Argus: Time-Domain PTA Inference

• Argus is a high-performance Python package for PTA gravitational-wave detection that models latent dynamics via a state-space formulation.
• It leverages Kalman filtering to recursively compute the likelihood in the time domain, achieving linear computational scaling with TOAs.
• Its JAX-based implementation enables just-in-time compilation, GPU acceleration, and automatic differentiation for efficient Bayesian sampling.

Argus is a high-performance Python package for detecting and characterising nanohertz gravitational waves in pulsar timing array (PTA) data. It provides a complete Bayesian inference framework based on state-space models, using Kalman filtering for efficient likelihood evaluation, and is implemented in JAX to support just-in-time compilation, GPU acceleration, and automatic differentiation (Kimpson et al., 13 Oct 2025). In PTA analysis, its central role is to offer a time-domain alternative to traditional frequency-domain workflows: instead of representing stochastic processes primarily through power spectral densities and large covariance matrices, it models the underlying stochastic dynamics directly in the time domain and evaluates the likelihood recursively. The package is positioned as a practical Bayesian inference engine for PTA gravitational-wave searches, especially for parameter estimation of the stochastic gravitational-wave background (GWB), while remaining extensible to other signal and noise components (Kimpson et al., 13 Oct 2025).

1. Scientific setting and purpose

PTAs measure pulse times-of-arrival (TOAs) from many millisecond pulsars and search for correlated deviations induced by passing gravitational waves. The target signal discussed for Argus is typically a nanohertz stochastic GWB, which appears as a low-frequency red process with a distinctive Hellings–Downs spatial correlation across pulsars (Kimpson et al., 13 Oct 2025). Traditional PTA analysis packages such as ENTERPRISE and TempoNest model timing noise and gravitational-wave signals as Gaussian processes in the frequency domain using power spectral densities, and then perform Bayesian inference with methods such as MCMC (Kimpson et al., 13 Oct 2025).

Argus offers a complementary route. The package is described as a modern, production-ready implementation of a state-space filtering approach to PTA inference. In this formulation, the analysis follows the specific time-ordered realization of noise and signal in the data through recursive filtering rather than fitting an ensemble power spectrum via full covariance-matrix operations (Kimpson et al., 13 Oct 2025). The paper identifies four principal consequences of this design: linear computational scaling with the number of TOAs, $\mathcal{O}(N)$; direct time-domain modeling of physical stochastic processes; non-stationary process handling; and Bayesian inference at PTA scale with modern accelerator support (Kimpson et al., 13 Oct 2025).

The interpretive significance of this framework is that it recasts PTA data analysis as a time-domain dynamical-systems problem. This suggests a change not only in numerical method but also in modeling emphasis, from spectral summaries to latent dynamical evolution.

2. State-space formulation of PTA inference

Argus is built around a state-space formulation in which measured timing residuals are generated by a hidden dynamical system together with observation noise (Kimpson et al., 13 Oct 2025). The latent state $x_k$ at observation time $t_k$ evolves as

$x_k = F_k x_{k-1} + w_k,$

where $F_k$ is the state transition matrix and $w_k$ is process noise. The paper states that this evolution may be derived from a linear stochastic differential equation representation of the underlying physical process (Kimpson et al., 13 Oct 2025).

The observed TOA residuals $y_k$ are modeled as

$y_k = H_k x_k + v_k,$

where $H_k$ is the observation matrix and $v_k$ is the measurement noise (Kimpson et al., 13 Oct 2025). The framework explicitly separates process noise from measurement noise. Process noise corresponds to physical stochastic evolution such as spin wandering, red noise, and GWB fluctuations, whereas measurement noise represents standard white-noise terms such as EFAC and EQUAD (Kimpson et al., 13 Oct 2025).

Within this formalism, Argus can represent multiple PTA-relevant components. The paper lists pulsar-intrinsic red noise, a stochastic GWB with Hellings–Downs correlations, dispersion measure (DM) variations, and white-noise parameters such as EFAC and EQUAD (Kimpson et al., 13 Oct 2025). It further states that stochastic processes such as pulsar-intrinsic red noise and the stochastic GWB are encoded as time-domain linear stochastic differential equations, and gives the Ornstein–Uhlenbeck process as an example for intrinsic red timing noise (Kimpson et al., 13 Oct 2025). These component processes are then compiled into a single state-space model.

A plausible implication is that the package is intended to preserve a close correspondence between astrophysical process definitions and statistical representation, since the latent-state dynamics are directly tied to physical stochastic-process models rather than introduced solely through spectral parameterizations.

3. Kalman filtering and likelihood construction

The central computational mechanism in Argus is the Kalman filter, which is used to evaluate the likelihood of the TOA data efficiently in the time domain (Kimpson et al., 13 Oct 2025). The standard recursion comprises prediction, innovation, and update steps. In the notation given for the framework, the prediction step is

$x_k$0

$x_k$1

where $x_k$2 is the state covariance and $x_k$3 is the process-noise covariance (Kimpson et al., 13 Oct 2025).

The innovation quantities are

$x_k$4

$x_k$5

with $x_k$6 the measurement-noise covariance (Kimpson et al., 13 Oct 2025). The update step is

$x_k$7

$x_k$8

$x_k$9

A key output of this recursion is the Gaussian log-likelihood accumulated over the data sequence:

$t_k$0

where $t_k$1 is the dimension of the observation at step $t_k$2 and $t_k$3 denotes the model parameters (Kimpson et al., 13 Oct 2025). The paper describes this as a JAX-jittable log-likelihood for PTA datasets.

