Issue 
A&A
Volume 682, February 2024



Article Number  A177  
Number of page(s)  10  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/202347221  
Published online  21 February 2024 
Deep learning forecasts of cosmic acceleration parameters from DECihertz Interferometer Gravitationalwave Observatory
^{1}
College of Physics, Chongqing University, Chongqing 401331, PR China
email: cqujinli1983@cqu.edu.cn
^{2}
Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, PR China
email: caoshuo@bnu.edu.cn
^{3}
Department of Astronomy, Beijing Normal University, Beijing 100875, PR China
Received:
18
June
2023
Accepted:
17
November
2023
Context. Validating the accelerating expansion of the universe is an important aspect in improving our understanding of the evolution of the universe. By constraining the cosmic acceleration parameter X_{H}, we can discriminate between the cosmological constant plus cold dark matter (ΛCDM) model and the Lemaître–Tolman–Bondi (LTB) model.
Aims. In this paper, we explore the possibility of constraining the cosmic acceleration parameter with the inspiral gravitational waveform of neutron star binaries (NSBs) in the frequency range of 0.1 Hz–10 Hz, which can be detected by the secondgeneration spacebased gravitational wave detector DECIGO.
Methods. We used a convolutional neural network (CNN) and a long shortterm memory (LSTM) network combined with a gated recurrent unit (GRU), along with a Fisher information matrix to derive constraints on the cosmic acceleration parameter, X_{H}.
Results. We assumed that our networks estimate the cosmic acceleration parameter without biases (the expected value of the estimation is equal to the true value). Under this assumption, based on the simulated gravitational wave data with a time duration of one month, we conclude that CNN can limit the relative error to 15.71%, while LSTM network combined with GRU can limit the relative error to 14.14%. Additionally, using a Fisher information matrix for gravitational wave data with a fiveyear observation can limit the relative error to 32.94%.
Conclusions. Under the assumption of an unbiased estimation, the neural networks can offer a highprecision estimation of the cosmic acceleration parameter at different redshifts. Therefore, DECIGO is expected to provide direct measurements of the acceleration of the universe by observing the chirp signals of coalescing binary neutron stars.
Key words: methods: statistical / binaries: close / stars: neutron / cosmological parameters / dark matter / dark energy
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
The verification of the acceleration of the cosmic expansion is a crucial subject in current astrophysical research. Cosmic acceleration refers to the phenomenon of the universe expanding at an increasingly fast rate and measuring its acceleration is essential for determining the ultimate fate of the universe. Observing the typeIa Supernovae has provided evidence to support the accelerating expansion of the universe, focusing on a distinctive correlation between the luminosity and distance of supernovae within a specific distance range (Riess et al. 1998; Aldering et al. 1998). Such a conclusion was further verified by the observations of cosmic microwave background radiation (de Bernardis et al. 2000) based on the results from Wilkinson Microwave Anisotropy Probe (WMAP) program (Spergel et al. 2003) and Planck Collaboration (Planck Collaboration VI 2020). The analysis of various observational data, including baryon acoustic oscillation, Hubble parameters derived from passively evolving galaxies (Cao et al. 2011a, 2015c; Cao & Liang 2013), strong gravitational lensing systems (Cao & Zhu 2012, 2014; Cao et al. 2012a,b, 2015a), and quasars calibrated as standard rulers and standard candles (Cao et al. 2015b, 2017a,b, 2018) has also suggested that the present universe is undergoing an accelerated phase of expansion. We refer to Yang & Gong (2020), Yoo et al. (2008) for the summary of recent observational progress made on such issue. Currently, several theoretical models have been proposed to explain the expansion of the universe. Dark energy and modified gravity theories are widely accepted as explanations for these observational facts. Dark energy’s significant influence on the evolution of the universe has played a pivotal role in its widespread acceptance within the scientific community. However, assuming that we are at the center of the universe and that the universe is no longer isotropic on large scales and becomes nonuniform can explain the current observational results without using a dark energy model or modifying gravity theory (Alnes et al. 2006; Yoo et al. 2010; GarciaBellido & Haugboelle 2008a,b, 2009; Bolejko & Wyithe 2009; Caldwell & Stebbins 2008; Clarkson & Regis 2011; Zhang & Stebbins 2011; Zibin et al. 2008; Cao et al. 2011b). Nevertheless, such an assumption clearly violates the Copernican principle. Therefore, directly detecting the acceleration of the universe’s expansion is an important means of verifying the current mainstream theory.
