different basis materials Johnson et al., 2007.

Different approaches have been developed to obtain material images: the image domain techniques are based on first reconstructing independently the energy dependent attenuation in each pixel Maaß et al., 2009. The majority of the proposed networks such as U-Net Nadkarni et al., 2022 and generative adversarial network (GAN) Shi et al., 2021 are trained with a supervised learning approach which requires the pair of energy reconstructed Dual-energy computed tomography (DECT) images and basis material segmented images in the dataset. However these methods do not account from the beam-hardening effect, caused by the poly-energetic nature of the X-ray source. Moreover this approach leads to propagate the estimation errors from the reconstruction to the subsequent material decomposition.

In order to account for the beam-hardening effect, one-step methods directly estimate basis materials images from Cai et al., 2013 measurements projects by leveraging a model-based optimization function to minimize. However because of the highly non-linear due to the energies X-ray source coupling, this leads to minimize non-convex cost functions which require high computational cost Long and Fessler, 2014.

An alternative approach is based on decomposing the high and low energy sinogram into two independent measurements which correspond to a single basis material. Different approximations of the decomposition function that convert the dual energy sinograms into  materials independent sinograms have been proposed in Alvarez and Macovski, 1976. Afterwards each material sinogram is converted into the image domain using model-based optimization methods Mechlem et al., 2018 with spatial regularization.

Recently, other works have exploited the paradigm of combining deep learning and the knowledge of the physics of the DECT model within the optimisation problem. The one-step material decomposition is implemented using supervised unrolling algorithms in Eguizabal et al., 2022, Perelli and Andersen, 2021 or using the Noise2Inverse framework which uses pair of sub-sampled noisy sinograms and training dataset Fang et al., 2021. One limitation of these methods is the computational cost of the iterative solver which hinders the usage in applications which require strict time constraints.

In this work, we propose a new optimization framework called End-to-End Material Decomposition (E2E-DEcomp) based on the idea to directly embedding the material decomposition function into the model-based optimization function. The optimization problem to solve becomes linear and the mapping is learned from the data during the iteration procedure of the solver.

Furthermore, the proposed method does not require any system calibration procedures to determine the material decomposition function as needed in previous approaches. The designed cost function contains the data consistency term in the material sinogram domain and a regularization term, acting in the material image domain, which is learned through an implicit denoising neural network-based function.

§ DUAL-ENERGY CT FORWARD MODEL

We introduce the forward mathematical model of the DECT system where two sources emit poly-energetic X-ray photons with spectrum $S_k(E)$, where $k\in\{e_1,e_2\}$, and the array of integrating detectors with dimension $N$ collects the photons after attenuation during the two independent acquisitions.

The linear attenuation is a spatially- and energy-dependent function $\mu(\boldr,E)$ at position $\boldr\in\R^n$ and energy  $E\in \R_+$. The energy-dependent image $\mu$ is sampled on a grid with $M$ basis pixel-functions $u_m$ such that $\mu(\boldr,E) = \sum_{m=1}^M \mu_m(E) u_m(\boldr)$ where $\mu_m(E)$ is the energy-dependent attenuation at pixel $m$. The photons' intensities $I_{n,k}$ detected for the $n$-th ray and energy source spectrum $k$, are a sum of Poisson random variables. The conditional mean of $I_{n,k}$ is

\begin{equation}\label{eq:cond_count}
\-E\left[I_{n,k} | \bm\mu\right] = \int_{0}^{\infty} S_k(E) e^{-\sum_{m=1}^M A_{n,m} \mu_m(E)} dE
\end{equation}

with $\boldA \in \R^{N\times M}$ defined as $[\boldA]_{n,m} = \int_{\calL_n} u_m(\boldr) \,\rmd \boldr$ and $\boldmu(E) = [\mu_1(E),\dots,\mu_J(E)]\transp \in \R_+^M$ is the discretized energy-dependent attenuation. In the normal dose case, the DECT measurements $\*y\in\-R^{N\times 2}$ collected from the scanner are obtained using the negative logarithm of the measured photons at each energy $\*y_n = [y_{n,e_1}, y_{n,e_2}] = \left[  -\log\left( I_{n,e_1} \right), -\log\left( I_{n,e_2} \right)\right]$. The conditional mean of the measurements over the linear attenuation becomes

\begin{equation}\label{eq:log_avg}
\-E\left[\*y_n | \bm\mu\right] = -\log\left(\int_{0}^{\infty} \*S(E) e^{-\sum_{m=1}^M A_{n,m} \mu_m(E)} dE\right)
\end{equation}

with $\*S(E) = \left[ S_{e_1}(E), S_{e_2}(E) \right]$ representing the spectrum of the two X-ray sources $\{e_1, e_2\}$ for each energy $E$.

From the model (<ref>) we can derive the conditional mean $\-E\left[\*y_n | \*x\right]$ over the material vectorized images $\*x\in\-R^{M\times 2}$, by exploiting the decomposition of the attenuation as a linear combination of the mass density of two basis materials

\begin{equation}\label{eq:m_decomp}
\mu_i(E) = x_{m,1}\varphi_1(E) + x_{m,2}\varphi_2(E)
\end{equation}

where $x_{m,s}$ (mg/cm$^3$) is the equivalent density for basis materials $s$ at voxel $m$ and $\varphi_s(E)$ (cm$^2$/mg) is the known energy-dependent mass attenuation function for basis material $s$. Then, by substituting Eq. (<ref>) into Eq. (<ref>), we have

\begin{equation}\label{eq:log_avg_mat}
h(\*p_n) = -\log\left(\int_{0}^{\infty} \*S(E) e^{-\*p_n\left(\bm\varphi(E)\right)^T} dE\right)
\end{equation}

where $\bm\varphi(E) = \left[\varphi_1(E), \varphi_2(E)\right]$, $\*p_n$ (mg/cm$^2$) is the material density projection defined as

\begin{equation}
\*p_n\left(\bm\varphi(E)\right) = \sum_{m=1}^M A_{n,m} \left(x_{m,1}\varphi_1(E) + x_{m,2} \varphi_2(E) \right)
\end{equation}

and $h: \-R^2 \rightarrow \-R^2$ is a vector-valued function which models the non-linear relationship between the material density projections and the expected attenuation. From this, we have $\-E\left[\*y_n |\*x \right] = h\left([\*A\*x]_n\right)$. The inverse function $h^{-1}: \-R^2 \rightarrow \-R^2$ is defined as $h^{-1}\left(h(\*p_n) \right) = \*p_n$ which represents the material decomposition in the sinogram domain since it converts the energy sinogram $h(\*p_n)$ into the material sinograms $\*p_n$.

