Detecting extreme-mass-ratio inspirals for space-borne detectors with deep learning

The Taiji detector is a space-based observatory that consists of three spacecrafts arranged in a triangular configuration. It uses lasers to measure distances between these spacecrafts, allowing it to detect gravitational waves in a frequency range of $10^{-4}$ Hz to $10^{-1}$ Hz. Currently, we utilize a simplified Keplerian orbit model to approximate the trajectories of the Taiji spacecrafts (SCs). It’s essential to note that these orbits are basic and do not capture the full complexity of actual orbital dynamics. The depiction of the orbital motion for each Taiji spacecraft (SCn, where n = 1, 2, 3) is outlined below:

[67]. The coordinates are defined within the solar system barycentric (SSB) frame. The angles $\beta_{n}$ are given by $(n-1)\frac{2\pi}{3}+\lambda$, and $\alpha(t)$ is defined as $\frac{2\pi}{1\text{ year }}t+\kappa$.

Initially, both $\lambda$ and $\kappa$ are set to zero. ‘a’ is 1 astronomical unit (AU), and the orbital eccentricity ‘e’ is calculated as $e=L/(2a\sqrt{3})$, where ‘L’ corresponds to the distance between two spacecrafts. The distance designed for the Taiji mission is 3 kilometers [25].

In the practical context of Taiji detection, we must take into account various factors that contribute to noise. To streamline our simulation process, we’ve opted for a simplified noise model consisting of two distinct components. The first component, denoted as $P_{\mathrm{oms}}(f)$, represents high-frequency noise originating from the optical metrology system. The second component addresses low-frequency noise, $P_{\mathrm{acc}}(f)$, resulting from the test mass’s acceleration. This noise can be effectively characterized through its power spectral density.

The PSD of the X channel for second-generation TDI can be mathematically expressed as follows [38]:

where $\omega=2\pi f/c$. The functions $P_{\mathrm{oms}}(f)$ and $P_{\mathrm{acc}}(f)$ can be described as follows:

In missions for a space-based detector, Time Delay Interferometry (TDI) plays a crucial role, serving as a vital technique to efficiently suppress laser frequency noise and attain the desired sensitivity levels. TDI’s fundamental principle involves precise time shifting and combination of measurements to create an interferometry with equal-arm configuration. First-generation TDI combinations excel in mitigating laser frequency noise in static unequal-arm configurations, while second-generation TDI combinations extend this noise reduction capability to scenarios involving relative motion. In this paper, we exclusively employ the second-generation TDI configuration. The Michelson combination in our paper is [68, 38]:

The variables Y and Z are obtained through cyclic permutation of the indices. An uncorrelated set of TDI variables, denoted as A, E, and T, can be derived from linear combinations of X, Y, and Z, as described by  [69]:

In this paper, all simulation data are processed through TDI, and we will specifically use the A and E channel data for analysis. We use fastlisaresponse(https://github.com/mikekatz04/lisa-on-gpu/tree/master) to generate the data after TDI in this work [70].

The Taiji gravitational wave detector possesses the capability to identify EMRI events involving a compact object (CO) and a massive black hole, covering a mass range from $10^{4}M\odot$ to $10^{7}M\odot$. The probability of detection increases as the system achieves a signal-to-noise ratio exceeding 20. Before the detection of EMRI signals, it is essential to have theoretical waveform models for EMRI.

In the analysis of EMRI events, researchers require rapid and precise waveform models. These models should be able to handle sources with various parameters. The AK model [71], while computationally efficient, simplifies gravitational wave radiation and provides only an estimation of strong field dynamics, deviating from the genuine signal. To address this, Ref [72] explored the AKK model, which extends fading effects until the frequency reaches a stable Kerr orbit [73]. The AAK waveform strikes a balance between accuracy and computational efficiency, enhancing our ability to model EMRI events effectively.