According to the standard cosmological model, the acceleration of the universe’s expansion leads to redshift drift, a crucial phenomenon that provides valuable information for understanding the evolution of the universe. Several methods can be used to observe redshift drift. One method involves using type Ia Supernovae as standard candles (Perlmutter et al. 1997; Riess et al. 1998). Another method involves using observations and calculations of cosmic microwave background radiation (Abazajian et al. 2004; Tegmark et al. 2004). In this study, we analyze the cosmic acceleration parameter by examining the redshift evolution resulting from cosmic accelerating expansion and the corresponding phase shift of gravitational waves. Such a phase shift caused by the cosmic acceleration was demonstrated by Seto et al. (2001), through a decade of observing gravitational wave signals generated by binary neutron stars. In this analysis, we chose the sensitive frequency range of 0.1 Hz–10 Hz for the DECIGO (Decihertz Interferometer Gravitationalwave Observatory) detector to capture gravitational waveforms generated during the inspiral phase of binary neutron stars. Yagi et al. (2012) used the covariance matrix derived from the Fisher information matrix to estimate the uncertainty of cosmic acceleration parameter. However, we should note that the phase shift in the lowfrequency part of the waveform can be easily overwhelmed by complex noise backgrounds. Therefore, a highprecision parameter estimation method is still required to obtain accurate results. Luckily, the deployment of machine learning algorithms in astronomy has demonstrated its efficacy in accelerating data processing and improving statistical inference. Specially, deep learning in gravitational wave data analysis is becoming crucial to quick and accurate estimation of parameters of interest (McLeod et al. 2022). In this paper we discuss the ability of different deep learning algorithms, a convolutional neural network (CNN), and a long shortterm memory (LSTM) network combined with a gated recurrent unit (GRU) to measure the cosmic acceleration parameters, based on the timeseries waveform of gravitational wave during the binary neutron star inspiral phase in DECIGO. Our results reveal that deep learning is able to provide measurements of the cosmic acceleration parameter with highprecision.
The paper is structured as follows. Section 2 is dedicated to the framework of the GW simulations produced for our analysis. Section 3 explains how the estimation of the cosmic acceleration parameter is performed, along with the deep learning results with different deep learning methods. Section 4 presents the results with Fisher information matrix for comparison. Finally, our main conclusions and final remarks are presented in Sect. 5.
2. Data simulation
2.1. Simulation of a gravitational wave signal
In this paper, we use natural units with c = G = 1 and select binary neutron stars as our sources of gravitational waves. Our simulation is based on the flat ΛCDM model, with the matter density parameter, Ω_{M} = 0.3, and the Hubble constant, H_{0} = 70 km s^{−1} Mpc^{−1}. The relation between redshift drift and the cosmic acceleration parameter is parameterized as (Seto et al. 2001): X_{H} = X(z)/H_{0}, and , with the Hubble parameter being .
The gravitational wave waveform in the presence of cosmic acceleration expansion is (Yagi et al. 2012; Seto et al. 2001; Takahashi & Nakamura 2005):
where the acceleration phase Ψ_{acc}(f) is derived from:
the waveform of stationary phase approximation without acceleration is given by (Berti et al. 2005; Cutler & Flanagan 1994):
where x = (πℳ_{z}f)^{2/3}, , ℳ_{z} = M(1+z_{c})η^{3/5} represents the chirp mass with redshift drift, z_{c} represents the redshift taking into account the cosmic acceleration expansion. The chirp mass is defined as ℳ = Mη^{3/5}, with the symmetric mass ratio η = m_{1}m_{2}/M^{2} and the total mass M = m_{1} + m_{2}. In the case of a flat universe, the luminosity distance to the source is:
We used the secondorder standard postNewtonian approximation method to construct the gravitational waveforms (Berti et al. 2005):
in the above phase expression, the first term includes the merger time, t_{c}, the second term includes the phase at the merger, ϕ_{c}, and the factor in the parentheses is the standard phase for quadrupole radiation in general relativity. The terms in the parentheses are the expansion terms in the postNewtonian approximation; then, β≲9.4 and σ≲2.5 represent the contributions from spinorbit coupling and spinspin coupling to the phase, respectively (Yagi et al. 2011).