§ MODEL-BASED OPTIMIZATION PROBLEM

In the normal X-ray dose case, we can approximate the Poisson distribution of the measurements $I_{n,k}$ with an anisotropic Gaussian distribution over $\*y_n$ with conditional average $\-E\left[\*y_n |\*x \right]$ and diagonal covariance $\*W$, where individual projections are conditional independent. The negative log-likelihood (NLL) $f(\*y, \*x) = -\log P(\*y | \*x)$ is expressed using the expression of the conditional mean (<ref>) as follow

\begin{equation}\label{eq:NLL_h}
f(\*y, \*x) = \frac{1}{2} \sum_{n=1}^N\| \*y_n - h\left([\*A\*x]_n\right) \|^2_{\*W_n} + C
\end{equation}

where $C$ is a normalizing constant, and $\*W_n = \~{Cov}^{-1}(\*y_n | \*x) = \diag([ w_{n,e_1}, w_{n,e_2} ])$ is the inverse covariance of $\*y_n$ with $w_{n,k} = [\~{Var}(\*y_{n,k} | \*x)]^{-1} \approx I_{n,k}, \; k \in \{ e_1, e_2 \}$.

Model-based iterative reconstruction (MBIR) approaches aim at optimising the cost function (<ref>), however this problem is computationally demanding to solve because $f(\*y, \*x)$ is non-linear and the inverse function $h^{-1}$ is challenging to compute since it requires an accurate calibration procedure.

§ END-TO-END MODEL-BASED DEEP LEARNING FOR MATERIAL DECOMPOSITION (E2E-DECOMP)

We describe the idea of embedding a learned material decomposition polynomial network $P_{\theta}$, whose parameters $\theta$ are learned from the sinogram data, into the negative log-likelihood (NLL) term. Using the definition $h^{-1}\left(h([\*A\*x]_n) \right) = [\*A\*x]_n
$ and the DECT model (<ref>), we obtain the estimated material sinogram

\begin{equation}
\hat{\*p}_n = h^{-1}(\*y)_n \approx \+P_{\theta}(\*y)_n
\end{equation}

where the last approximation is obtained by substituting the vector function $\+P_{\bm\theta} \colon \R^{N\times 2}_+ \to \R^{N\times 2}_+$ with learned parameters $\bm\theta$ for the non-linear material decomposition function $h^{-1}$. $\+P_{\theta}$ is a vector polynomial function which takes as input the energy sinogram $\*y\in\R^{N\times 2}$ and output the material sinogram of the same dimension as follow

\begin{equation*}
\+P_{\theta}(\*y)_n =[\hat{p}_{1,n}, \hat{p}_{2,n}] = \left[
\begin{array}{c}
\sum_{i=0}^I \sum_{j=0}^J \theta_{i,j,e_1} y_{n,e_1}^i y_{n,e_2}^j \\
\sum_{i=0}^I \sum_{j=0}^J \theta_{i,j,e_2} y_{n,e_1} y_{n,e_2} \\
\end{array}
\right]
\end{equation*}

where the matrix of parameters $\bm\theta$ is trained through a polynomial regression network. The NLL function, which guarantees consistency in the material sinogram domain, is

\begin{equation}\label{eq:data_cons}
f(\*y, \*x) = \sum_{n=1}^{N} \|\+P_{\bm\theta}(\*y)_n - [\*A \*x]_n \|^2_{\*B_n}
\end{equation}

where $\*B_n = [\nabla \+P_{\bm\theta}(\*y)_n]^{-1}\, \*W_n\, [\nabla \+P_{\bm\theta}(\*y)_n]^{-T}\in \R^{2\times 2}$ is the statistical covariance matrix of the problem in the new domain which is obtained using the first order Taylor expansion as in Zhang et al., 2013 using the polynomial network $\+P_{\theta}$, where $\nabla \+P_{\bm\theta}(\cdot)$ can be efficiently computed using back-propagation.

The material decomposition of the images $\*m\in\R^{M\times 2}$ is formulated as a minimization of the following cost function $\*x^* = \argmin_{\*x\in\R^{M\times 2}_+} f(\*y,\*x) + \lambda R(\*x)$ where $f(\*y,\*x)$ represents the DECT data consistency based on the mathematical negative log-likelihood model in (<ref>), $\lambda$ is the regularization parameter and the regularization $R(\*x) = \|\+R_{\bm\rho}(\*x)\|^2$ is learned from the materials' data. We constrained the optimisation to non-negative material images. We design $\+R_{\bm\rho}$ to be a learned convolutional neural network (CNN) estimator of noise of the material images, with parameters $\bm\rho$

\begin{equation}\label{eq:reg_r}
\+R_{\bm\rho}(\*x) = (\*I - \+D_{\bm\rho})(\*x) = \*x - \+D_{\bm\rho}(\*x)
\end{equation}

where $\+D_{\bm\rho}(\*x)$ is the denoised version of $\*x$, after the removal of noise. By formulating the problem with (<ref>)-(<ref>), we are able to decouple the learning procedures which are applied in different domain since the data consistent term contains the learning module $\+P_{\bm\theta}(\cdot)$ for the material decomposition in the sinogram domain while the learning of the image data prior $\+R_{\bm\rho}(\cdot)$ is applied within the regularization term as

\begin{equation}\label{eq:opt_fully}
\*x^* = \arg\min_{\*x\in\R^{M\times 2}_+} \sum_{n=1}^{N} \|\+P_{\bm\theta}(\*y)_n - [\*A \*x]_n \|^2_{\*B_n} +\lambda \| \*x - \+D_{\bm\rho}(\*x) \|^2
\end{equation}

Solving (<ref>) with any first order solvers would require computing the Jacobian $\*J$ of the denoiser $\+D_{\bm\rho}(\cdot)$ which is computationally demanding. We adopt the same approach as in Aggarwal et al., 2018 by approximating the non-linear term $\+D_{\bm\rho}(\*x)$ using the first order Taylor approximation at the $k$-th iteration which leads to $\| \*x - \+D_{\bm\rho}(\*x^k + \nabla\*x) \|^2 \approx \| \*x - \*z^k \|^2 + \| \*J^k\Delta\*x \|^2$. For small perturbations around $\*m^k$, the term $\| \*J^k\Delta\*x  \|^2$ can be approximated to zero. Therefore the problem (<ref>) becomes

\begin{eqnarray}\label{eq:opt_split}
\*x^k &=& \argmin_{\*x\in\R^{M\times 2}_+} \sum_{n=1}^{N} \|\+P_{\bm\theta}(\*y)_n - [\*A \*x]_n \|^2_{\*B_n} +\lambda \| \*x - \*z^k \|^2 \nonumber \\
\*z^{k+1} &=& \+D_{\bm\rho}(\*x^k)
\end{eqnarray}

The optimization problem in the first step of problem (<ref>) has a quadratic form therefore the associated linear problem

\begin{equation}\label{eq:data_consist}
\left[\*A^T(\*B\odot \*A) +\lambda\*I\right]\*x^k = \left( \*A^T \+P_{\bm\theta}(\*y) + \lambda \*z^k \right)
\end{equation}

where $\*B=\left[ \diag(\*B_1) ; \ldots ; \diag(\*B_N)\right]\in\R^{N\times 2}$ is the concatenation of the diagonal element of the covariance weighting matrices $\*B_n$. Eq. (<ref>) can be solved efficiently using using the Conjugate Gradient (CG) iterative algorithm Nazareth, 2009 which does not require any computation of the matrix inverse.

