Searches for gravitational waves (GWs) from isolated supermassive black hole binaries (SMBHBs) in pulsar timing array (PTA) data require simultaneous estimation of signal and noise parameters, so the dimensionality of the fit scales with the number of observed pulsars. This computational difficulty is exacerbated when source evolution from GW emission is included, since retaining both Earth and pulsar terms introduces the unknown pulsar distances. Existing frequentist methods such as the ℱ-statistic, restricted so far to non-evolving sources, effectively, imply a circular analysis, which may lead to biased estimators. We present a Generalized Likelihood Ratio Test (GLRT) and the associated 𝒯-statistic that overcomes the aforementioned limitations. The formulation of the GLRT extends earlier work in which the dimensionality of the fitting problem was drastically reduced by semi-analytical maximization of the likelihood over the pulsar phase parameters, followed by efficient global optimization over the remaining parameters using Particle Swarm Optimization. Our simulations demonstrate that for an evolving SMBHB signal with chirp mass ℳ=10^9.2 M_⊙ and signal-to-noise ratio 20, this detection statistic achieves a 100% detection probability at a false-alarm probability of 0.06 in a 30-pulsar timing array, which is characterized by a 100 ns root-mean-square white noise residual and pulsar-specific red noise.

The pulsar timing array (PTA) utilizes the ultra-high precision timing characteristics of millisecond pulsars to construct a galaxy-scale detector, aiming to usher in a new era of nano-Hertz gravitational wave (GW) astronomy. This unique observation window allows us to study the stochastic gravitational wave background (SGWB) generated by the merger processes of numerous supermassive binary black hole (SMBBH) systems in the universe, the continuous gravitational waves (CGW) emitted by individual, stronger SMBBH systems, as well as the gravitational wave signals produced by phase transitions in the early universe and cosmic strings. Detecting and studying these signals holds significant scientific importance. They not only directly reveal the history of galaxy mergers and evolution, tracing the growth paths of supermassive black holes in the universe, but also provide crucial clues for understanding the dynamical behavior of SMBBHs, including addressing the "final parsec problem", and testing fundamental physical theories such as general relativity under strong gravitational field conditions. Global collaboration has brought together various regional arrays, including North America's NANOGrav, Europe's EPTA, Australia's PPTA, India's InPTA, and South Africa's MPTA. These arrays, along with China's CPTA, are collectively committed to continuously improving the detection sensitivity of the International Pulsar Timing Array (IPTA). This review will systematically summarize the latest progress, challenges, and future prospects of these major international PTA collaborations in searching for nano-Hertz SGWB signals, particularly the recent significant evidence, as well as in detecting CGW from individual SMBBHs.

This study investigates methods for detecting anisotropies across different frequency bands in the stochastic gravitational wave background (SGWB) using Pulsar Timing Arrays (PTA), with analyses based on simulated data. The North American Nanohertz Observatory for Gravitational Waves (NANOGrav) has identified a 3 sigma-4 sigma significant correlation between the SGWB signal and the Hellings-Downs (HD) spatial correlation curve in its latest 15-year dataset, providing strong evidence for the existence of the SGWB. One possible explanation for the imperfect match between the observed SGWB signal and the HD curve is the presence of anisotropy in the background. We employ a per-frequency optimal statistic (PFOS) and spherical harmonic decomposition techniques to perform frequency-resolved detection of anisotropy in the gravitational wave background within pulsar timing data, while simultaneously reconstructing the sky distribution of the background power. Simulations demonstrate that in backgrounds containing only a single anisotropic point source, the detectability of anisotropy depends on the anisotropy level, the choice of detection statistic, and the source's position. Given NANOGrav's current timing precision, for a one-point background, detectable background anisotropies require an angular power spectrum C_(l=1)>= 0.3C(l=0) However, in more realistic astrophysical backgrounds generated by populations of supermassive black hole binaries, anisotropy may still be detectable even when the first-order angular power spectrum in the two lowest frequency bins is as low as C-_l=1=0.1C(l=0) owing to the presence of multiple point sources.