Due to the timedomain signals detected by DECIGO, timedomain signals are more direct and convenient in terms of representation and processing, without the need for additional transformations or processing steps. Furthermore, timedomain signal processing is typically faster than frequencydomain signal processing, which is crucial for handling a large volume of gravitational wave data. In addition, the direct information contained in timedomain signals includes important features and information of gravitational wave signals, such as duration, amplitude, phase, and shape, which are vital for the identification and classification of gravitational wave signals. Moreover, preprocessing and data processing of timedomain signals are relatively simple, allowing the utilization of various filtering techniques, noise reduction methods, and data cleaning techniques to improve the signaltonoise ratio (S/N) and extract useful features. It is necessary to impose an inverse Fourier transform on the frequencydomain gravitational wave to the timedomain. To simplify the calculations, we employed the average response function R_{DECIGO} of the DECIGO detector (Zhang et al. 2020) as:
where , L = 1000 km is the arm length of the DECIGO detector (Kawamura et al. 2006) and c is the speed of light. Considering that our frequencydomain data are obtained from numerical simulations, we use the 1D discrete inverse Fourier transform,
where h(t) is the GW strain in the timedomain signal, N is the sample number, and is the GW strain in the frequencydomain, R_{DECIGO}(f) represents the average response function, reflecting the detector’s sensitivity to signals in different frequencies.
2.2. Simulation of noise data
The onesided noise power spectral density for DECIGO is given by (Kawamura et al. 2011; Yagi & Seto 2011):
where f_{p} = 7.36 Hz. The three terms correspond to short noise, radiation pressure noise, and acceleration noise, respectively. Additionally, we need to consider confusion noise from the galactic and extragalactic white dwarfs (Nelemans et al. 2001; Farmer & Phinney 2003):
the noise from the galactic and extragalactic white dwarfs is multiplied by a factor of ℱ = exp{−2(f/0.05 Hz)^{2}}, which corresponds to the highfrequency cutoff. We also considered the superposition of gravitational waves from many binary neutron star systems, which contribute to the background noise as (Yagi & Seto 2011; Cutler & Harms 2006):
Given the overall noise sources, we could obtain the unilateral noise power spectral density (PSD) of DECIGO as (Yagi et al. 2011):
where κ ≃ 4.5, T_{obs} is the observation time and dN/df is the number density of white dwarfs in the galactic per unit frequency:
The factor of 0.01 in represents the proportion of neutron star noise that cannot be removed. The sensitive frequency range of DECIGO is from f_{min} = 10^{−3} Hz to f_{max} = 100 Hz. In Fig. 1, we show the noise spectral density of different GW detectors, from which one could see that gravitational waves generated by binary neutron stars falls within the observation range of DECIGO instead of LISA.
Fig. 1. Sensitivity curves for DECIGO (blue line) and LISA (orange line), as well as the gravitational wave amplitude of a binary neutron star (1.4 + 1.4) M_{⊙} at a luminosity distance of D_{L} = 3Gpc and a frequency range of 10^{−5} Hz ∼ 100 Hz (green line). The arrows on the graph indicate that when observing at the same highfrequency of 1 Hz, the lowest observational frequency differs depending on the observation time. Specifically, for an observation time of T_{obs} = 5 years, the lowest observational frequency is f_{min} = 0.073Hz. For an observation time of T_{obs} = 1 year, the lowest observational frequency is f_{min} = 0.133 Hz, and for an observation time of T_{obs} = 1 month, the lowest observational frequency is f_{min} = 0.4036 Hz. 