The workflow of the E2E-DEcomp algorithm at inference is shown in Fig. <ref>, and the structure of the E2E-DEcomp algorithm for inference is reported in Table <ref>.

[node distance=1.5cm,scale=1]
[punkt] (P) $\+P_{\bm\theta}$;
[fig_n, left= of P, inner sep=0pt] (IN)
[->] (IN.east) edge node [above] $\*y$ (P.west);
[punkt] (P) $\+P_{\bm\theta}$;
[fig_n, right= of P, inner sep=3pt] (DEC)
[CG, right= of DEC, text width=3em] (DC) $\+{DC}$;
[->] (P.east) edge node  (DEC.west);
[Denoi, right= of DC, text width=3em] (Denoi) $\+D_{\bm\rho}$;
[->] (DC.east) edge node  (Denoi.west);
[below= of Denoi, text width=0em, text centered] (EMPTY)  ;
[->] (EMPTY.south) edge node  (Denoi.south);
[->] (DEC.east) edge node  (DC.west);

[right= of Denoi, text width=4em, text centered] (Iter) repeat

[CG, right= of Iter, text width=3em, text centered] (DC1) $\+{DC}$;
[->] (Denoi.east) edge node  (Iter.west);
[->] (Iter.east) edge node  (DC1.west);
[Denoi, right= of DC1, text width=3em] (Denoi1) $\+D_{\bm\rho}$;
[->] (DC1.east) edge node  (Denoi1.west);
[right= of Denoi1, text width=0em, text centered] (OUT) ;
[->] (Denoi1.east) edge node [above] $\*x^K$ (OUT.west);
[below= of Denoi1, text width=0em, text centered] (EMPTY1)  ;
[->] (EMPTY1.south) edge node  (Denoi1.south);
[right= of Denoi1, text width=0em, text centered] (OUT) ;
[Denoi, below= of DC1, text width=8.5em, minimum height=2em, fill=yellow!60] (W)  Shared Weights $\bm\rho$;
[-] (W.east)+(0, 0.12) edge node  (EMPTY1.south);
[-] (W.west)+(0, 0.12) edge node  (EMPTY.south);
[Denoi, below= of P, minimum height=2em, fill=yellow!60] (theta)  Weights $\bm\theta$;
[->] (theta.north) edge node  (P.south);
[above= of DC, text width=0em, text centered] (EMPTY2)  ;
[->] (EMPTY2.south) edge node  (DC.north);
[above= of DC1, text width=0em, text centered] (EMPTY3)  ;
[->] (EMPTY3.south) edge node  (DC1.north);
[-] (EMPTY2.south) edge node  (EMPTY3.south);
[-] (P.east)+(0.5,0) |- (EMPTY2.south);
background[node distance=1cm]
[background,fit=(P) (DC) (DC1) (Denoi) (Denoi1) (W) (theta) (EMPTY2) (OUT), fill=orange!20, label=above:End-to-End Material Decomposition (E2E-DEcomp) Training] (end_to_end) ;
[background,fit=(DC) (Denoi), fill=red!30, label=above:Iteration 1] (iter1) ;
[background,fit=(DC1) (Denoi1), fill=red!30, label=above:Iteration K] (iter1) ;

End-to-End Material Decomposition (E2E-DEcomp) material decomposition training workflow. The network is obtained by unfolding $K$ iterations of the algorithm and training end-to-end from energy sinograms $\*y$ to material images $\*x^K$. Each iteration is constituted by the data consistency block $\-{DC}$ and the denoising module $\-D_{\bm\rho}$ whose trainable parameters $\bm\rho$ are shared through the iterations.

End-to-End Material Decomposition (E2E-DEcomp) for Material Decomposition

[1]
energy sinogram $\*y$, parameters $\bm\rho$ and $\bm\theta$, number of iterations $K$
initial state $\*z^0 (= \*0) \in \R^{M\times 2}$, $\*x_c = \*A^T \+P_{\bm\theta}(\*y)$
$k = 0, \ldots, K-1$
$\*x^k \colon \left\{\begin{array}{l} \left[\*A^T(\*B\odot \*A) +\lambda\*I\right]\*x^k = \*A^T \+P_{\bm\theta}(\*y) + \lambda \*z^k \\ \mbox{solved using CG algorithm} \end{array}\right.$
$\*z^{k+1} = \+D_{\bm\rho}(\*x^k)$

§ NUMERICAL RESULTS

We have tested the E2E-DEcomp algorithm using the AAPM Dual-energy CT scans for a breast model containing adipose and fibroglandular tissues Sidky and Pan, 2023 and simulating the DECT sinogram with the ASTRA Toolbox Van Aarle et al., 2016 using a parallel-beam geometry with a detector of $1024$ elements. We used $I_0=10^5$ X-ray photons with the spectrum from Sidky and Pan, 2023 and adding Poisson noise. The spatial resolution of the images is set to $512\times 512$. We deploy an end-to-end strategy to train the E2E-DEcomp algorithm where we fix a-priori the number of iteration $K=3$ and we train the overall unrolled iterations by calculating the mean square error (MSE) between the material estimates and the ground truth $\*x^*$.

In order to reduce the number of learnable parameters we utilise the same architecture for the denoising module $\+D$ at each iteration $k$ with shared parameters $\bm\rho$. In Fig. <ref> it is shown the qualitative comparison on a test material image of the adipose tissue using filtered back projection (FBP) and E2E-DEcomp while in Fig. <ref> is is reported the PSNR error for a set of 10 testing images for the 2 material decomposition.

Ground truth image

FBP $30$ angles

FBP $60$ angles

FBP $90$ angles

FBP $180$ angles

E2E-DEcomp $30$ views

E2E-DEcomp $60$ angles

E2E-DEcomp $90$ angles

E2E-DEcomp $180$ angles
Qualitative comparison between the material decomposition for adipose using E2E-DEcomp and FBP using different number of angular projections.

xlabel=Angles $\theta$,
ylabel=PSNR (dB),
xtick=30, 90, 180, 360, 512,
grid = major,
legend columns=1,
legend cell align=left,
legend entries=E2E-DEcomp, FBP,
legend style=at=(0.7,0.3),anchor=north
[color=blue, mark=* , dashed] table[x=Ang, y=MoDL] Noise_sim_MoDL_FBP.txt;
[color=red, mark=square*, dashed] table[x=Ang, y=FBP] Noise_sim_MoDL_FBP.txt;

Comparison of DE-MoDL and FBP for 2 materials decomposition using noisy DECT acquisition with photon counts $I_0 = 10^5$. The PSNR metric is calculated for different number of DECT angular projections.