Bayesian methods for the detection of continuous gravitational waves (CGWs) in pulsar timing array (PTA) data incur substantial computational costs that grow rapidly due to the number of noise and signal parameters characterizing the fitted model being proportional to the size of the PTA. This computational burden limits the scalability of these methods for large-scale PTAs comprising hundreds of pulsars anticipated from next-generation radio astronomy facilities. In this work, we introduce a computationally efficient frequentist method designed to circumvent this challenge. This is achieved by combining an adaptive spline fitting algorithm that nonparametrically suppresses red noise, thereby eliminating the need for complex noise modeling inherent to Bayesian methods, with a novel scheme for optimizing the subsets of pulsars included in the search. We quantify the performance of our method on a simulated dataset based on the NANOGrav 15-yr data release and find that it achieves a performance comparable to that of Bayesian analysis: for a CGW signal with a signal-to-noise ratio of approximate to 10, our method yields a relative characteristic strain error of 1.0% and a frequency error of 0.072% from the injected values by using the optimal pulsar selections, while the same errors are 1.7% and 0.16%, respectively, for the standard Bayesian analysis. At the same time, our analysis completes in less than 5 h, in contrast to the 1-2 days required by Bayesian methods. This allows us to perform a rigorous study of our method using multiple data realizations and signal parameters, establishing it as an efficient and scalable tool for CGW searches with large-scale PTAs.

TianQin is a proposed space-borne laser interferometer which aims to detect gravitational waves in the low frequency band(10-4 Hz-100 Hz).However,for space-borne interferometry detectors the laser phase noise is expected to be 7-8 orders of magnitude higher than the gravitational wave signals,due to unequal and time-varying arm lengths.Time delay interferometry(TDI)is an effective method to cancel laser phase noise by synthesizing virtual equal arm interferometric measurements with time delayed Doppler data combinations.In previous work it has been shown that TDI variables can also be understood within the context of principal component analysis(PCA),as combinations of eigenvectors of the total noise covariance matrix.It is therefore possible to generate the same TDI variables using a PCA approach-by computing the eigenvectors of the noise covariance matrix at a specific time.In this paper we extend significantly the previous work on a PCA approach to generating TDI variables by presenting TDI sensitivity curves for TianQin approximated as both a static and moving constellation of satellites.We compare the covariance matrix for the static and moving case,from the point of view of performing statistical inference using PCA.We also demonstrate the equivalency between the'conventional'second generation TDI variables and those obtained using PCA.

Extremely large mass-ratio inspirals (XMRIs), consisting of a brown dwarf orbiting a supermassive black hole, emit long-lived and nearly monochromatic gravitational waves in the millihertz band and constitute a promising probe of strong-field gravity and black-hole properties. However, dedicated data-analysis pipelines for XMRI signals have not yet been established. In this work, we develop, for the first time, a hierarchical semi-coherent search pipeline for XMRIs tailored to space-based gravitational-wave detectors, with a particular focus on the TianQin mission. The pipeline combines a semi-coherent multi-harmonic ℱ-statistic with particle swarm optimization, and incorporates a novel eccentricity estimation method based on the relative power distribution among harmonics. We validate the performance of the pipeline using simulated TianQin data for a Galactic center XMRI composed of a brown dwarf and Sgr A*. For a three-month observation, the pipeline successfully recovers the signal and achieves high-precision parameter estimation, including fractional uncertainties of <10^-6 in the orbital frequency, ≲10^-3 in the eccentricity, ≲2×10^-3 in the black-hole mass, and ≲10^-3 in the black-hole spin. Our framework establishes a practical foundation for future XMRI searches with space-based detectors and highlights the potential of XMRIs as precision probes of stellar dynamics and strong-field gravity in the vicinity of supermassive black holes.