Based on the onesided noise power spectral density (PSD), we could obtain the timedomain noise signal from the onesided PSD; in this paper, we utilized the Python function pycbc.noise.gaussian.noise_from_psd (Nitz et al. 2023), which takes a PSD as input and returns colored Gaussian noise, to simulate the timedomain noise received by the DECIGO detector.
2.3. Distribution of BNS and numerical settings
For the probability density function of the distribution of neutron stars, we adopted the following form (Zhao et al. 2011):
where the comoving distance is and the evolution of the inflation rate with time is quantified as (Schneider et al. 2001; Cutler & Holz 2009; Cai & Yang 2017):
the above probability density function is normalized as:
with a normalization factor of . The distribution function of redshift is given by:
with the probability density function of redshift ρ(z) and the distribution function P(z) are shown in Fig. 2a. We divide the redshift range into 20 intervals according to the distribution function P(z) and randomly select 1000 redshift values that fall into these intervals. The resulting distribution of 1000 GW sources, based on the distribution function P(z) are presented in Fig. 2b.
Fig. 2. Redshift distribution function and the samples generated according to this distribution. Panel a: probability density function and distribution function of redshift. Panel b: redshift distribution of 1000 sources. The xaxis and yaxis denote the range of each interval and the number of GW sources in each interval. 
In our simulation, we set the masses of neutron stars to m_{1} = m_{2} = 1.4 M_{⊙}, the merger time to t_{c} = 0, the coalescence phase ϕ_{c} = 0, and the angles to β = σ = 0. Moreover, we do not include the effects of the spins of the merging bodies (S = 0). The high and low frequencies of the gravitational waves and noise are determined as f_{fin} = (f_{ISCO},f_{end}) and f_{in} = max(f_{obs},f_{min}). Here f_{ISCO} = (6^{3/2}πM)^{−1}, and f_{end} = 1 Hz, within the sensitive range of the DECIGO detector. is the lowest frequency that can be observed during the corresponding observation time Δt_{0} = 1 month (Feng et al. 2019). At the luminosity distance of D_{L} = 3 Gpc, the lowest observational frequency is 0.4036 Hz. Therefore, the frequency range of signal and noise is set as 0.4036 ∼ 1 Hz, as shown in Fig. 3.
Fig. 3. Frequencydomain amplitude of gravitational wave signal observed for 1 month (yellow line) and onesided noise power spectral density of DECIGO (blue line), with a frequency range of 0.4036 ∼ 1 Hz. The binary neutron star with the mass of (1.4 + 1.4) M_{⊙} is located at a luminosity distance of D_{L} = 3 Gpc. 
3. Estimating the acceleration parameters with deep learning
3.1. Construction of the dataset
We generated 1000 samples of gravitational wave data covering the redshift range of 0 < z < 2. The actual input to the network is the feature amplitude extracted from the timedomain data s(t) = h(t)+n(t), using a singlelayer convolutional neural network, where h(t) is the inverse Fourier transform of and n(t) is the timedomain noise. The sampling frequency of the data is taken as 2 Hz, the sampling time is one month and the data length is 1 × 5 184 000. However, due to the observation time, resulting in an excessive data length, the total size of our dataset comprising 1000 samples amounts to 38.624 gigabytes (GBs). That leads to significant challenges for training neural networks. Therefore, we employed a 1D convolutional neural network with a single layer for feature extraction from our raw data. Table 1 represents the feature extraction network architecture. Following feature extraction, each sample has a length of 1 × 2000. In Fig. 4b, we illustrate one sample after feature extraction. We divide the 1000 samples into training and testing dataset, and our testing subset contains 30% of the original sample size. Then we deploy different machine learning algorithms on the training test. The input data are (x_train, y_train), (x_test, y_test), where x_train and x_test are timedomain data in the training and testing datasets. y_train and y_test are the parameters space here is (ln𝒜, t_{c}, ϕ_{c}, lnℳ, lnη, lnℳ_{z}, X_{H}) that the neural network needs as labels. . The S/N values of these 1000 data are shown in Fig. 5.