It is worth noting that the improvement in the decomposition accuracy are consistent, around 5 dB, across different levels of dose, i.e. from sparse views to higher number of projections. We have also compared the E2E-DEcomp framework with the FBP ConvNet method Jin et al., 2017 and Fig. <ref> shows how E2E-DEcomp can achieve a faster convergence in training using fewer epochs.

ylabel=PSNR (dB),
xtick=20, 40, 60, 80, 100, 120,
grid = major,
legend columns=1,
legend cell align=left,
legend entries=FBP ConvNet, E2E-DEcomp,
legend style=at=(0.7,0.3),anchor=north
[color=green, mark=* , dashed] table[x=Epochs, y=PSNR] Training_DW.txt;
[color=blue, mark=square*, dashed] table[x=Epochs, y=PSNR] Training_DWDC.txt;

Comparison of the PSNR training error between the FBP ConvNet and the E2E-DEcomp algorithms.

§ CONCLUSION

This work proposed a direct method for DECT material decomposition using a model-based optimization able to decouple the learning in the measurement and image domain. Numerical results show the effectiveness of the proposed E2E-DEcomp compared to other supervised approaches since it has fast convergence and excellent performance on low-dose DECT which can lead to further study with clinical dataset.

§ COMPLIANCE WITH ETHICAL STANDARDS

This is a numerical simulation study for which no ethical approval was required.

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Millimeter-Wave UAV Coverage
in Urban Environments

Seongjoon Kang^†   Marco Mezzavilla^†   Angel Lozano^♭    Giovanni Geraci^♭
William Xia^†   Sundeep Rangan^†     Vasilii Semkin^♯   Giuseppe Loianno^†
^†NYU Tandon School of Engineering, Brooklyn, NY, USA
^♯VTT Technical Research Centre of Finland Ltd, Finland
^♭Univ. Pompeu Fabra, Barcelona, Spain
S. Rangan, W. Xia, S. Kang, and M. Mezzavilla were supported by NSF grants  1302336,  1564142,  1547332, and 1824434,  NIST, SRC, and the industrial affiliates of NYU WIRELESS. A. Lozano and G. Geraci were supported by ERC grant 694974, by MINECO's Project RTI2018-101040, by the Junior Leader Fellowship Program from “la Caixa" Banking Foundation, and by the ICREA Academia program. The work of V. Semkin was supported by the Academy of Finland.


Email]

# Gravitational Waves Through Time:
Scientific Significance, Detection Techniques, and Recent Breakthroughs

Chris Jia  [<EMAIL_ADDRESS>Physics and Astronomy Department at the
University of Texas at Austin

###### Abstract

This paper provides a detailed review of gravitational waves. We begin with a
thorough discussion regarding the history of gravitational waves, beginning
even before Albert Einstein’s theory of general relativity, highlighting
important developments and milestones in the field. We then discuss the
scientific significance of gravitational wave detections such as the
verification of general relativity and key properties of black holes/neutron
stars. We extend our analysis into various detection techniques including
interferometer-based detectors (LIGO, Virgo, GEO600), pulsar timing arrays,
and proposed space-based detectors (LISA, DECIGO, BBO). Finally, we conclude
our review with a brief examination of the captivating event GW190521.

††preprint: APS/123-QED

##  I Introduction

Across our endless universe, a myriad of mysteries and cosmic phenomena have
captivated the imagination of humanity. Over the centuries, our understanding
of the universe has evolved drastically, with one of the most recent
revolutionary breakthroughs being the detection of gravitational waves—a
phenomenon predicted by Albert Einstein’s theory of general relativity over a
century ago [1].

Gravitational waves (GW) are ripples in spacetime itself, caused by the
disturbance of compact consolidations of energy [2]. Although only extremely
massive objects such as neutron stars and black holes are able to produce
gravitational waves at discernible levels for our current detectors,
theoretically the acceleration of any matter can cause gravitational waves.
Furthermore, gravitational waves propagate at the speed of light, adding an
intriguing dimension to their nature and implications [1].

With the first detection of gravitational waves on September 14, 2015 from
LIGO (Laser Interferometer Gravitational-Wave Observatory), countless
achievements have been made within the fields of physics and astronomy [1].
Gravitational waves tied up one of the last loose ends in the theory of
general relativity at a remarkably high precision ($>$96%) for recent
detections [3]. Our understanding of black holes and neutron stars also rely
heavily on data collected from GW. Significant advancements have been achieved
in understanding the properties of these massive entities, including their
formation rates, angular momenta, and mass limits due to the discovery of
gravitational waves [1]. Furthermore, gravitational waves may provide
scientists with an avenue of connecting quantum mechanics with general
relativity through a hypothetical elementary particle known as gravitons [4].
Although this review will not dive into the fascinating theory of gravitons,
we suggest the following articles for interested readers [5][6].

In Section II of this paper, we provide a brief history of gravitational
waves, beginning even before Einstein’s theory of general relativity.
Subsequently, in Section III, we delve into the scientific implications and
significance of detecting gravitational waves. Section IV is dedicated to an
exploration of various detectors and analytical techniques employed in the
study of gravitational waves. In Section V, we shine a spotlight on a recent
and particularly captivating gravitational wave event, denoted as GW190521.

##  II A Brief History of Gravitational Waves

We begin our brief history of gravitational waves in 1876, roughly 40 years
before Einstein’s theory of general relativity. One of the first recorded
ideas similar to the concept of GW was written by William Kingdon Clifford in
his paper titled “On the Space-Theory of Matter”. In his paper he notes [7],

> “(1) That small portions of space are in fact of a nature analogous to
> little hills on a surface which is on average flat; namely that the ordinary
> laws of geometry are not valid for them. (2) That the property of being
> curved or distorted is continually being passed on from one portion of space
> to another after the manner of a wave. (3) That this variation of the
> curvature of space is what really happens in that phenomenon which we call
> the motion of matter, whether ponderable or etherial.”

The first explicit mention of gravitational waves, however, was made by
Poincaré in 1905. He postulated that special relativistic gravity could have
waves that proliferated at the speed of light [8].

Finally in 1915, Einstein’s proposed his theory of general relativity. Seen as
a continuation of his earlier theory of special relativity, this new theory
would make waves within the physics community. Fundamentally, Einstein’s
theory of general relativity asserted that gravity was not a force, as
interpreted under Newton’s Law, but rather a curvature of spacetime itself
[3].

After finalizing his theory, Einstein theorized that the existence of
gravitational waves might be possible—perhaps in ways similar to
electromagnetic waves. One way of forming electromagnetic waves is by rotating
an electric dipole (formed by a positive and a negative charge separated at a
distance). This oscillation would, in turn, form electromagnetic waves. For
gravity, however, there is no “opposite” to a positive mass. Without a
“gravitational dipole”, Einstein ultimately pursued other avenues of
possibility that could produce gravitational waves [3].