Galactic compact binaries are expected to form a dominant foreground in the millihertz band of the Laser Interferometer Space Antenna (LISA). Residual power from injected sources that do not meet the adopted recovery criteria can bias stochastic gravitational-wave background (SGWB) inference or increase its uncertainty. We use LISA Data Challenge 2A Sangria injections and the Erebor comparison table to construct a catalog residual spectrum between 0.4 and 6.0 mHz with orbit-averaged long-wavelength Michelson X source powers. The source power concentration in each frequency bin determines the excess kurtosis of a random-phase source sum; instrumental noise and fiducial SGWB power strongly reduce the resulting excess kurtosis in most bins. The residual spectrum also overlaps an isotropic power-law SGWB in the mean binned power. We use a fixed covariance obtained by summing independent Fourier-mode power variances. For a frequency-independent SGWB with fiducial amplitude Ω_0=10^-11, marginalizing over the dimensionless residual-power factor β increases the Ω_0 uncertainty by 13.6% when the residual power is distributed uniformly over the Fourier frequencies in each bin. The largest Gaussian prior standard deviation on β that limits this increase to 10% is 0.0073. More concentrated distributions of the residual power among Fourier frequencies reduce the increase, reflecting unresolved frequency structure. Omitting the fiducial residual with the covariance held fixed shifts the best-fitting Ω_0 by 119.5 times the uncertainty obtained with β fixed. This projection of the residual spectrum onto the SGWB spectrum is not a posterior detection significance. The numerical values are conditional on the catalog-level scalar power model, fixed instrumental noise, and independent mode-power covariance.

TianQin is a future space-based gravitational wave (GW) observatory targeting the frequency window of 10−4–1 Hz. A large variety of GW sources are expected in this frequency band, including the merger of massive black hole binaries, the inspiral of extreme/intermediate mass ratio systems, stellar-mass black hole binaries, Galactic compact binaries, and so on. TianQin will consist of three Earth orbiting satellites on nearly identical orbits with orbital radii of about 105 km. The satellites will form a normal triangle constellation whose plane is nearly perpendicular to the ecliptic plane. The TianQin project has been progressing smoothly following the ‘0123’ technology roadmap. In step ‘0’, the TianQin laser ranging station has been constructed and it has successfully ranged to all the five retro-reflectors on the Moon. In step ‘1’, the drag-free control technology has been tested and demonstrated using the TianQin-1 satellite. In step ‘2’, the inter-satellite laser interferometry technology will be tested using the pair of TianQin-2 satellites. The TianQin-2 mission has been officially approved and the satellites will be launched around 2026. In step ‘3’, i.e. the TianQin-3 mission, three identical satellites will be launched around 2035 to form the space-based GW detector, TianQin, and to start GW detection in space.

The present study aimed to explore the mediating role of poor academic achievement and the moderating role of cumulative lifestyle risk factors (CLRF) in the relationship between mobile phone addiction and depression. In the cross-sectional study, a sample of 21,481 adolescents (mean age = 15.42 years, SD = 1.79) were recruited through a multi-stage cluster sampling method. Information was collected on demographic characteristics, Problematic Mobile Phone Use (PMPU), academic achievement, CLRF, and depressive symptoms. Mediation and moderation analyses were performed with the PROCESS macro v3.4 for SPSS. The results found that PMPU was significantly and positively associated with depressive symptoms, and the association was found to be mediated by poor academic achievement. In addition, CLRF moderated the pathways from PMPU to depressive symptoms. The present study enhances our understanding of how and under what conditions PMPU affects depressive symptoms. Practitioners should pay more attention to students with poor academic achievement and provide them with help and encouragement to reduce their depressive symptoms. Moreover, a variety of healthy lifestyle practices can be incorporated as a supplement in intervention programs to reduce depressive symptoms in adolescents.