Fig. 4. Panel a: a timedomain data sample. Noise represented as the blue curve represents, while the yellow curve shows the gravitational wave (a). The sample’s S/N is 15.739. The observation time for the signal is 1 month, with a frequency range of 0.4036 Hz–1 Hz. The gravitational wave source is a binary neutron star system with a combined mass of (1.4 + 1.4) M_{⊙} and a luminosity distance of D_{L} = 3 Gpc. Other parameters are set as described in Sect. 2.3. Panel b: corresponding feature of (a), obtained through a singlelayer convolutional neural network convoluting with raw data. The horizontal axis of the (b) represents the data length, while the vertical axis represents the normalized feature amplitude. 
Fig. 5. Variation in the S/N with different observation time. The blue line represents a 5year observation, the green line represents a 1year observation, and the yellow line represents a 1month observation. 
Feature extraction network architecture.
Now our analysis will be carried out on all simulated and datasets with two deep learning algorithms (CNN and LSTMGRU), as well as the Fisher information matrix estimation.
3.2. Convolutional neural networks
Convolutional neural networks (CNN) offer several advantages for parameter estimation in the analysis of timedomain gravitational wave data. Firstly, a CNN enables automatic feature learning, alleviating the need for manual design of feature extractors (Christensen & Meyer 2022). Secondly, it possesses local perception capabilities, allowing it to extract features at different positions within the input data through convolutional filters (Edwards 2021). This is crucial for capturing local structures and temporal features in timedomain gravitational wave data, facilitating effective discrimination of different gravitational wave signals. Third, CNN employs multiple layers for hierarchical feature representation by stacking convolutional and pooling layers. This progressive learning enables the network to capture abstract features in timedomain gravitational wave data, enhancing the accuracy of parameter estimation. Fourth, CNN exhibits robustness and generalization abilities, acquired through training on largescale datasets. It demonstrates resilience against noise and nonideal conditions, which is vital for accurate parameter estimation in the presence of noise and interference in real detectors (George & Huerta 2018). Lastly, CNN excels in handling largescale datasets, a critical advantage for processing the extensive gravitational wave data (Dreissigacker et al. 2019) expected from the nextgeneration DECIGO detector. As DECIGO is planned to produce hightemporalresolution data, efficient data processing methods are essential and CNN is capable of effectively managing largescale data. Collectively, these advantages establish CNN as a powerful tool for parameter estimation in analysis of DECIGO’s timedomain gravitational wave data. Based on these characteristics, we chose a 1D CNN model for parameter estimation of gravitational wave signals, with our network structure shown in Table 2.
CNN architecture and hyperparameter settings.
3.3. LSTMGRU hybrid network
The LSTMGRU hybrid network provides numerous advantages for parameter estimation of timedomain gravitational wave data obtained from the DECIGO detector. Specifically designed to handle sequential data, long shortterm memory (LSTM) and gated recurrent unit (GRU) are variants of recurrent neural networks (RNN) that effectively capture the longterm dependencies often observed in DECIGO gravitational wave data. The LSTMGRU hybrid network demonstrates robustness and generalization capabilities, making it resilient against noise and incomplete training data (Cahuantzi et al. 2021; Elsayed et al. 2019). Given that DECIGO gravitational wave data may contain measurement errors or missing information, the hybrid network leverages its gating mechanisms and memory units to capture crucial features, enabling reliable parameter estimations even in the presence of such challenges. Additionally, the network exhibits strong generalization ability, allowing accurate estimation of parameters for unseen data samples. Moreover, the LSTMGRU hybrid network excels in capturing both local and global information. While LSTM captures longterm dependencies by utilizing gating mechanisms and memory cells, GRU can swiftly capture shortterm local patterns through its update and reset gates. This combination facilitates comprehensive understanding and modeling of DECIGO timedomain gravitational wave data by enabling the network to focus on both local and global features. Furthermore, The LSTMGRU hybrid network’s nonlinear activation functions and gating mechanisms contribute to its powerful nonlinear modeling capabilities (Lewis & Stevens 1991). Traditional linear models often fail to accurately capture the complex nonlinear relationships present in timedomain gravitational wave data. In contrast, the LSTMGRU hybrid network, with its flexibility in capturing nonlinear patterns and complex data features, enhances the accuracy of parameter estimation. In conclusion, the LSTMGRU hybrid network offers a range of advantages, including longterm dependency modeling, longterm memory and forgetting capabilities, robustness and generalization capabilities, the ability to capture both local and global information, and powerful nonlinear modeling capabilities. These attributes make the LSTMGRU hybrid network a valuable tool for parameter estimation of timedomain gravitational wave data acquired from the DECIGO detector. In this paper, we use LSTM and GRU to construct a hybrid model for parameter estimation and prediction. Our LSTMGRU hybrid network structure is shown in Table 3.