After many years of deliberation, Einstein ultimately reached a conclusion in
his 1936 paper co-authored alongside Nathan Rosen titled “Do gravity waves
exist”. The paper concluded with a startling statement to many scientists at
the time: gravitational plane waves cannot exist [8]. Fortunately, the story
doesn’t end there (or else we would not be discussing it now).

While the aforementioned paper was waiting to be published, Einstein was made
aware of some issues with the resulted obtained by the paper from his new
assistant, Leopold Infeld. After some modification, he arrived at a conclusion
almost opposite of his original findings. The title was revised to “On
Gravitational Waves” and provided the world one of the first mathematical
solutions for gravitational waves [3][8]. In the abstract, Einstein states
[9],

> “We investigate rigorously the case of cylindrical gravitational waves. It
> turns out that rigorous solutions exist and that the problem reduces to the
> usual cylindrical waves in euclidean space.”

Now that gravitational waves had been theorized and a mathematical solution
presented, experiments were needed to confirm this fascinating phenomenon. In
1957, the Chapel Hill conference began the race to detect gravitational waves.
Over the course of the six-day conference, a motley of topics were discussed,
including cosmology, radio astronomy, general relativity, and, notably, the
elusive gravitational waves [3].

Shortly after the Chapel Hill conference, Joseph Weber began experimenting
with ideas and designs for a gravitational wave detector. In 1966, he and his
team built an “antenna” surrounded by a belt of detectors. The “antenna” was a
massive aluminum cylinder and weighed roughly 3000 kg (Figure 1). To ensure
the issue of “local disturbances” such as storms or power supply fluctuations
wouldn’t cause false signals for the detector, he built two detectors 950 km
apart and connected them with a high-speed phone line. This would ensure that
the collected data could be relatively trusted and not just a result of
outside factors [3].

Figure 1: Snapshot from Cervantes-Cota 2016 [3] depicting a sketch of Weber’s
cylindrical detector and a photo of Joseph Weber at the antenna.

In 1969, Weber published a paper claiming he found evidence of gravitational
waves. Figure 2 displays a table of several “coincidences” between detectors.
He concluded [10],

> “The probability that all of these coincidences were accidental is
> incredibly small. Experiments imply that electromagnetic and seismic effects
> can be ruled out with a high level of confidence. These data are consistent
> with the conclusion that the detectors are being excited by gravitational
> radiation.”

After much debate within the scientific community, many believed that Weber’s
discoveries were unreliable. His frequency of observations seemed unlikely and
also appeared to be connected to extremely irregular events [3].

By the mid-1970s, many new GW detectors were constructed across the world. In
stark contrast to Weber’s previous discoveries, these new GW detectors all
arrived at a similar conclusion: there were no gravitational wave signals. The
lack of signals, however, did not deter researchers but rather invigorated
them even further to try and locate gravitational waves [3].

Figure 2: Snapshot from Weber 1969 [10] presents a table that lists some of
the “coincidences” along with the period of time which must elapse for a
coincidence with equal to or greater amplitudes to occur accidentally.

At the dawn of the 21st century, several state-of-the-art gravitational wave
detectors, now integral to our scientific landscape, reached completion. For
interferometer-based detectors, notable names include LIGO in the USA, Virgo
in Italy, and GEO600 in Germany [3]. Other types of detectors such as pulsar
timing arrays and space-based detectors have also become more prevalent in
recent years and provide yet another avenue for gravitational wave detection.
In Section IV, we will delve into a comprehensive exploration of each
individual detector. Before this, however, we must focus on the scientific
significance inherent in the study of gravitational wave detections.

##  III Scientific Significance of Gravitational Wave Detections

As we delve into the intricacies of gravitational wave research, this section
strives to elucidate the diverse spectrum of discoveries that GW have ushered
into our scientific understanding. From confirming the predictions of
Einstein’s theory of general relativity to understanding black holes and
neutron star mergers, the exploration of gravitational wave detections opens
unprecedented avenues for humanity’s understanding of the cosmos.

###  III.1 Verification of general relativity

Given Albert Einstein’s pivotal role as the “father” of gravitational waves,
it is only fitting that we delve into the intricacies of how these waves serve
as a validation for the principles laid out in general relativity.

Importantly, tests of general relativity (such as solar system experiments and
cosmological measurements) have primarily been in low-velocity or weak-field
regimes [11]. Gravitational waves, on the other hand, allow researchers to
test general relativity in the high-speed and strong-field regime, an avenue
previously unexplored [12].

In 2020, the LIGO and Virgo collaborations examined 8 gravitational wave
events within the past few years. Across 4 different tests (residuals tests,
IMR consistency, parameterized tests, and modified dispersion relations), they
ultimately concluded [13],

> “We find no evidence for new physics beyond general relativity, for black
> hole mimickers, or for any unaccounted systematics.”

While a comprehensive exploration of each test lies beyond the scope of this
paper, we will provide a brief overview of two specific tests: residuals tests
and parameterized tests. To view the full results and efforts taken to verify
general relativity using gravitational waves, we recommend the following
references [12][13].

####  III.1.1  Residuals Test

A residuals test involves examining the differences between the observed data
and the model predictions (in this case, general relativity). If we assume
general relativity to be correct, subtracting the astrophysical signal should
reveal residuals that are comparable with instrumental noise [12].

When analyzing these residuals, however, our models of general relativity are
not completely perfect (although they are very accurate). Furthermore, we also
don’t have a complete certainty about signals in the data. Both of these
factors taken together requires scientists to calculate something called the
“fitting factor” (FF) that must have a lower bound because of the
uncertainties involved [12].

$\displaystyle FF=\frac{||h||}{\sqrt{||h||^{2}+||s||^{2}}}$ (1)

$\displaystyle FF_{90}=\frac{SNR_{GR}}{\sqrt{SNR_{GR}^{2}+SNR_{90}^{2}}}$ (2)

Equations (1) and (2) are given by [12]. Equation (1) represents a generic
form to calculate the fitting factor with h as the model and s as the signal.
Equation (2) is a more specific form of Equation (1) with “90” referring to
the 90% chance that the Signal-to-Noise Ratio (SNR) of the leftover signal
(residual) after removing the best-fit GR template is less than or equal to a
specific value denoted as SNR90.

Figure 3: Snapshot from Abbott et al. 2021 [13] showing SNR90 for each GW
event, as a function of the SNR maximum-likelihood template (SNRGR), with the
corresponding p-values.

In Figure 3, the set of p-values show no distinct correlation between SNR90
and SNRGR. This implies that the SNR90 magnitude is solely influenced by the
instrumental properties associated with each event and not influenced by the
subtracted template (general relativity). This lack of correlation provides
remarkably clear evidence in favor of general relativity [13].