We integrate the key results of our previous studies [8, 9, 11], providing a unified view on multimessenger and multiband (dual-line) observations of inspiraling double neutron stars (DNSs) in our Galaxy. Future space-based gravitational wave (GW) detectors, such as LISA, TianQin, and Taiji, are poised to bridge the detection gap of tight DNSs with orbital periods of approximately 10 min, which remain challenging to detect with radio telescopes. Our investigation will first explore GW and radio follow-up detection capabilities for Galactic inspiraling DNSs. Furthermore, next-generation ground-based GW observatories, such as Cosmic Explorer and Einstein Telescope, which are projected to be operational in the mid-2030s concurrently with LISA, TianQin, and Taiji, are expected to detect high-frequency GWs emitted by spinning NSs. This development will advance the study of dual-line GWs from DNSs. We then focus on GW waveform modeling for the spinning NS in a tight DNS and consider its potential role in inferring binary geometry and NS physics parameters by dual-line GW detection.

The opening of the gravitational wave window has significantly enhanced our capacity to explore the Universe's most extreme and dynamic sector. In the mHz frequency range, a diverse range of compact objects, from the most massive black holes at the farthest reaches of the Universe to the lightest white dwarfs in our cosmic backyard, generate a complex and dynamic symphony of gravitational wave signals. Once recorded by gravitational wave detectors, these unique fingerprints have the potential to decipher the birth and growth of cosmic structures over a wide range of scales, from stellar binaries and stellar clusters to galaxies and large-scale structures. The TianQin space-borne gravitational wave mission is scheduled for launch in the 2030s, with an operational lifespan of five years. It will facilitate pivotal insights into the history of our Universe. This document presents a concise overview of the detectable sources of TianQin, outlining their characteristics, the challenges they present, and the expected impact of the TianQin observatory on our understanding of them.

BACKGROUND:Limited research has examined the relationship between heatwaves and adolescent mental health, particularly depression and anxiety. This study aimed to explore the relationship between heatwaves and depression and anxiety. METHODS:We conducted a cross-sectional study including 19,852 adolescents (mean age 15.16 years; 50.2 % females). Air temperature data were from the fifth generation European ReAnalysis-Land (ERA5-Land) dataset. Heat exposure was assessed using three heatwave metrics: The excess heat factor-based (HWM1), maximum temperature-based (HWM2), and minimum temperature-based (HWM3) heatwave magnitude indices. Depression and anxiety were assessed using the PHQ-9 and GAD-7 scales. Subgroup analyses evaluated interactions with sex, grade and region of school. RESULTS:Depression and anxiety prevalence were 19.37 % and 16.27 %, respectively. Heatwaves were associated with depression (OR [95 % CI]: 1.13 [1.09-1.17]) and anxiety (OR [95 % CI]: 1.12 [1.08-1.16]) based on HWM1. Significant associations existed for depression alone (OR [95 % CI]: 1.14 [1.09-1.20]), anxiety alone (OR [95 % CI]: 1.13 [1.06-1.21]), and comorbid depression and anxiety (OR [95 % CI]: 1.13 [1.09-1.18]). Associations using HWM2 and HWM3 showed consistent directions but varied significance. We observed significant interactions between heatwaves and sex for anxiety alone, and between heatwaves and region of school for both depression alone and comorbid depression-anxiety (all P for interaction <0.05). LIMITATIONS:The cross-sectional design constrained our capacity to draw causal inferences. CONCLUSIONS:Heatwaves were associated with a significant of prevalence depression, anxiety, and their comorbidity, with males and rural students potentially more susceptible to these effects.