Assuming the spin of the compact object is negligible, an EMRI can be described by 14 parameters. 7 is the intrinsic parameters :$\left(\mu,M,a,e_{0},\iota_{0},\gamma_{0},\psi_{0}\right)$. $M$ is the mass of the massive black hole, $\mu$ is the mass of the CO. ‘a’ is spin parameter of the MBH, ‘p’ is the semi-latus rectum, ‘e’ is eccentricity, $\iota$ is the orbit’s inclination angle from the equatorial plane. $\gamma:=\tilde{\gamma}-\beta$ and $\beta(\hat{\mathbf{R}},\hat{\mathbf{S}},\hat{\mathbf{L}})=\beta\left(\theta_{S},\phi_{S},\theta_{K},\phi_{K},\iota,\alpha\right)$ is an azimuthal angle in the orbital plane. True anomaly $\psi$ is the angle that describes an object’s position in an elliptical orbit relative to the central body and the periapsis. Rest parameters are the extrinsic parameters: $\left(p_{0},\theta_{S},\phi_{S},\theta_{K},\phi_{K},\alpha_{0},D\right)$. ‘p’ is the semi-latus rectum, $\theta_{S}$ and $\theta_{K}$ are the polar and azimuthal sky location angles.  $\theta_{K}$ and $\phi_{K}$ are the azimuthal and polar angles describing the orientation of the spin angular momentum vector of the MBH. The time dependence of the orbital orientation is confined to $\alpha~{}(t)$, $D$ is the luminosity distance [72]. The EMRI signals in the training data for this study have all been generated using the AAK method through the EMRI_Kludge_Suite [72].

In order to employ a Convolutional Neural Network for detecting EMRI signals within the Taiji detector data, we start by dividing the dataset into two groups, each comprising half of the samples. One of these sets contains data encompassing both signals and noise ($d=h+n$) and is assigned the label 1, whereas the other set exclusively contains noise ($n$) and is assigned the label 0. Subsequently, the dataset is further divided into two sets: training data and testing data. The training data is employed for model training, while the testing data serves as an evaluation for the trained CNN.

We make use of the AAK model to create EMRI signals, with the range of the parameters specified in Table 1. The signals extend for a duration of 0.5 years and are sampled every 10 seconds ($dt$). Following this, we apply the analytical Power Spectral Density to simulate noise, ensuring it aligns with the generated signal, which leads to the generation of both datasets $d$ and $n$.The total number of data generated for both $d$ and $n$ amounts to 6000.

Subsequently, we incorporate Time Delay Interferometry (TDI) into the dataset, getting the A, E, and T channel data. For our training and testing purposes, we select data from the A and E channels. The datasize is (2,1560000) for both $d$ and $n$ . After TDI, we proceed to compute the Signal-to-Noise Ratio values, carefully maintaining them in the certain range.

We define the characteristic strain $h_{\mathfrak{c}}$ and noise amplitude $h_{n}$ as follows [74]:

So that we can express the SNR in equation 7. We utilize $h_{\mathfrak{c}}(f)$ and $h(f)$ in the SNR calculation. When these values are visualized using a logarithmic scale, the area between the source and detector curves represents the SNR. This SNR value indicates the detectability of the source. We consider an EMRI signal to be a “golden” EMRI when its SNR exceeds 50. Consequently, we adjust the distance to ensure that the SNR remains within the range of (50, 100). We use $\varrho$ to represent the SNR value in each channel:

When taking two channels into account, we get the target SNR using the following equation:

Time-frequency analysis has proven to be highly effective in the EMRI data analysis. In Ref. [52], a time-frequency plot depicting a typical EMRI signal, wherein distinct frequency tracks clearly reveal the presence of dominant harmonics. The EMRI signals show a marked concentration of frequency components within these tracks, displaying substantial intensity. Therefore, we have chosen to employ the Constant Q Transform (CQT) for data preprocessing in this deep learning project. CQT is the preferred choice due to its advantages in this particular endeavor.

In signal processing, time-frequency analysis encompasses techniques that examine a signal in both the time and frequency domains simultaneously. The use of time-frequency analysis offers several key advantages. Firstly, it allows for the adjustment of the trade-off between time and frequency resolution, providing greater flexibility in resolution settings. Additionally, time-frequency representations can extract relevant features crucial for deep learning and classification tasks. In comparison to time series analysis, time-frequency analysis significantly enhances feature extraction capabilities, making it a valuable tool. Time-frequency analysis techniques encompass a variety of methods such as Short-Time Fourier Transform (STFT), Wavelet Transform, Continuous Wavelet Transform (CWT), and more.