Hyperparameters of LSTMGRU hybrid network.
We present the results (under the assumption of unbiased estimation) obtained from the two neural networks in Fig. 6. By comparing the two deep learning methods, we find that the two neural networks perform similarly in terms of parameter estimation in timedomain data. However, due to the fewer trainable parameters in the CNN, the training speed of CNNs is faster. For the task of accelerating parameter estimation with the same dataset, the CNN is more efficient than LSTMGRU.
Fig. 6. Comparison of X_{H} error estimation using CNNs and LSTMGRU hybrid networks, for binary neutron stars with one month of observations. 
4. Estimating the acceleration parameter with a Fisher information matrix
In this section, we describe our use of the Fisher information matrix(FIM) for cosmic acceleration parameter estimation. It is based on the methodology of matched filtering to detect gravitational wave signals using fixed templates. The methodology was firstly proposed in Finn (1992), Kucukelbir et al. (2017). Matched filtering convolves the detector output signal with the gravitational wave template in the library to find the template with the maximum correlation. The parameter information of the detection signal is given by the template with the maximum correlation. However, typically the parameter information obtained for the detection signal is not accurate. If the background noise of the signal is Gaussian, the estimated parameters of the signal will have a Gaussian distribution around the actual parameter values.
We assume that λ^{i} denotes the actual value of the parameter, λ^{i} + Δλ^{i} denotes the measured value, and the rootmeansquare of Δλ^{i} follows a Gaussian distribution: p(Δλ^{i}) ∝ exp(−Γ_{ij}Δλ^{i}Δλ^{j}/2). Here Γ_{ij} is the matrix element of the Fisher information matrix:
where , is the frequencydomain form of the gravitational wave signal, and is the complex conjugate of . The rootmeansquare error of the ith parameter is , where Σ^{ii} = [Γ^{−1}]^{ii} is the covariance matrix element for parameter i. The rootmeansquare error of the ith parameter is . The offdiagonal elements in the Σ matrix represent the size of the correlation between the ith and jth parameters, denoted as c^{ij} and calculated as . The parameter space here is (ln𝒜, t_{c}, ϕ_{c}, lnℳ, lnη, lnℳ_{z}, X_{H}), where , and all parameter settings are given in Sect. 2. The S/N is calculated as:
where N_{int} = 8 represents the number of effective interferometer arms of DECIGO. The S/N values for observation times of 1 month and 5 years are shown in Fig. 5.
It should be noted that the results obtained by our neural networks are based on the assumption of unbiased estimation. Here, ΔX_{H} represents the standard deviation of X_{H} estimated by our model. For our neural networks model, we performed 1000 predictions, resulting in 1000 samples of X_{H} at each redshift. Subsequently, the standard deviation ΔX_{H} was computed using Eq. (21). In contrast, for our Fisher method, we directly derived the standard deviation by calculating the covariance matrix and taking the square root of its diagonal elements.
where x_{i} represents our samples, n = 1000, which is the size of samples.
Given a one month of observations with DECIGO, the estimated results of ΔX_{H} are shown in Table 4. We further studied the estimated uncertainty ΔX_{H} with 5year observation of DECIGO. The results in Table 5 are well consistent with those obtained in the previous works (Yagi et al. 2012). Based on five years of observation, we can essentially estimate the acceleration parameter with reasonable uncertainties, allowing for a rough assessment of the cosmic accelerated expansion. Our finding indicate that increasing the observation time can improve the S/N and thus reduce the uncertainties of X_{H}. Such a conclusion is strongly supported by the results shown in Fig. 7, from which we may also find that the estimated error gradually increases with the luminosity distance.