Future space-based laser interferometric detectors, such as the Laser Interferometer Space Antenna (LISA), will be able to detect gravitational waves (GWs) generated during the inspiral phase of stellar-mass binary black holes (SmBBHs). These detections contain a wealth of important information concerning astrophysical formation channels and fundamental physics constraints. However, the detection and characterization of GWs from SmBBHs poses a formidable data analysis challenge, arising from the large number of wave cycles that make the search extremely sensitive to mismatches in signal and template parameters in a likelihood-based approach. This makes the search for the maximum of the likelihood function over the signal parameter space an extremely difficult task, with grid-based deterministic global optimization methods becoming computationally infeasible. We present a data analysis method that addresses this problem using both algorithmic innovations and hardware acceleration driven by graphics processing units. The method follows a hierarchical approach in which a semicoherent .T-statistic is computed with different numbers of frequency domain partitions at different stages, with multiple particle swarm optimization (PSO) runs used in each stage for global optimization. An important step in the method is the judicious partitioning of the parameter space at each stage to improve the convergence probability of PSO and avoid premature convergence to noise-induced secondary maxima in the semicoherent .T-statistic. The hierarchy of stages confines the semicoherent searches to progressively smaller parameter ranges, with the final stage performing a search for the global maximum of the fully coherent .T-statistic. We test our method on 2.5 yr f a single LISA time delay interferometry combination and find that for an injected SmBBH signal with a signal-to-noise ratio between X11 and X14, the method can estimate (i) the chirp mass with a relative error of 0.01%, (ii) the time of coalescence within X100 s, (iii) the sky location within X0.2 deg2, and (iv) orbital eccentricity at a fiducial signal frequency of 10 mHz with a relative error of 1%.

The coming era of gravitational wave (GW) astronomy, enabled by spacebased laser interferometric detectors, heralds unprecedented opportunities to study compact binaries, in particular double white dwarfs (DWDs), double neutron stars (DNSs), and binary black holes, through their GW and electromagnetic observations, providing important insights into binary evolution, NS physics, and the overarching architecture of the Universe. In this paper, we provide an overview of our recent progress in the scientific objectives and data analysis relevant to these compact binaries within the scope of the space-based missions, emphasizing the multi-messenger observations of DWDs and DNSs in our Galaxy that synergize the capabilities of TianQin, LISA, and current and future optical and radio telescopes, and the promising multi-wavelength (dual-line) observations of the low-frequency GWs due to the binary motion of a DNS and the high-frequency GWs due to the spin of the aspherical NS component. We also briefly discuss the challenge of data analysis for the stellar-mass binary black holes for space-based detectors and present our preliminary solution, which involves both algorithmic innovations and hardware acceleration driven by graphics processing units (GPUs).

Noise in Pulsar Timing Array (PTA) data is commonly modeled as a mixture of white and red noise components. While the former is related to the receivers, and easily characterized by three parameters (EFAC, EQUAD and ECORR), the latter arises from a mix of hard to model sources and, potentially, a stochastic gravitational wave background (GWB). Since their frequency ranges overlap, GWB search methods must model the non-GWB red noise component in PTA data explicitly, typically as a set of mutually independent Gaussian stationary processes having power-law power spectral densities. However, in searches for continuous wave (CW) signals from resolvable sources, the red noise is simply a component that must be filtered out, either explicitly or implicitly (via the definition of the matched filtering inner product). Due to the technical difficulties associated with irregular sampling, CW searches have generally used implicit filtering with the same power law model as GWB searches. This creates the data analysis burden of fitting the power-law parameters, which increase in number with the size of the PTA and hamper the scaling up of CW searches to large PTAs. Here, we present an explicit filtering approach that overcomes the technical issues associated with irregular sampling. The method uses adaptive splines, where the spline knots are included in the fitted model. Besides illustrating its application on real data, the effectiveness of this approach is investigated on synthetic data that has the same red noise characteristics as the NANOGrav 15-year dataset and contains a single non-evolving CW signal.