####  III.1.2  Parameterized Tests

Parameterized tests provide another avenue of verifying general relativity. In
essence, these tests entail the systematic adjustment of parameters linked to
the detected signals. The objective is to ascertain whether the observed
gravitational wave events align with the predictions stipulated in the
framework of general relativity [13].

In the case of gravitational wave detections, scientists employ parameterized
tests to evaluate various aspects of the detected signals, such as their
waveform, amplitude, and frequency. These tests often involve comparing the
observed data with the predictions derived from general relativity. By
systematically varying specific parameters in the theoretical models and
comparing the results with the actual observations, researchers can assess the
consistency of the detected signals with the predictions of general relativity
[13].

The detailed investigation carried out by Abbott et al. not only reaffirms the
validity of general relativity but also contributes valuable insights into the
precision and accuracy of current gravitational wave detection techniques
[13].

###  III.2 Insights into the nature of black holes and neutron stars

Some of the most major events regarding gravitational waves have been mergers
of black holes or binary systems of neutrons stars. The following brief
sections cover the study of black holes and neutron stars from a gravitational
wave perspective.

####  III.2.1  Black Holes

The recent discoveries of major black hole merger events at LIGO (such as
GW150914 and GW190521) have ignited renewed interest into the study of
primordial black holes (PBHs). One of the major areas of interest is how black
hole binary systems actually come about [14].

Two main ideas have been explored for how these pairs of black holes might
form. The first involves chance encounters between PBHs within a dense
environment, which works in a more recent universe. The other idea involves
the tidal perturbation of distant PBHs, which could have happened in the early
universe during the radiation-dominated epoch [14].

These investigations demonstrate the immense usefulness of gravitational
waves, as it provides researchers with another avenue besides electromagnetic
waves for examining the characteristics of PBHs, including their mass and
abundance. As ongoing experiments receive upgrades along with the construction
of future detectors, instruments will have greatly enhanced sensitivity across
various frequencies. As such, the collection of more comprehensive
information, including details such as eccentricity, redshift, and spatial
inhomogeneities, will become readily accessible to uncover the true nature of
PBHs and black hole binary systems [14].

Figure 4 depicts a graph of our current and future GW detector’s range of data
collection in addition to the amplitude and frequency range for the first
detected binary black hole coalescence (GW150914) [15].

For a deeper exploration of the intricate information that gravitational waves
can unveil about black holes, we highly recommend referring to the publication
titled Black holes, gravitational waves and fundamental physics: a roadmap
[16].

####  III.2.2  Neutron Stars

Just as black holes release immense gravitational waves, neutron stars also
prove to be exceptional sources of these waves. Exotic processes are triggered
within a neutron star’s nucleus due to the tremendous pressure, which can
forcefully release mass at relativistic speeds. Consequently, gravitational
waves become a vital means of measuring these fascinating astrophysical
objects and the dynamic processes they undergo [17].

To generate discernible gravitational waves from rotating neutron stars, a
crucial requirement is the presence of asymmetrical deformations in the
neutron star. Notably, alternative mechanisms, such as pulsar
“glitches”—instances where neutron stars undergo a sudden “jump” in spin
frequency—or magnetar flares, where a neutron star with an exceptionally high
magnetic dipole experiences magnetic decay, can also induce substantial
gravitational waves. [17].

Specifically, our primary emphasis will be on the asymmetrical deformations
within the neutron star. Magdalena Sieniawska and Michał Bejger in 2019
provide a set of equations to estimate a GW strain amplitude h0 from such
neutron stars. They reach a succinct result which is given by Equation (3)
[15].

$\displaystyle h_{0}\approx 10^{2}\frac{G\epsilon I_{3}\nu^{2}}{c^{4}d}$ (3)

In Equation (3), $\epsilon$ represents a dimensionless parameter that depends
on the level of asymmetry for the neutron star. The moment of inertia along a
rotational axis is I3 which is dependent on the neutron star’s mass M and
radius R (I3 $\approx$ MR2). $\nu$ is the neutron star’s rotational frequency
and d represents the distance to the source (in this case, the neutron star)
[15].

Figure 4: Snapshot from Sieniawska and Bejger 2019 [15] depicting different
gravitational wave detectors and their respective range of data collection.
The pink outline represents the GW150914 signal strain’s amplitude ($h_{0}$)
and frequency range, whereas the blue region represents the ranges for
pulsars.

As shown in Equation (3), by measuring the amplitude of these gravitational
waves and utilizing other properties we already know about a specific neutron
star, we can obtain new information or verify existing information for a wide
collection of neutron stars.

For more information about other particulars concerning the emission of
gravitational waves from neutron stars, such as pulsar glitches and magnetar
flares, we highly recommend [17][15].

##  IV Detection Techniques

###  IV.1 Interferometer-based detectors

In Section II, we presented the first gravitational wave detector made by
Joseph Weber. This sort of detector was known as a “bar” detector, and by the
1990s, 5 major bar detectors across the globe were operational. However, these
bar detectors were only able to detect gravitational waves within a certain
frequency range ($\sim$700 Hz to $\sim$900 Hz) which inspired the development
of other types of detectors. By 2010, interferometer-based detectors had
finally finished construction (although some were in the process of receiving
upgrades), and would provide a much wider scope in detectable amplitudes and
frequencies of GW compared to bar detectors [18]. Figure 4 provides a solid
illustration of such ranges.

In order to detect gravitational waves through interferometers, lasers and
extremely sensitive instruments are needed to detect minuscule changes. As a
GW travels through space, they distort spacetime, inducing subtle stretching
and compressing along perpendicular directions. Upon interacting with the
detector, the wave momentarily modifies the lengths of the perpendicular arms,
leading to corresponding alterations in the path length traveled by the laser
beam.

Upon recombination, the beams may either amplify or counteract each other,
leading to a noticeable shift in the interference pattern. These alterations
are precisely gauged using photodetectors, which signals the presence of a
gravitational wave and includes details about its characteristics such as
amplitude, frequency, and arrival time [19]. Figure 5 provides a simple
schematic of how GW interferometers are designed.

$\displaystyle\delta L=hL$ (4)

Equation (4) shows a straightforward method of calculating the gravitational
wave amplitude h. L refers to the length of the laser arms and
correspondingly, $\delta$L represents the resulting length difference between
the two arms of the interferometer due to the gravitational wave passing
through [20].

Figure 5: Snapshot from Caltech/MIT/LIGO Lab [19] depicting a diagram of
LIGO’s interferometers with a gravitational waves approaching the detector

The following sections will provide an elementary description of various
interferometer-based detectors (LIGO, Virgo, and GEO600) along with major
differences between them.

