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We propose a novel approach which implements the relativistic calculations of the photon travel time into a robust timing model for pulsars orbiting supermassive black holes. We demonstrate the inability of current timing codes based on the post-Newtonian expansion of General Relativity to correctly estimate the relativistic time of arrival of the emitted pulses at an Earth-based observatory. We also show how a misestimation of the pulsar parameters can lead to the appearance of phase-dependent residuals, which hints at a tremendous constraining power of the binary and intrinsic parameters for timing observations of potential pulsars at the Galactic Center.
Pulsar timing in the Galactic Center
On the theoretical side, the analysis of Times Of Arrival (TOAs) of pulses emitted by pulsars, usually known as pulsar timing analysis, might potentially improve all previous tests of General Relativity in the strong field regime. The methodology for the numerical computation of fully relativistic propagation times for photons emitted in a generic spherically symmetric spacetime was introduced in previous works. This can be applied to estimate the observational impact on the timing residuals of a change in the fundamental nature of the central object or the underlying theory of gravity. This further demonstrates the paramount importance of pulsars orbiting SMBH as strong-field probes of gravity and motivates scientific and technical efforts required to successfully discover and time these kinds of objects. Forthcoming telescopes will benefit from a substantial increase in the collection areas and, among them, SKA has been demonstrated to potentially be able to achieve an accuracy on the pulse TOA measurement of the order of 100 microseconds for normal pulsars at frequencies above 15 GHz even when all possible limiting effects on the precision of TOA measurements of young pulsars in the Galactic Center environment are considered (i.e. the signal-to-noise ratio of the measured pulses, the intrinsic pulse phase jitter and the changes in pulse shape caused by interstellar scintillation).
We aim to introduce a different approach for studying the TOAs of pulsars orbiting the SMBH in the Galactic Center which is based on our methodology to compute the photon propagation time in black hole spacetimes. First, we will assume that pulsars at the Galactic Center actually exist and will be successfully detected and timed within the accuracy goal of 100 microseconds per TOA. With these hypotheses in mind, we can estimate the corresponding TOAs received by a distant observer using both the standard pulsar timing techniques based on post-Newtonian approximations and using our novel methodology.
The Metric: we will adopt the Schwarzschild solution in harmonic coordinates
Methodology: Using pulsar timing techniques, one can measure the TOAs of pulses emitted by a pulsar far away from the observer, monitor them on a timescale of years, and fit these TOAs to a model, namely a timing model. The latter is then used to link the measured TOA and the proper time of emission at the pulsar, allowing the computation of the pulse phase of emission, accounting for the inherent variations in its period. Matching the predictions of the timing model with the observed TOAs, for pulsars in binary systems, one obtains a very precise estimation of the pulsar intrinsic parameters, e.g. intrinsic period and spin-down rate, and of the orbital parameters, e.g. masses and orbital period.
Low-mass-ratio pulsar binary systems do not belong to a strong relativistic regime. Therefore, the evolution of such systems can be studied through a post-Newtonian treatment of both the orbital motion and photon propagation, which returns a timing model whose precision is below the experimental uncertainty on the TOAs. For this reason, all the pulsar timing codes available nowadays adopt this approach.
Conversely, high-mass ratio binary pulsars are in a strong gravitational field, and 1PN approximation of both the orbital motion and the photon propagation time cannot provide a timing model that accurately reproduces the TOAs. As a consequence, residuals of the timing models are not adequately computed, undermining the fitting capabilities of usual timing codes. We propose a novel approach which implements the relativistic calculations of the photon travel time into a more robust timing model for pulsars orbiting SMBHs. We will perform a proof-of-concept analysis to investigate the failure of weak field approximation and the advantages of our methodology in the framework of parameter estimation for pulsars at the Galctic Centrer with SKA.
Results: To appreciate how the shape and amplitude of the timing residuals change by over- (under) estimating the orbital and intrinsic parameters of a pulsar, we perform a qualitative analysis of the TOA deviations. For example, we considered the effect of a misestimation of the intrinsic pulsar period P and its derivative and show that both drastically alter the time of arrivals by linearly and quadratically drifting the residuals, respectively. More specifically, if we change the intrinsic pulsar period by only 1e-9% the timing residuals exceed the sensitivity threshold after only two orbital periods. We point out that changing intrinsic parameters, such as the pulsar period and its derivative, changes the shape and amplitude of the residuals but does not alter the orbit of the pulsar (nor the photon travel time). Those changes in the shape and amplitude of the residuals do not depend on the particular system. All results are summarised in the Table below.
Our procedure to compute TOAs for pulsars around a SMBH described by the Schwarzschild spacetime also allows for an interesting comparison between the results of the fully relativistic procedure and the ones obtained using formulas that rely on the 1PN approximation, and are implemented in all current codes devoted to TOA analysis. A useful way to assess the difference between the two methodologies is to consider the maximum amplitude, over the considered orbital period, of the difference between the PN and the fully relativistic photon propagation time. Due to the way current timing codes compute the pulse phase, and thus the residuals, from the coordinate time of observation of any pulse, whenever the discrepancy that we are computing surpasses the pulsar intrinsic period, it implies the failure of the PN-based timing formulas to correctly identify the emission pulse within the pulse sequence, and thus, the failure of the whole timing procedure in obtaining phase-connected residuals. The results of this analysis are shown in Figure 6, which illustrates the maximum discrepancy over one orbital period between the PN approximation and the fully relativistic approach for pulsar timing around a SMBH. The results consider the mass of Sgr A* and include two cases: circular orbits (left panel) and a highly eccentric S2-like orbit with e = 0.88 (right panel). For both cases, discrepancies are calculated across a range of semi-major axes, [2,1300] AU for Sgr A*, and for three orbital inclinations. A key feature is the significant impact of orbital inclination on timing discrepancies, with edge-on orbits showing the largest deviations due to pronounced strong lensing effects that are not correctly accounted for by 1PN photon propagation formulas. Even for an orbit with the same semi-major axis and inclination as the S2 star, the resulting discrepancy is of order 0.1 s, three orders of magnitude above the nominal SKA sensitivity, and thus potentially able to spoil the ability to perform timing with current techniques. This emphasizes the importance of fully relativistic modelling for precise timing near SMBHs and the failure of the PN approximation to appropriately account for the fully nonlinear effects of General Relativity.
Timing gravity with pulsars in the strong field
We propose a novel approach for the timing of pulsars orbiting a supermassive black hole, which implements the fully relativistic calculations of the photon travel time into a robust timing model. We generate realistic mock catalogues of pulsar times-of-arrival for several putative pulsars on tight orbits around the Galactic Center supermassive black hole, Sagittarius A* (Sgr A*). Then, we perform a proof-of-concept sensitivity analysis to forecast the accuracy that future observational facilities, like the SKA, will achieve in the characterization of the parameters of our timing model.
The inverse timing formula depends parametrically on the following parameters
We perform a Monte Carlo Markov Chain (MCMC) analysis to fit our timing model to the TOAs mock data. We adopt uniform large priors on all the parameters of our analysis, which are heuristically based on the qualitative analysis of the precision achieved with SKA (results are shown above). Our analysis shows how the observation of pulsars at the Galactic Center will open an incredibly promising avenue for the characterization of the physical properties of Sgr A*, which can improve by at least three orders of magnitude the current constraints on the black hole's mass achieved with the S-stars and event-horizon scale observations.
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