The successful detection of continuous gravitational waves from spinning neutron stars (NSs) will shape our understanding of the physical properties of dense matter under extreme conditions. Binary population synthesis simulations show that forthcoming space-borne gravitational wave detectors may be capable of detecting some tight Galactic double NSs with 10-min orbital periods. Successfully searching for continuous waves from the individual NS in such a close binary demands extremely precise waveform templates considering the interaction between the NS and its companion. Unlike the isolated formation channel, double NS systems from the dynamical formation channel have moderate to high orbital eccentricities. To accommodate these systems, we generalize the analytical waveforms from triaxial nonaligned NSs under spin-orbit coupling derived by Feng et al. [Phys. Rev. D 108, 063035 (2023)] to incorporate the effects of the orbital eccentricity. Our findings suggest that, for binaries formed through isolated binary evolution, the impact of eccentricity on the continuous waves of their NSs can be neglected. In contrast, for those formed through dynamical processes, it is necessary to consider eccentricity, as high-eccentricity orbits can result in a fitting factor of less than or similar to 0.97 (0.9) within approximately 0.5 (1) to 2 (5) yr of a coherent search (at wave frequencies of 100 and 200 Hz). Once the continuous waves from spinning NSs in tight binaries are detected, the relative measurement accuracy of eccentricity can reach Delta e/e similar to O(10(-7)) for a signal-to-noise ratio of O(100) based on the Fisher information matrix, bearing significant implications for understanding the formation mechanisms of double NS systems.

Gravitational wave (GW) searches using pulsar timing arrays (PTAs) are commonly assumed to be limited to a GW frequency of less than or similar to 4x10(-7) Hz given by the Nyquist rate associated with the average observational cadence of 2 weeks for a single pulsar. However, by taking advantage of asynchronous observations of multiple pulsars, a PTA can detect GW signals at higher frequencies. This allows a sufficiently large PTA to detect and characterize the ringdown signals emitted following the merger of supermassive binary black holes (SMBBHs), leading to stringent tests of the no-hair theorem in the mass range of such systems. Such large-scale PTAs are imminent with the advent of the FAST telescope and the upcoming era of the Square Kilometer Array (SKA). To scope out the data analysis challenges involved in such a search, we propose a likelihood-based method coupled with Particle Swarm Optimization and apply it to a simulated large-scale PTA comprised of 100 pulsars, each having a timing residual noise standard deviation of 100 nsec, with randomized observation times. Focusing on the dominant (2,2) mode of the ringdown signal, we show that it is possible to achieve a 99% detection probability with a false alarm probability below 0.2% for an optimal signal-to-noise ratio (SNR) >10. This corresponds, for example, to an equal-mass non-spinning SMBBH with an observer frame chirp mass M-c = 9.52x10(9)M(circle dot) at a luminosity distance of D-L = 420 Mpc.

This paper provides a comprehensive guide to gravitational wave data processing, with a particular focus on signal generation, noise modeling, and optimization techniques. Beginning with an introduction to gravitational waves and the detection techniques used by LIGO and Virgo, the manual covers the essentials of signal processing, including Fourier analysis, filtering, and the generation of quadratic chirp signals. The analysis of colored Gaussian noise and its impact on interferometric detectors like LIGO is explored in detail, alongside signal detection methods such as the Generalized Likelihood Ratio Test (GLRT). The paper also delves into optimization techniques like Particle Swarm Optimization (PSO), which can be applied to improve signal estimation accuracy. By providing MATLAB-based implementations, this manual serves as both a theoretical and practical resource for researchers in the field of gravitational wave astronomy.

Neutron star (NS) binaries can be potentially intriguing gravitational wave sources, with both high- and low-frequency radiation from the possibly aspherical individual stars and the binary orbit, respectively. The successful detection of such a dual-line source could provide fresh insights into binary geometry and NS physics. In the absence of electromagnetic observations, we develop a strategy for inferring the spin-orbit misalignment angle using the tight dual-line double NS system under the spin-orbit coupling. Based on the four-year joint detection of a typical dual-line system with the Laser Interferometer Space Antenna and Cosmic Explorer, we find that the misalignment angle and the NS moment of inertia can be measured with subpercentage and 5% accuracy, respectively.