####  IV.1.1 LIGO

LIGO consists of two identical interferometers located in the United States:
one in Livingston, Louisiana, and the other in Hanford, Washington. Each
observatory features two perpendicular arms measuring 4 km in length with
lasers operating at around 40 W. Power recycling mirrors, unique suspension
systems, and seismic isolation techniques (passive and active) are all
utilized to ensure that the detection of gravitational waves can be precisely
measured. LIGO is perhaps most well-known for being the first laboratory to
detect gravitational waves (GW150914) from an astrophysical source.
Subsequently, the facility has continued to achieve numerous monumental
detections, including GW190521, which will be further explored in Section V
[19].

####  IV.1.2 Virgo

Similar to LIGO, the Virgo interferometer in Italy exhibits comparable
performance with some distinct differences. Some of these differences include
a shorter arm lengths (3 km versus 4 km), increased laser power (17 W versus
10 W), and seismic isolation measures. Furthermore, Virgo also excels in
detecting frequencies below 40 Hz due to its seismic isolation, whereas LIGO
boasts greater sensitivity around 150 Hz. Virgo’s heightened sensitivity at
lower frequencies also allows Virgo to detect neutron stars that LIGO can’t,
like Vela [18][21]. Importantly, the information provided here was taken in
2013, and since then, both LIGO and Virgo have received or are in the process
of receiving upgrades which aim to improve both their sensitivity and laser
power. For example, Advanced Virgo can now produce laser power up to 50 W [19]
whereas aLIGO (advanced LIGO) produces laser power up to $\sim$40 W [22].

####  IV.1.3 GEO600

The GEO600 interferometer features folded arms spanning 600 meters and employs
a 12 W input laser. Despite being constructed within a relatively constrained
budget (and thus lacks the sensitivity that LIGO and Virgo provide), it
supplies a multifaceted role within gravitational wave detection. First and
foremost, it serves as a vigilant observatory during periods when LIGO and
Virgo are inactive, with the potential to validate exceptionally intense event
candidates. Furthermore, it also functions as a testing ground for advanced
LIGO technology. By 2012, GEO600 had been primarily geared towards detecting
nearby galactic supernovae [18][23].

###  IV.2 Pulsar timing arrays

Pulsar timing arrays (PTAs) provide a different method for detecting low-
frequency gravitational waves, as shown by Figure 6. As gravitational waves
travel through space, they distort the fabric of spacetime just enough such
that pulsar signals arrive at slightly different times on Earth. Since these
pulsars provide exceptionally regular “clocks” over an extended period of
time, any minuscule deviations may hint at the presence of a gravitational
wave [18].

The 3 primarily PTA collaborations include EPTA (European Pulsar Timing
Array), NANOGrav (North American Nanohertz Observatory of Gravitational
Waves), and PPTA (Parkes Pulsar Timing Array). Each of the telescopes hosts a
PTA collaboration and conducts routine timing programs occurring at intervals
ranging from 1 week to 1 month. These three collaborations collectively form
the IPTA (International Pulsar Timing Array), under which they share data sets
and jointly publish findings [24].

In 2016, S. R. Taylor et al. noted in their publication that while PTAs have
set upper limits on the amplitude of gravitational waves in the nanohertz
range, the prospect of small pulsar groups detecting gravitational waves
within the next few decades appears unlikely. Nevertheless, researchers
anticipate a significantly increased likelihood of detection by employing more
extensive pulsar arrays, as discussed in [25].

Just 6 years later, the IPTA partially proved this prediction correct. They
found evidence of a low-frequency gravitational wave pattern with am amplitude
and spectral index that aligned with expectations from supermassive black hole
mergers. However, there was no clear evidence confirming these waves’
gravitational origin [26].

Continued research employing PTAs promises to enhance our understanding of
gravitational waves. The collaborative efforts of international initiatives
through the IPTA showcase the power of global collaborations, and the future
holds great promise with ongoing advancements in technology, data-sharing
practices, an expanding array of pulsars, and increased sensitivity all in
hopes of detecting elusive gravitational waves [26].

###  IV.3 Space-based detectors

Space-based detectors represent a groundbreaking frontier in the realm of
gravitational wave astronomy, offering a unique vantage point beyond the
constraints of Earth’s atmosphere. Unlike ground-based counterparts, space-
based detectors are not hindered by terrestrial noise and environmental
factors, enabling them to probe the cosmos with unprecedented precision. The
following section will primarily focus on the Laser Interferometer Space
Antenna (LISA) with a brief mention to other proposed space-based
gravitational-wave detectors such as DECIGO and BBO.

####  IV.3.1 LISA

Figure 6: Snapshot taken from NASA [27] depicting the various detection
techniques used to probe the gravitational wave spectrum.

Within the expansive realm of gravitational wave frequencies, the proposed
LISA mission focuses primarily on the lower frequency spectrum from around 0.1
mHz to 1 Hz (as seen in Figure 6). As such, LISA will be able to better detect
sources with wider orbit and greater mass than LIGO, thereby expanding the
range of observable gravitational wave phenomena [27].

LISA’s planned operational configuration involves three spacecraft forming an
equilateral triangle in space, with extended “arms” (lasers) spanning about a
million miles (Figure 7). Because some gravitational waves have wavelengths
longer than Earth, the long “arms” of LISA would allow us to probe various new
astrophysical sources and phenomenon such as ultra-compact binaries and
supermassive black hole mergers. These gravitational wave events cause the
spacecraft to slightly alter their respective position, which in turn produces
a specific pattern from the lasers that can provide crucial insights into the
gravitational wave’s origin, such as the source’s location and physical
characteristics [27].

Figure 7: Snapshot taken from Danzmann and Rudiger 2003 [28] illustrating the
proposed LISA mission. In the figure, the the triangle arms are scaled by a
factor of 10.

Although the LISA mission still doesn’t have a definitive launch date,
numerous researchers remain optimistic about the expansive capabilities that
space-based detectors could potentially offer. For example, one primary
objective of LISA is to identify a stochastic gravitational wave background.
This background not only provides insights into our early universe right after
the Big Bang, but may also reveal key details of other astrophysical sources
[29]. Additional information we may obtain from LISA include supernovae
processes, galaxy formation, and the nature of dark matter/dark energy [28].

####  IV.3.2 Other proposed space-based missions

In addition to LISA, other notable proposed space-based detectors include
DECIGO (DECI-hertz Interferometer Gravitational-wave Observatory) and BBO
(Big-Bang Observer). These detectors also primarily focus in the 0.1 - 1 Hz
range and would observe primordial gravitational wave background and neutron
stars/black hole mergers, similar to LISA [30].

In the current preliminary design for both DECIGO and BBO, four clusters of
satellites, each comprising of three spacecraft, will orbit the Sun with a
sidereal year period. Similar to LISA, each spacecraft within a cluster will
be connected by lasers. Two of these clusters are positioned closely together,
while the other two are extensively separated along Earth’s orbit. This
arrangement allows for increased sensitivity to the stochastic GW background
and improves our ability to identify the location of neutron star binaries.
Researchers predict up to $10^{5}$ \- $10^{6}$ neutron star binaries could be
observed per year with these space-based detectors [30] [31].