The significance of this construction lies in the replacement of large covariance-matrix manipulations by recursive filtering. The paper explicitly identifies this as the source of the claimed $t_k$4 scaling in the number of TOAs (Kimpson et al., 13 Oct 2025).

4. Bayesian inference and JAX-based implementation

Argus is explicitly described as a complete Bayesian inference framework (Kimpson et al., 13 Oct 2025). Its inferential objective is to recover posterior distributions over physical and noise parameters $t_k$5 from the TOA data $t_k$6:

$t_k$7

Here, $t_k$8 is supplied by the Kalman-filter likelihood and $t_k$9 encodes prior information (Kimpson et al., 13 Oct 2025).

A major emphasis of the package is its JAX implementation. The paper highlights four software properties as crucial for scalable Bayesian inference: just-in-time compilation, GPU acceleration, automatic differentiation, and integration with modern probabilistic programming through NumPyro and BlackJAX (Kimpson et al., 13 Oct 2025). JIT compilation turns Python-level model definitions into optimized executable kernels for repeated likelihood evaluation. The same code can run on GPUs as well as CPUs and TPUs. Automatic differentiation yields end-to-end gradients of the log-likelihood with respect to model parameters. The JAX-native likelihood can then be used directly with advanced samplers such as NumPyro’s NUTS and BlackJAX without rewriting inference code (Kimpson et al., 13 Oct 2025).

The paper treats these implementation choices as methodologically consequential rather than incidental. In particular, it states that differentiability enables gradient-based samplers, Hamiltonian Monte Carlo, and efficient posterior-geometry exploration, and emphasizes that these software features are what make state-space PTA inference feasible at realistic scale (Kimpson et al., 13 Oct 2025).

5. Relation to frequency-domain PTA pipelines

The paper contrasts Argus with classical frequency-domain PTA pipelines. Traditional methods model red timing processes via power spectral densities and infer parameters through covariance-matrix-based Gaussian-process likelihoods (Kimpson et al., 13 Oct 2025). These methods are described as mature and widely used, but they involve large matrix operations and an ensemble-statistical description of the noise.

Against this background, Argus is presented as a complementary rather than replacement methodology. It models the temporal evolution of latent variables directly, uses recursive filtering rather than full covariance inversions, has computational cost $x_k = F_k x_{k-1} + w_k,$0 in the number of TOAs, and naturally supports non-stationary or physically structured processes (Kimpson et al., 13 Oct 2025). The paper further stresses an interpretive distinction: the Kalman filter tracks the actual realization of intrinsic timing noise in a given pulsar, rather than averaging over a distribution of possible realizations through a power spectral density fit (Kimpson et al., 13 Oct 2025). This is said to help separate intrinsic noise from GW-induced correlations.

This distinction also motivates Argus’s role as an independent cross-check relative to ENTERPRISE and TempoNest. The paper states that it should retain the same astrophysical ingredients while introducing different numerical and systematic failure modes (Kimpson et al., 13 Oct 2025). A common misconception would therefore be to view the package as merely a software reimplementation of existing pipelines; the paper instead frames it as a distinct time-domain inference strategy.

6. Parameterization, validation, and prospective extensions

The paper identifies several classes of parameters represented in the framework. These include GWB parameters such as amplitude and turnover frequency, written in a figure caption as $x_k = F_k x_{k-1} + w_k,$1 and $x_k = F_k x_{k-1} + w_k,$2; pulsar red-noise parameters such as $x_k = F_k x_{k-1} + w_k,$3 and $x_k = F_k x_{k-1} + w_k,$4; and white-noise parameters EFAC and EQUAD (Kimpson et al., 13 Oct 2025). These quantities were demonstrated in a corner plot for a pulsar in the second IPTA mock data challenge.

The validation example described in the paper uses the second IPTA mock data challenge to show posterior distributions for the GWB amplitude and turnover parameters, red-noise parameters for one pulsar, and white-noise terms EFAC and EQUAD (Kimpson et al., 13 Oct 2025). The caption states that the posteriors were inferred using NUTS through NumPyro, taking advantage of the package’s JAX-native likelihood and automatic differentiation, and that the posteriors were unimodal (Kimpson et al., 13 Oct 2025). The paper presents this as evidence that the likelihood supports stable Bayesian recovery of physically meaningful parameters and that gradient-based sampling works well in practice.

Although the paper is not described as a benchmark study with extensive timing tables, it repeatedly claims practical efficiency benefits: linear scaling with TOA count, suitability for PTA-scale Bayesian inference, and the ability to run on accelerator hardware via JAX (Kimpson et al., 13 Oct 2025). It also emphasizes that Argus is a flexible foundation rather than a closed model. Planned or natural extensions listed in the paper include Bayes-factor model selection, deterministic continuous-wave sources such as individual supermassive black hole binaries, more advanced noise models including chromatic noise and non-Gaussian components, custom modular state-space models, and integration with larger PTA analysis ecosystems such as ENTERPRISE (Kimpson et al., 13 Oct 2025).

Taken together, these features define Argus as a JAX-based, state-space Bayesian inference framework for PTA gravitational-wave searches whose methodological identity lies in time-domain latent-process modeling and Kalman-filter likelihood evaluation. The paper presents it as a practical software package for Bayesian parameter estimation in PTA gravitational-wave detection with demonstrated mock-data performance and explicit pathways toward broader model coverage (Kimpson et al., 13 Oct 2025).