Fig. 7. Comparison of X_{H} estimation from 1month and 5year observations of binary neutron stars, based on Fisher information matrix. 
Estimation of ΔX_{H} with 1month observation of binary neutron stars at different redshifts, based on Fisher information matrix.
Estimation of ΔX_{H} with 5year observation of binary neutron stars at different redshifts, based on Fisher information matrix.
Finally, Fig. 8 shows the results of both methods. Table 6 provides Estimation of ΔX_{H} using Fisher information matrix with a 5year observation and deep learning with a 1month observation. We can also calculate the relative error and visualize the results in Fig. 9. Based on the simulated gravitational wave data with a time duration of 1 month, the CNN can limit the relative error to 15.71%, while the LSTM network combined with GRU can limit the relative error to 14.14%. Additionally, using Fisher information matrix for gravitational wave data with a 5year observation can limit the relative error to 32.94%. Therefore, the neural networks can give a highprecision estimation of the acceleration parameter at different redshifts. In this case, DECIGO is expected to provide direct measurements of the acceleration of the universe, by observing the chirp signals of coalescing binary neutron stars.
Fig. 8. X_{H} estimation using deep learning and Fisher information matrix. Panel a: X_{H} estimation using deep learning with 1month observation and Fisher information matrix with 5year observation. Panel b: X_{H} estimation using deep learning with 1month observation and Fisher information matrix with 1month and 5year observation. 
Fig. 9. Error estimated by the Fisher matrix for 5year observation data, shown as a green solid line. The orange solid line represents the error value estimated by the LSTMGRU networks for one month of observation data, while the blue solid line represents the error value estimated by the CNN networks for one month of observation data. 
Estimation of ΔX_{H} using Fisher information matrix with a 5year observation and deep learning with a 1month observation.
5. Summary and discussion
In this paper, we explore the possibility of constraining the cosmic acceleration parameters with the inspiral gravitational waveform of neutron star binaries (NSBs) in the frequency range of 0.1 Hz–10 Hz, which can be detected by the secondgenerattion spacebased gravitational wave detector DECIGO. We use a convolutional neural network (CNN), a long shortterm memory (LSTM) network combined with a gated recurrent unit (GRU), and Fisher information matrix to derive constraints on the cosmic acceleration parameter, X_{H}. Under the assumption of unbiased estimation, based on the simulated gravitational wave data with a time duration of 1 month, we conclude that the CNN can limit the relative error to 15.71%, while the LSTM network combined with GRU can limit the relative error to 14.14%. Additionally, using Fisher information matrix for gravitational wave data with a 5year observation can limit the relative error to 32.94%. Therefore, DECIGO is expected to provide an unprecedented opportunity for highprecision detection of cosmic acceleration, by observing the chirp signals of coalescing binary neutron stars (Seto et al. 2001; Kawamura et al. 2021).
We should stress that the present paper is only an interesting example of extensive applications of deep learning in cosmological studies (Li et al. 2021; He et al. 2019; LucieSmith et al. 2020). Still, there are several remarks that remain to be clarified as follows. Firstly, the deep learning models used can be further optimized and enhanced to improve the measurements of cosmic acceleration parameters. We can explore the use of other neural networks or combinations of multiple networks to achieve more stringent cosmological constraints (Wen et al. 2023; Wang et al. 2020; GómezVargas et al. 2023). Combining deep learning methods with other approaches, such as Markov chain Monte Carlo (MCMC) method, could contribute to resolving such an important issue. Secondly, the GW observations provide a powerful and novel method to detect the cosmic acceleration in a cosmologicalmodelindependent way. This strengthens the probative power of such method to inspire new observing programs in the framework of DECIGO, focusing on a large number of neutronstar binaries in inspiraling phases. Finally, in future works, we can apply deep learning methods to constrain the cosmological parameters associated with the selected scientific objectives encompassed by the DECIGO (Cao et al. 2021, 2022b, 2019, 2022a; PiórkowskaKurpas et al. 2021; Zhang et al. 2022; Geng et al. 2020; Liu et al. 2020; Bian et al. 2021; Hou et al. 2022). This will open up a new window for gravitationalwave cosmology.