Figure 8: Snapshot taken Cutler and Holz 2009 [31] portraying the preliminary
design for BBO. DECIGO would follow an identical composition, except the
“arms” would be 50 times shorter than BBO’s.

Regrettably, considerable effort and funding are still required before the
launch of these missions. While the potential for groundbreaking technology to
yield vast amounts of new information is promising, the journey toward
achieving this goal is expected to encounter several challenges along the way.

##  V Recent Breakthrough: GW 190521

In this last subsection, we delve into the remarkable scientific breakthrough
represented by the gravitational wave event GW190521. This groundbreaking
discovery, made possible by advanced gravitational wave observatories, was
detected on May 21, 2019 by both LIGO and Virgo. Most notably, GW190521 was
caused by the massive merger of two black holes: one at $\sim$85 solar-masses
and the other at $\sim$66 solar-masses. If true, this would be the most
massive black hole merger observed by our detectors to date [32].

In addition to GW190521 being an extremely massive black hole merger, it also
challenges our understanding of how black holes form. Currently, there’s a
predicted “mass gap” between 65 $\textup{M}_{\odot}$ and 135
$\textup{M}_{\odot}$ for black holes formed from dying stars [33]. Stars
within this mass range are thought to undergo pair instability at the end of
their life, an unstable process in which a star leaves no remnants behind
after death [34]. Although it is still possible for black holes within this
mass range to form (such as through the gradual merging of smaller black holes
or theories beyond the Standard Model), GW190521 provides an interesting
avenue to test such hypotheses [33].

Furthermore, another intriguing event may be associated with GW190521.
Observations by the Zwicky Transient Facility (ZTF) regarding this event noted
a flash of light, a completely unexpected result as the merger of two black
holes are typically invisible to light-detecting telescopes. This suggests one
of two possibilities: the black hole merger event occurred near a third
supermassive black hole accretion disk or it may just be the result of another
astrophysical process. If it is the first possibility, Graham et al. provides
an explanation of how a burst of light could occur. After black holes merge,
they “kick” off in a random direction. If this were to happen near an
accretion disk of a supermassive black hole, the newly formed black hole
shoots through the cloud of gas and dust, releasing an electromagnetic flare
[35].

The total energy transmitted to the accretion disk $E_{b}$ can be described in
Equation (5) from Graham et al. [35].

$\displaystyle E_{b}=1/2M_{b}v_{k}^{2}=3/2Nk_{B}T_{b}$ (5)

$M_{b}$ = $Nm_{H}$ represents the mass of the bound gas (with N defined as the
number of atoms of hydrogen with mass $m_{H}$), $v_{k}$ is the “kick” velocity
of the merged black holes, $k_{B}$ is the Boltzmann constant, and $T_{b}$
signifies the average temperature of the gas “post-shock”. Furthermore, we can
also estimate the luminosity associated with the accretion disk as the
Bondi–Hoyle–Lyttleton (BHL) luminosity $L_{BHL}$ as seen in Equation (6) where
$\eta$ is the radiative efficiency [35].

$\displaystyle L_{BHL}=\eta\dot{M}_{BHL}c^{2}$ (6)

$\displaystyle\dot{M}_{BHL}=\frac{4\pi G^{2}M_{BBH}^{2}\rho}{v_{rel}^{3}}$ (7)

In Equation (7), we calculate for the mass accretion rate $\dot{M}_{BHL}$.
$M_{BBH}$ is the mass of the binary black hole merger with $\rho$ representing
the gas density. $v_{rel}$ is the relative velocity which is equal to
$v_{k}+c_{s}$ with $c_{s}$ as the gas sound speed [35].

For a more technical discussion regarding GW190521, we highly recommend the
following papers [34][36][37].

##  VI Conclusion

In conclusion, the journey through the exploration of gravitational waves has
been nothing short of extraordinary. The initial notion of gravitational waves
is attributed to William Kingdon Clifford in 1876, but it was Henri Poincaré
in 1905 who explicitly postulated their existence. The concept gained momentum
with Einstein’s formulation of general relativity, but even Einstein himself
was initially skeptical regarding their existence. In 1937, he revised his
stance after discovering mathematical solutions for gravitational waves with
help from his assistant, Leopold Infeld. The pursuit of experimental
verification began in earnest in 1957, leading to Joseph Weber’s pioneering
efforts with a gravitational wave detector in 1966. Despite Weber’s claims of
gravitational waves detections in 1969, subsequent detectors worldwide in the
mid-1970s found no evidence of gravitational waves, fueling a renewed
commitment to their discovery. By the 21st century, advanced detectors like
LIGO, Virgo, and GEO600 emerged, marking a new era in gravitational wave
astronomy and culminating in the first historic detection, GW150914.

We then discuss the profound scientific significance of gravitational waves,
confirming Einstein’s general relativity in the relativistic and strong-field
regime with complex tests, such as residual and parameterized tests.
Additionally, gravitational waves provide insights into black hole and neutron
star mergers, shedding light on primordial black hole formation and the
dynamic processes within neutron stars.

The landscape of gravitational wave detection techniques includes
interferometer-based detectors, pulsar timing arrays, and space-based
detectors, each offering unique perspectives and capabilities. Interferometer-
based detectors, exemplified by LIGO, Virgo, and GEO600, utilize lasers and
sensitive instruments to measure minute changes in spacetime caused by passing
gravitational waves. PTAs, such as EPTA, NANOGrav, and PPTA, employ the
precise timing of pulsar signals to detect low-frequency gravitational waves.
LISA, a space-based detector, also aims to explore the low frequency spectrum
(although not as low as PTAs), offering unique observational advantages. The
LISA mission, along with proposed detectors like DECIGO and BBO, presents an
opportunity to study diverse astrophysical phenomena, but many challenges and
substantial efforts lie ahead in realizing the full potential of these
detection methods.

The gravitational wave event GW190521, detected by LIGO and Virgo
observatories in 2019, marked a significant breakthrough in astrophysics. The
merger involved two black holes, one approximately 85 solar masses and the
other around 66 solar masses, representing the most massive black hole merger
ever recorded by our detectors. This discovery challenges existing models of
black hole formation, particularly the predicted “mass gap” between 65 and 135
solar masses. Furthermore, the unexpected observation of a flash of light
associated with GW190521 by the ZTF introduces intriguing possibilities,
suggesting the merger either occurred near a third supermassive black hole
accretion disk or involved an unforeseen astrophysical process. This
unexpected light emission, if associated with the merger, raises questions
about the dynamics of black hole interactions and their potential impact on
surrounding environments.

In essence, the study of gravitational waves has not only solidified our grasp
of fundamental physics but has also ushered in a new era of discovery. As we
continue to probe the universe with ever-improving technologies and
methodologies, the future promises even more revelations and a deeper
understanding of spacetime itself.

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