In our research, we have decided to employ the Constant Q Transform (CQT) for data processing in our deep learning project. We have chosen CQT over the Short Time Fourier Transform (STFT) for several reasons. Firstly, CQT offers adaptable time and frequency resolution, unlike STFT, which has a fixed resolution determined by window size and window overlap. Generally, CQT provides finer frequency details for higher frequencies and vice versa. Secondly, CQT’s flexible time and frequency resolution, along with its exceptional ability to accurately pinpoint both time and frequency information, makes it the preferred choice for visualizing and analyzing signals like EMRI, especially when compared to STFT.

CQT is a special case of Variable Q transform(VQT). Both of them are related to complex Morlet wavelet transform. The definition is

where $\delta f_{k}$ is the bandwidth of the k-th filter, $f_{min}$ is the central frequency of the lowest filter, and n is the number of filters per octave (https://en.wikipedia.org/wiki/Constant-Q_transform).

Define the quality factor Q for CQT:

With this factor, we can define window length:

Hamming window is one of the most commonly used window, take this for example:

With things above, define Q transform as:

We ues librosa(https://librosa.org/doc/latest/index.html) to do the process. In Fig. 3(a), we see the signal after the Q-transform. The EMRI signal is strong, with a high SNR of 100. In Figure 3(b), we can observe the signal amidst the background noise of the detector. It’s noticeable that only a few of its features remain preserved after adding noise. In Figure 4, we can see the pure noise denoted as $n$, and the data $d$ that includes both the signal and the noise after undergoing the Q-transform. In Figure 4(b), where the signal-to-noise ratio is 50, the distinctive features of the EMRI are obscured by the noise, making it difficult to distinguish between Figure 4(a) and Figure 4(b) by visual inspection alone. However, employing convolutional neural networks (CNN) can make this differentiation quite easy.

We store the Q-transform results as RGB images and resize them to (255,255) matrixs to use them as input for the CNN network.

This study adopts the basic framework of a CNN network architecture, which includes Conv2d layers with filter sizes of 16 and 32, followed by two fully connected layers with sizes of 128 and 64. Standard ReLU activation functions are employed as non-linearities between layers, and the convolutional layers have kernel sizes of $3\times 3$ and $3\times 3$. The network incorporates max-pooling layers. We uses a stride of 1 for convolution layers and a stride of 2 for pooling layers. Specific details about the network can be found in Table 2.

In our approach, we utilize the cross-entropy loss function, which is a widely adopted choice in classification tasks due to its effectiveness in speeding up the convergence of deep learning models. This loss function quantifies the difference between predicted probabilities and actual labels, encouraging the model to improve its predictive accuracy. The mathematical expression for this loss function is as follows:

Specifically, for our binary classification task, which involves distinguishing EMRI signals from noise, the cross-entropy loss measures the discrepancy between the predicted probabilities of EMRI and the actual EMRI labels in the training data. By minimizing this loss, our model aims to align its predictions with the true EMRI labels, ultimately enhancing its ability to classify EMRI signals accurately and reliably.

Additionally, we implement the Adam optimizer with a learning rate equal to 0.01 to update the model’s weights during training. Over the course of 200 training epochs for training data, we monitor both loss and accuracy, periodically saving the model with the lowest loss after the convergence as our best performing model. Then we test the saved model on test data.

In this section, we use receiver operating characteristics (ROC) to see how well our CNN model can find the signal in our testing data. The ROC curve is a visual tool that helps us see how well our model can find the signal in our data without messing up. The ROC curve serves as a graphical representation that provides insights into our model’s performance. It focuses on two critical aspects: the true positive rate, indicating how frequently the model correctly identifies the signal, and the false positive rate, highlighting instances where the model incorrectly detects a signal when none exists. To provide an overarching assessment of our model’s capabilities, we rely on the Area Under the ROC Curve (AUC). AUC is like a score which can helps us see how well our model is doing its job. A higher AUC value, closer to 1, signifies superior performance, indicating that the model excels in detecting the signal with precision.

In Figure 5, there are two separate ROC curves, one in (a) and the other in (b). These curves represent different SNR values in the test data. In (a), the SNR values range from 50 to 100, while in (b), it’s a constant value of 50. The AUC values shown in the figures give us a numerical measure of how good the model is at accurately spotting EMRI signals. To make it easier to see the detection performance, especially at low false positive rates, there’s a smaller inset figure inside Fig. 5 that zooms in on the TPR from 0 to 0.01. This helps us understand how well our CNN network performs in detecting EMRI signals across different SNR values.