Acknowledgments
This work is supported by the National Key Research and Development Program of China (Grant No. 2021YFC2203004); the National Natural Science Foundation of China under Grants Nos. 12021003, 12347101, 11690023, and 11920101003; the Natural Science Foundation of Chongqing (Grant No. CSTB2023NSCQMSX0103); the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB23000000; and the Interdiscipline Research Funds of Beijing Normal University.
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All Tables
Estimation of ΔX_{H} with 1month observation of binary neutron stars at different redshifts, based on Fisher information matrix.
Estimation of ΔX_{H} with 5year observation of binary neutron stars at different redshifts, based on Fisher information matrix.
Estimation of ΔX_{H} using Fisher information matrix with a 5year observation and deep learning with a 1month observation.
All Figures
Fig. 1. Sensitivity curves for DECIGO (blue line) and LISA (orange line), as well as the gravitational wave amplitude of a binary neutron star (1.4 + 1.4) M_{⊙} at a luminosity distance of D_{L} = 3Gpc and a frequency range of 10^{−5} Hz ∼ 100 Hz (green line). The arrows on the graph indicate that when observing at the same highfrequency of 1 Hz, the lowest observational frequency differs depending on the observation time. Specifically, for an observation time of T_{obs} = 5 years, the lowest observational frequency is f_{min} = 0.073Hz. For an observation time of T_{obs} = 1 year, the lowest observational frequency is f_{min} = 0.133 Hz, and for an observation time of T_{obs} = 1 month, the lowest observational frequency is f_{min} = 0.4036 Hz. 

In the text 
Fig. 2. Redshift distribution function and the samples generated according to this distribution. Panel a: probability density function and distribution function of redshift. Panel b: redshift distribution of 1000 sources. The xaxis and yaxis denote the range of each interval and the number of GW sources in each interval. 

In the text 
Fig. 3. Frequencydomain amplitude of gravitational wave signal observed for 1 month (yellow line) and onesided noise power spectral density of DECIGO (blue line), with a frequency range of 0.4036 ∼ 1 Hz. The binary neutron star with the mass of (1.4 + 1.4) M_{⊙} is located at a luminosity distance of D_{L} = 3 Gpc. 

In the text 
Fig. 4. Panel a: a timedomain data sample. Noise represented as the blue curve represents, while the yellow curve shows the gravitational wave (a). The sample’s S/N is 15.739. The observation time for the signal is 1 month, with a frequency range of 0.4036 Hz–1 Hz. The gravitational wave source is a binary neutron star system with a combined mass of (1.4 + 1.4) M_{⊙} and a luminosity distance of D_{L} = 3 Gpc. Other parameters are set as described in Sect. 2.3. Panel b: corresponding feature of (a), obtained through a singlelayer convolutional neural network convoluting with raw data. The horizontal axis of the (b) represents the data length, while the vertical axis represents the normalized feature amplitude. 

In the text 
Fig. 5. Variation in the S/N with different observation time. The blue line represents a 5year observation, the green line represents a 1year observation, and the yellow line represents a 1month observation. 

In the text 
Fig. 6. Comparison of X_{H} error estimation using CNNs and LSTMGRU hybrid networks, for binary neutron stars with one month of observations. 

In the text 
Fig. 7. Comparison of X_{H} estimation from 1month and 5year observations of binary neutron stars, based on Fisher information matrix. 

In the text 
Fig. 8. X_{H} estimation using deep learning and Fisher information matrix. Panel a: X_{H} estimation using deep learning with 1month observation and Fisher information matrix with 5year observation. Panel b: X_{H} estimation using deep learning with 1month observation and Fisher information matrix with 1month and 5year observation. 

In the text 
Fig. 9. Error estimated by the Fisher matrix for 5year observation data, shown as a green solid line. The orange solid line represents the error value estimated by the LSTMGRU networks for one month of observation data, while the blue solid line represents the error value estimated by the CNN networks for one month of observation data. 

In the text 
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